This paper presents a p-version least squares finite element formulation (LSFEF) for two-dimensional, incompressible, non-Newtonian fluid flow under isothermal and non-isothermal conditions. The dimensionless forms of the differential equations describing the fluid motion and heat transfer are cast into a set of first-order differential equations using nowNewtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal-order, Co-continuous, p-version hierarchical approximation functions. The application of least squares minimization to the set of coupled first-order non-linear partial differential equations results in finding a solution vector { h } which makes the partial derivatives of the error functional with respect to ( 6 ) a null vector. This is accomplished by using Newton's method with a line search.The paper presents the implementation of a power-law model for the non-Newtonian Viscosity. For the non-isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid-driven cavity) are presented for isothermal power-law fluid flow. The Couette shear flow problem and the 4: 1 symmetric sudden expansion are used to present numerical results for non-isothermal power-law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.

This paper presents a p-version least-squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled. The dimensionless form of the differential equations describing the problem are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C" element approximation. The element properties are derived by utilizing p-version approximation functions in both space and time and then minimizing the error functional given by the space-time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space-time coupled least-squares minimization procedure.A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. The space-time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space-time coupled approach. For non-linear problems the non-linear algebraic equations resulting from the least-squares process are solved using Newton's method with a line search. This procedure results in a symmetric Hessian matrix. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance.The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection-diffusion and Burgers equations. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space-time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements. The work presented in this paper provides the basis for the extension of the space-time coupled least-squares minimization concept to two-and three-dimensional unsteady fluid flow. Graduate Student

SUMMARYThis paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier-Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier-Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of Co element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the spac*time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space-time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {d} which makes gradient of error functional with respect to {a} a null vector. This is accomplished by using Newton's method with a line search.A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space-time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space-time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space-time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.

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