A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part I

Summary From the continuum mechanical point of view, most of practical fibre-reinforced composites may be considered to be transversely isotropic, orthotropic, tetratropic, hexatropic, or octotropic according to material symmetries. Engineering applications of fibre-reinforced materials have already been well established. Based on tensor function representations, the formulation of constitutive equations for fibre-reinforced composites constitutes a consistent mathematical basis for modelling the mechanical behaviour of these oriented materials. Although we have good knowledge of transversely isotropic and orthotropic tensor function representations, only integrity bases for tetratropic and hexatropic polynomials could be learnt up to now in very limited cases. In this paper, we present the complete and irreducible representations for tetratropic, hexatropic and octotropic scalr-, symmetric tensor-, skew-symmetric tensor- and vector-valued functions (not only polynomials) of any finite number of symmetric tensors, skew-symmetric tensors and vectors in plane problems; a generalization of these results is also given. By use of these representations we shall perform the formulation of constitutive equations of fibre-reinforced composites in plane problems of elasticity, heat conduction, asymmetric elasticity, plasticity and continuum damage mechanics in a continued paper (Part II).

---

Formulierung von Stoffgleichungen für faserverstärkte Verbundwerkstoffe bei ebenen Problemen: Teil I

Übersicht Im Hinblick auf Materialsymmetrien kann man für den praktischen Einsatz transversal-isotrope, orthotrope, tetratrope, hexatrope oder auch oktotrope faserverstärkte Verbundwerkstoffe unterscheiden. Für Anwendungen im Ingenieurbereich sind sie von größter Bedeutung. Die Darstellungstheorie tensorwertiger Funktionen bildet die Grundlage zur Formulierung von Stoffgleichungen anisotroper Stoffe, zu denen auch die oben erwähnten Verbundwerkstoffe gehören. Tensorfunktionen speziell für transversal-isotrope und orthotrope Stoffe sind hinreichend bekannt. Hingegen fehlen entsprechende Untersuchungen zum tetratropen und hexatropen Verhalten. Daher sollen im folgenden vollständige und irreduzible Darstellungen für derartige Materialorientierungen entwickelt werden. Dazu müssen vektor- und tensorwertige Funktionen mit unterschiedlichen Tensor- und Vektorargumenten in Betracht gezogen werden. Eine weiterführende Untersuchung (Teil II), in der u. a. auch anisotrope Werkstoffschädigungen berücksichtigt werden, ist in Vorbereitung.

---

---

Paper presented at the XVIIIth ICTAM Congress in Haifa, Israel, August 1992.     

A least squares finite element method for viscoelastic fluid flow problems

Viscoelastic flows remain a demanding class of problems for approximate analysis, particularly at increasing Weissenberg numbers. Part of the difficulty stems from the convective behavior and in the treatment of the stress field as a primary unknown. This latter aspect has led to the use of higher-order piecewise approximations for the stress approximation spaces in recent finite element research. The computational complexity of the discretized problem is increased significantly by this approach but at present it appears the most viable technique for solving these problems. Motivated by recent success in treating mixed systems and convective problems, we formulate here a least squares finite element method for the viscoelastic flow problem. Numerical experiments are conducted to test the method and examine its strengths and limitations. Some difficulties and open issues are identified through the numerical experiments. We consider the use of high degree elements (p refinement) to improve performance and accuracy.     

On constitutive equations for fiber-reinforced nonlinearly viscoelastic solids

In this paper, we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation. These invariants are analyzed, and we obtain restrictions such as positivity of some of them.     

Practical comparison of differential viscoelastic constitutive equations in finite element analysis of planar 4:1 contraction flow

Finite element modeling of planar 4:1 contraction flow (isothermal incompressible and creeping) around a sharp entrance corner is performed for favored differential constitutive equations such as the Maxwell, Leonov, Giesekus, FENE-P, Larson, White-Metzner models and the Phan Thien-Tanner model of exponential and linear types. We have implemented the discrete elastic viscous stress splitting and streamline upwinding algorithms in the basic computational scheme in order to augment stability at high flow rate. For each constitutive model, we have obtained the upper limit of the Deborah number under which numerical convergence is guaranteed. All the computational results are analyzed according to consequences of mathematical analyses for constitutive equations from the viewpoint of stability. It is verified that in general the constitutive equations proven globally stable yield convergent numerical solutions for higher Deborah number flows. Therefore one can get solutions for relatively high Deborah number flows when the Leonov, the Phan Thien-Tanner, or the Giesekus constitutive equation is employed as the viscoelastic field equation. The close relationship of numerical convergence with mathematical stability of the model equations is also clearly demonstrated.     

