Finite element analysis of die swell of a highly elastic fluid

A mixed finite element method is applied to the die swell calculation of ana Oldroyd fluid B. The use of large entry lengths together with the presence of the retardation time in the constitutive equations allow us to reach values of the recoverable shear as high as four for the flow emerging from slit and circular dies, with swelling ratios of the order of 2. The numerical results are in good agreement with some available experimental data.     

Numerical simulation of delayed die swell

In a recent paper, Joseph et al. showed that, for a number of viscoelastic fluids, one can observe the phenomenon of delayed die swell beyond a critical extrusion velocity, or beyond a critical value of the viscoelastic Mach number. Giesekus had also observed that delayed die swell is a critical phenomenon.In the present paper, we find a set of material and flow parameters under which it is possible to simulate delayed die swell. For the viscoelastic flow calculation, we use the finite element algorithm with sub-elements for the stresses and streamline upwinding in the discretized constitutive equations. For the free surface, we use an implicit technique which allows us to implement Newton's method for solving the non-linear system of equations. The fluid is Oldroyd-B which, in the present problem, is a singular perturbation of the Maxwell fluid. The results show very little sensitivity to the size of the retardation time. We also show delayed die swell for a Giesekus fluid.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

One dimensional finite element solutions for a Maxwell fluid

The one-dimensional flow of a Maxwell fluid is used to study several aspects of the numerical simulation of viscoelastic fluids. The numerical results are produced via a Galerkin finite element method of the mixed type and are compared with an analytic solution. The accuracy of the numerical solutions are evaluated with respect to variations in mesh refinement and changes in the interpolation order for the stress variables. Various convergence and accuracy problems associated with the chosen numerical schemes are noted and a tentative explanation for these difficulties is provided through use of an eigenvalue analysis.     

Finite element simulation of solidification problems

The modelling of liquid-solid phase change phenomena is extremely important in many areas of science and engineering. In particular, the solidification of molten metals during various casting methods in the foundry, provides a source of important practical problems which may be resolved economically with the aid of computational models of the heat transfer processes involved. Experimental design analysis is often prohibitively expensive, and the geometries and complex boundary conditions encountered preclude any analytical solutions to the problems posed. Thus the motivation for numerical simulation and computer aided design (CAD) systems is clear, and several mathematical/computational modelling techniques have been brought to bear in this area during recent years.This paper reports on the application of the finite element method to solidification problems, principally concerning industrial casting processes. Although convective heat transfer has been modelled, the work herein considers only heat conduction, for clarity. The heat transfer model has also been coupled with thermal stress analysis packages to predict mechanical behaviour including cracking and eventual failure, but this is reported elsewhere.Following the introduction, the mathematical and computational modelling tools are described in detail, for completeness. A discussion on the handling of the phase change interface and latent heat effects is then presented. Some aspects of the solution procedures are examined next, together with special techniques for dealing with the mold-metal interface. Finally, some numerical examples are presented which substantiate the capabilities of the finite element model, in both two and three dimensions.Nomenclature c heat capacity - C capacitance matrix - f time function - F loading term - h heat convection coefficient - H specific enthalpy - |J| Jacobian determinant - || patch approximation to |J| - k thermal conductivity - K conductance matrix - L latent heat - unit outward normal - N _i nodal shape function - q known heat flux - R _i nodal heat capacity - S phase change interface - t time - T temperature - known boundary temperature - T vector of nodal temperatures - T _a ambient temperature - T _c solidification temperature - T _L liquidus temperature - T ₀ initial temperature - T _s solidus temperature - x space coordinates - interface heat transfer coefficient - iteration parameter - boundary of domain - T solidification range - t timestep magnitude - vector gradient operator - convergence tolerance - timestepping parameter - t known vector in alternating-direction formulation - Laplace modifying parameter - (, ) local space coordinates - density - time limit - () shape function factor - () shape function factor - domain of interest     

