Purpose -Most studies of power-law fluids are carried out using stress-based system of Navier-Stokes equations; and least-squares finite element models for vorticity-based equations of power-law fluids have not been explored yet. Also, there has been no study of the weak-form Galerkin formulation using the reduced integration penalty method (RIP) for power-law fluids. Based on these observations, the purpose of this paper is to fulfill the two-fold objective of formulating the least-squares finite element model for power-law fluids, and the weak-form RIP Galerkin model of power-law fluids, and compare it with the least-squares finite element model. Design/methodology/approach -For least-squares finite element model, the original governing partial differential equations are transformed into an equivalent first-order system by introducing additional independent variables, and then formulating the least-squares model based on the lower-order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least-squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation. Findings -The non-Newtonian fluids require higher-order polynomial approximation functions and higher-order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power-law indices compared to higher power-law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill-conditioned especially for lower values of power-law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters. Originality/value -Vorticity-based least-squares finite element models are developed for power-law fluids and effects of linearizations are explored. Also, the weak-form RIP Galerkin model is developed.

No abstract

In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C 0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Kármán nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.which makes it more important to study these materials. The theory of viscoelastic behavior is long been established, the reader can refer to Lakes [2009] for a materials perspective and Reddy [2008], Flügge [1975], Christensen [1982], Findley et al. [1976] and Lockett [1975] for a continuum perspective. Within the continuum purview there are analytical methods like integral transforms, Laplace transforms and correspondence principle, etc., to study the mechanical response of structures made of viscoelastic materials. But as with many analytical methods, they are limited to simple cases of geometry and loading. In such scenarios numerical techniques like the finite element method becomes very useful in testing and predicting the properties of a material without actually fabricating them. Numerical methods can be used to obtain approximate solution with desired accuracy.Many researchers [Taylor et al., 1970;Oden and Armstrong, 1971;Holzapfel and Reiter, 1995;Henriksen, 1984;Hartmann, 2002;Roy and Reddy, 1988] have used the finite element method to study viscoelastic materials. The main difficulty with the viscoelastic finite element models is the approximation of convolution integrals that come from viscoelastic constitutive equations. Most of these finite element models try to circumvent the problem with the time-dependent convolution integrals by transforming them to a set of discrete algebraic equations in space. Taylor et al. [1970] and Oden and Armstrong [1971] used recurrence relations such that only the deformation history from last few iterations is needed to be stored instead of entire deformation from beginn...

Abstract-This study deals with the use of high-order spectral/hp approximation functions in finite element models of various of nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order 4  p ) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., 3 < p ) finite elements. For instance, higher-order spectral/hp finite element procedures allow us to develop robust structural elements such as beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. For fluid flows, when combined with least-squares variational principles, the higher-order spectral/hp technology allows us to develop efficient finite element models that always yield a symmetric positive-definite (SPD) coefficient matrix and, hence, robust iterative solvers can be used. Also, the use of spectral/hp finite element technology results in a better conservation of physical quantities like dilatation, volume, and mass, and stable evolution of variables with time for transient flows. The present study considers the weak-form based displacement finite element models elastic shells and the least-squares finite element models of the Navier-Stokes equations governing flows of viscous incompressible fluids. Numerical solutions of several nontrivial benchmark problems are presented to illustrate the accuracy and robustness of the developed finite element technology.