A modified finite element method for solving the time-dependent,incompressible Navier-Stokes equations. Part 1: Theory

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.     

A modified finite element method for solving the time-dependent,incompressible Navier-Stokes equations. Part 2: Applications

Three examples will be presented to demonstrate the performance of the scheme described in Part 1 of this paper.¹ Two are isothermal (T = 0) and two-dimensional, and one of these is steady and the other time-dependent. The third example involves buoyancy effects, is time-dependent and three-dimensional, and is presented in less detail. The paper concludes with a short discussion and some conclusions from both Parts 1 and 2.     

On instabilities of single-integral constitutive equations for viscoelastic liquids

Various types of instabilities are exposed in this paper for time-strain separable single-integral viscoelastic constitutive equations (CE's). They were distinguished into two groups and defined as Hadamard and dissipative type of instabilities. As for the Hadamard-type, previously obtained criteria are found to be necessary only. They are necessary and sufficient only for thermodynamic stability. Improved, stricter Hadamard stability criteria are described briefly in this paper, and then applied to study of stability of several CE's. It is shown that the Currie potential with the K-BKZ equation and the model proposed by Papanastasiou et al. are Hadamard unstable. In the case of dissipative stability, the necessary and sufficient condition for stress boundedness in any regular flow with a given history, is proved. Then, this criterion was applied to the neoHookean, Mooney, and Yen and McIntire specifications of the general K-BKZ model, to exhibit unbounded solutions. In addition, Larson-Monroe potential which is later proved to be Hadamard unstable but satisfies the above criterion of boundedness, is shown to have unstable decreasing branch in steady simple shear flow. At present, to the authors' knowledge, there is no viscoelastic single-integral CE of factorable type proposed in the literature which can satisfy all the Hadamard and dissipative stability criteria.     

On a class of constitutive equations for viscoelastic liquids

Simple rheological equations that describe non-linear viscoelastic phenomena in polymeric liquids have long attracted the attention of many rheologists. Although there are many ways of deriving such equations, only one concept is considered here. This concept is based on the introduction of an internal parameter, the recoverable strain tenson, and arises from a special kinematic study together with the formalism of irreversible thermodynamics. The main part of the paper sets out the theory for a single mode but a multimode extension is demonstrated towards the end and is compared with experimental data. Finally some of the problems that remain unsolved in the theory are discussed.The aim of this paper is to acquaint rheologists with the author's views as the model rheological equations he developed have recently been discussed in the literature without his participation.     

A fast and efficient iterative scheme for viscoelastic flow simulations with the DEVSS finite element method

We present a new fast iterative solution technique for the large sparse-matrix system that is commonly encountered in the mixed finite-element formulation of transient viscoelastic flow simulation: the DEVSS (discrete elastic-viscous stress splitting) method. A block-structured preconditioner for the velocity, pressure and viscous polymer stress has been proposed, based on a block reduction of the discrete system, designed to maintain spectral equivalence with the discrete system. The algebraic multigrid method and the diagonally scaled conjugate gradient method are applied to the preconditioning sub-block systems and a Krylov subspace iterative method (MINRES) is employed as an outer solver. We report the performance of the present solver through example problems in 2D and 3D, in comparison with the corresponding Stokes problems, and demonstrate that the outer iteration, as well as each block preconditioning sub-problem, can be solved within a fixed number of iterations. The required CPU time for the entire problem scales linearly with the number of degrees of freedom, indicating O(N) performance of this solution algorithm.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

A new decoupled finite element algorithm for viscoelastic flow. Part 1: Numerical algorithm and sample results

This paper presents an algorithm for two-dimensional Steady viscoelastic flow Simulation in which the Solution of the momentum and continuity equations is decoupled from that of the constitutive equations. The governing equations are discretized by the finite element method, with 3 × 3 element subdivision for the stress field approximation. Non-consistent Streamline upwinding is also used. Results are given for flow through a converging channel and through an abrupt planar 4:1 contraction.     