断裂过程的有限元模拟

讨论了材料断裂过程的有限元模拟技术。基于自适应有限元的一般原理，并针对多相材料的裂纹扩展的特点，提出了一种简化的高精度和高效率有限元网格的动态重新划分策略。裂纹被假设沿着单元之间的路径连续扩展，利用节点力释放技术生成新的裂纹自由表面，发展了一种可随裂尖连续移动的网格动态加密和释放方法。这种方法已在各种裂纹问题中得以实现与应用。     

Finite element simulation of vacuum arc remelting

Vacuum arc remelting is a process for producing homogeneous ingots of reactive and macrosegregation-sensitive alloys. A mathematical model of the transport phenomena in the ingot melt is presented together with a discussion of the various simplifying assumptions and approximations that make the problem tractable, with particular attention on transport in the interdendritic mushy zone and on the magnetohydrodynamics. The finite element method is used to discretize the equations for the non-isothermal flow problem and the quasi-static electromagnetic problem. Coupling of the finite element solutions for the two field problems is accomplished using the Parallel Virtual Machine software. An analysis of the fluid flow and heat transport in the melt pool of the solidifying ingot shows some of the factors that influence ingot quality during quasi-steady growth conditions. © 1997 John Wiley & Sons, Ltd.     

Finite element simulation of localized plastic deformation

Summary A rational finite element algorithm for capturing localized plastic deformation due to softening and/or non-associated plastic flow is devised. The incremental relations are based on implicit integration. In each increment it is important to use a carefully designed modified Newton iteration procedure in conjunction with a start solution whose calculation is based on a diagnostic bifurcation analysis. The algorithm performs successfully for a finite element mesh that is biased in the sense that the element sides are prealigned with the anticipated localization zone which is demonstrated for a slope stability problem.

---

Finite-Element-Berechnung von örtlichen plastischen Verformungen

Übersicht Die Konstruktion eines Algorithmus, der im Zusammenhang mit der Methode der Finiten Elemente örtliche plastische Verformungen als Folge von Materialentfestigung und/oder nichtassoziiertem plastischen Fließen erfaßt, wird beschrieben. Die inkrementellen Gleichungen ergeben sich aus der Verwendung eines impliziten Integrationsverfahrens. In jedem Inkrement wird ein sorgfältig entwickeltes, modifiziertes Newton-Iterationsverfahren verwendet, dessen erste Lösungsabschätzung auf einer diagnostischen Analyse des Verzweigungsproblems beruht. Am Beispiel einer Böschungsstabilitätsungtersuchung wird gezeigt, daß der Algorithmus erfolgreich ist, wenn das verwendete Finite-Elemente-Netz so ausgerichtet ist, daß die Elementseiten der erwarteten Versagenzone folgen.

---

---

Presented at the workshop on Numerical Methods for Localization and Bifurcation of Granular Bodies, held at the Technical University of Gdansk (Poland), September 25–30, 1989     

Finite difference solution of a Newtonian jet swell problem

A finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary-fitted co-ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free-surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.     

Stability of a Maxwell fluid in hydromagnetics

The character of equilibrium of a Maxwell fluid in the presence of a magnetic field has been investigated. It is shown that the solution is characterized by a variational principle when the direction of magnetic field is either horizontal or vertical. For both directions, an approximate solution has been developed for a fluid layer of finite depth and exponentially varying density. It is found that the stability criterion remains unaffected by the viscosity parameters, although they influence the rate at which the unstable stratification departs from the equilibrium position. The question of excitation of waves has also been considered in detail.     

Finite element simulation of penetration into geological targets

The finite element method has been applied to study the projectile penetration process into geological targets. To illustrate the solution procedure, two example problems involving conical nose steel penetrators into sea ice and antelope tuff are presented. Details of the numerical treatment using the computer code PRONTO 2D are given. Good agreements between numerical and experimental results are observed. Although the examples are limited to normal impact problems in which the axis of the penetrator is perpendicular to the target surface and where no angle of attack for the penetrator is allowed, some thoughts regarding ways to deal with nonnormal three-dimensional penetration problems are discussed.     

A model for interpreting die swell of polymer melts

Summary We propose here a new model to interpret the most peculiar characteristics of die swelling of polymer melts.The model, which applies to short capillaries, is based on the observations that the deformation undergone by the melt entering the capillary from the reservoir is essentially elastic and that the deformation within the capillary is essentially viscous.Under these assumptions we can derive an equation for the deformation of the polymer emerging from the capillary, which is a measure of swelling.In order to check the validity of this equation swelling experiments were performed on high density polyethylene by using a technique of annealing the strands in silicon oil after extrusion.Quite satisfactory agreement between theory and experiments was found, provided that the characteristic relaxation time,, was properly evaluated. In fact it turned out that is of the order of magnitude of the time at which the shear stress fully relaxes in an experiment of stress relaxation after steady state shear flow.

---

Zusammenfassung Es wird ein neues Modell zur Erklärung der wesentlichsten Züge der Strahlaufweitung bei Polymerschmelzen vorgeschlagen. Dieses Modell, das auf kurze Kapillaren beschränkt ist, basiert auf der Beobachtung, daß die von der Schmelze beim Einströmen aus dem Reservoir in die Kapillare ausgeführte Deformation im wesentlichen elastisch, die Deformation beim Durchgang durch die Kapillare dagegen im wesentlichen viskos ist.Unter diesen Voraussetzungen wird eine Gleichung für die Deformation des aus der Kapillare austretenden Polymers angegeben, welche ein Maß für die Aufweitung liefert.Zur Prüfung dieser Gleichung werden Strahlaufweitungsexperimente mit hochdichtem Polyäthylen ausgeführt, wobei das Extrudat in Siliconöl getempert wird.Zwischen Theorie und Experimenten wird eine recht befriedigende Übereinstimmung gefunden, vorausgesetzt daß die charakteristische Relaxationszeit in angepaßter Weise bestimmt wird. Es stellt sich heraus, daß von der Größenordnung derjenigen Zeit ist, bei der die Schubspannung nach voraufgegangener stationärer Scherströmung vollständig relaxiert ist.

---

---

With 7 figures and 3 tables     

Finite element computations for unsteady fluid and elastic membrane interaction problems

The interaction between the hydrodynamic forces of a flow field and the elastic forces of adjacent deformable boundaries is described by elastohydrodynamics, a coupled fluid–elastic membrane problem. Direct numerical solution of the unsteady, highly non-linear equations requires that the dynamic evolution of both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. This paper describes a numerical algorithm based on the deformable spatial domain space–time (DSD/ST) finite element method for the unsteady motion of an incompressible, viscous fluid with elastic membrane interaction. The unsteady Navier–Stoke and elastic membrane equations are solved separately using an iterative procedure by the GMRES technique with an incomplete lower-upper (ILU) decomposition at every time instant. One-dimensional, two-dimensional and deformable domain model problems are used to demonstrate the capabilities and accuracy of the present algorithm. Both steady state and transient problems are studied. © 1997 John Wiley & Sons, Ltd.     

A boundary element investigation of extrudate swell

Numerical studies of die swell have until now dealt perimarily with the Maxwell or Oldroyd-B viscoelastic models. However, these models exhibit features that often make them unsuitable for numerical work. Furthermore, they are not realistic representations of actual viscoelastic fluids. In this report a comparison is made between the behaviour of a variety of different viscoelastic models when applied to the die swell problem. A wide range of elongational and shear behaviour is exhibited by the models examined. Both types of behaviour are shown to be important in the die swell problem, and the observed swelling is related to these characteristics of the models.     

Decay of a potential vortex in a Maxwell fluid

The velocity field and the associated tangential tension corresponding to a potential vortex in a Maxwell fluid are determined by means of the Hankel transform. The similar solutions for a Newtonian fluid appear as a limiting case of our solutions.     