A new decoupled finite element algorithm for viscoelastic flow. Part 2: Convergence properties of the algorithm

We present the results of some numerical experiments which were carried out in order to investigate the general characteristics of the algorithm described in Part I of this paper.     

Comparative evaluation of some existing rate-type constitutive equations for a viscoelastic fluid undergoing wiggle flow

A comparative evaluation of existing rate-type constitutive equations is provided for a viscoelastic fluid undergoing accelerated flow. To this end, accurate point velocity and stress birefringence data previously obtained by laser Doppler anemometry and stress birefringence are utilized. For each constitutive equation, the numerical values of constants which yield the best fit with experimental data are determined via non-linear regression analysis. The best agreement between experimental and calculated normal stress differences is obtained with the White-Metzner equation. The success of this equation is attributed to the deformation rate dependence of its viscosity and time constant.     

Boundary element method for solving dynamical response of viscoelastic thin plate (I)

I.lntroducti0nThedynamicresponseofviscoelasticstructuresisoneofimportantresearchdirectionsinsolidmechanics.BecauseofthecompIexityoftheconstitutiverelations0fviscoeIasticmaterials,theproblemofsolvingthedynamicresponseisverydifflcult.Therearesomeavailablenu…     

Block element method for solving integrated equations of contact problems in wedge-shaped domains

This paper describes the block element method for spatial integral equations with a difference kernel in the boundary-value problems of continuum mechanics and mathematical physics. The basis of the proposed method is the Wiener — Hopf method, whose generalization for a spatial case is called integral factorization method. The block element method is applied to solve problems in domains with piecewise smooth boundaries containing corner points. The developed method was used to solve the contact problem for a wedge-shaped stamp occupying the first quadrant. This paper describes in detail the methods of obtaining various characteristics of the solution constructed by reversing the system of one-dimensional linear integral equations typical for dynamics and static contact problems for stamps in the form of a strip.     

A new mixed finite element for calculating viscoelastic flow

A new mixed finite element has allowed us to calculate flows of Maxwell-B and Oldroyd-B fluids at very high values of the Deborah number, De. The element is divided into several bilinear sub-elements for the stresses, while streamline-upwinding is used for discretizing the constitutive equation. The method is applied to the stick-slip problem, the flow through a tapered contraction and the flow through four-to-one abrupt plane and circular contractions. Important corner vortices develop at high values of De in the circular contraction. We have not encountered upper limits for the Deborah number in our calculations with Oldroyd-B fluids.     

A pressure-smoothing scheme for incompressible flow problems

A pressure-smoothing scheme for Stokes and Navier–Stokes flows of Newtonian fluids and for Stokes flow of Maxwell fluids is described. The stress deviator obtained from the calculated velocity field is substituted into the governing equilibrium equation. The resulting equation is then solved to obtain a new, smoothed pressure by a least square finite element method.     

A consistent numerical scheme for the von Mises mixed-hardening constitutive equations

Analytical solutions are derived for the von Mises mixed-hardening elastoplastic model under rectilinear strain paths, and the concept of response subspace is introduced such that the original five-dimensional problem in deviatoric stress space is reduced to a more economic two-dimensional problem, of which two coordinates (x,y) suffice to determine normalized active stress. Furthermore, in this subspace a Minkowski spacetime can be endowed, on which the group action is found to be a proper orthochronous Lorentz group SO_o(2,1). The existence of a fixed point attractor in the normalized active stress space is demonstrated by the long-term behavior deduced from the analytical solutions, which together with the response stability is further verified by Lyapunov's direct method. Two numerical schemes based on a nonlinear Volterra integral equation and on a group symmetry are derived, the latter of which exactly preserves the consistency condition for every time step. The consistent scheme is stable, accurate and efficient, because it updates the stress point automatically on the yield surface at each time step without any iteration. For the purpose of comparison and contrast, numerical results calculated by the above two schemes as well as by the radial return method were displayed for several loading examples.     

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC （finite incremental calculus） pressure stabilization process and the DEVSS （discrete elastic viscous stress splitting） method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU （streamline-upwind） method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP （extended Pom-Pom） constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4：1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of the Weissenberg number, particularly the capability in reproducing the results, which can be used to explain the ＂die swell＂ phenomenon observed in the polymer injection molding process.