Finite element technique to solve the elastic strain for Leonov fluid flow

The finite element method is used to find the elastic strain (and thus the stress) for given velocity fields of the Leonov model fluid. With a simple linearization technique and the Galerkin formulation, the quasi-linear coupled first-order hyperbolic differential equations together with a non-linear equality constraint are solved over the entire domain based on a weighted residual scheme. The proposed numerical scheme has yielded efficient and accurate convective integrations for both the planar channel and the diverging radial flows for the Leonov model fluid. Only the strain in the inflow plane is required to be prescribed as the boundary conditions. In application, it can be conveniently incorporated in an existing finite element algorithm to simulate the Leonov viscoelastic fluid flow with more complex geometry in which the velocity field is not known a priori and an iterative procedure is needed.     

Finite element simulation of the loss of stable resistance in a foundation-soil system

Summary The fundamentals of the methodology for numerical simulation of non-linear behavior of the foundation-subsoil system by the finite element method are described. Constitutive laws governing the evolution of plastic zones in soil and crack development in concrete are used. These processes are known to proceed monotonously as the load increases, but can suddenly become avalanche-like. The latter phenomenon can terminate in the exhaustion of bearing capacity of the foundation subsoil system or in the stabilization of this system's state, and may be followed by an increase in resistance to loads. The developed algorithms and a code allow simulation of both phenomena. Examples of analysis are given.

---

Finite-Element-Modellierung des Verlusts stabilen Widerstandes des Systems Fundament-Bodengründung

Übersicht Die Grundlagen für numerische Modellierung des nichtlinearen Verhaltens vom System Fundament-Bodengründung nach der Finite-Element-Methode werden dargelegt. Angewandt werden Stoffgesetze, die die Entwicklung plastischer Bereiche im Baugrunde und Rißbildung im Beton berücksichtigen. Bekanntlich entwickeln sich diese Erscheinungen in der Regel monoton mit der Belastungssteigerung, können sich aber auch lawinenartig entwickeln. Letztere Erscheinung kann mit der Erschöpfung der Tragfähigkeit des Systems Fundament-Bodengründung oder mit der Stabilisierung des Systems endan, worauf ein weiterer Widerstand gegen die steigende Belastung möglich ist. Ausgearbeitete Algorithmen und ein Programm erlauben es, beide Erscheinungen zu modellieren. Rechenbeispiele sind angeführt.

---

---

Presented at the Workshop on Numerical Methods for Localisation and Bifurcation of Granular Bodies, held at the Technical Univeristy of Gdansk (Poland), September 25–30, 1989     

Finite element simulation of sheet forming based on a planar anisotropic strain-rate potential

Many finite element (FEM) formulations have been based on stress potentials defined in the stress field. Nevertheless, there are formulations where potentials defined in the strain-rate field are especially convenient to implement. These include rigid-plastic formulations based on minimum plastic work paths, which can be used for process design as well as for process analysis. Based on a strain-rate potential recently proposed for anisotropic materials exhibiting orthotropic symmetry, a formulation for sheet forming process analysis has been developed using a Cartesian coordinate system in this paper. An efficient formulation to account for material rotation is also included. Earing predictions made for a cup drawing test of a 2090-T3 aluminum-lithium alloy sheet showed good agreement with experiments. However, some discrepancies were observed between predicted and experimental thickness strain and cup height directional trends. The cause of the discrepancies was discussed using a simple analysis based on Lankford (or plastic strain ratio, r) values.     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

Finite element simulation of deformation behaviour of cellular rubber components

This paper presents a new prospect of investigating the mechanical behaviour of cellular rubber using a porous hyperelastic material model within the framework of homogenisation method to consider pore volume fraction. There are number of hyperelastic material models to describe the behaviour of homogeneous elastomer, but very few to characterise the complex properties of cellular rubber. The analysis of dependence of material behaviour on pore density using the new material model is supported with experiments to characterise the actual material behaviour. The finite element simulations are then followed by compression load tests to validate the material model.     

Finite element simulation of viscoelastic fluids of the integral type

In the present paper, we develop an algorithm for calculating the steady flow of viscoelastic fluids of the integral type. The calculation is based upon a simple integration of the strains along the streamlines, and the method remains valid for arbitrary elements. The technique is applied to the flow of a Maxwell fluid through a wedge and to the calculation of the hole-pressure error for the Doi-Edwards fluid.