Viscoelasticity

As part of an NSF research grant we are investigating least squares finite element methods for several applications in solid and fluid mechanics. In the current phase of the project we are interested in establishing robust numerical methods with rigorous accompanying theoretical foundations for non-Newtonian fluids of Oldroyd type. These materials describe a fluid with significant elastic characteristics and are often used to model blood, paints, oils, polymer melts, etc.

The equations describing these fluids resemble the Navier-Stokes equations, but with a stress tensor decomposed into solid and fluid components and a more complicated constitutive equation. Numerically, these have been historically very difficult to treat. We have identified 3 challenges common to all numerical methods: (1) the nonlinear iteration often fails, (2) corner singularities introduce errors difficult to handle, and (3) the resulting linear systems are difficult to handle. We believe our approach holds great promise to combat these challenges.

(1) We employ a nested-iteration strategy to handle the nonlinearity. In short, we let the nonlinear iteration work in tandem with adaptive mesh refinement. This is a very natural arrangement for a least-squares formulation. Not only does this make the iteration more efficient, but also supplies the nonlinear iteration with good initial guesses on each new mesh. The nonlinear and linear least-squares functionals provide reliable information on where to refine and when the nonlinear error is small relative to the discretization error. (2) Our weighted norm approach removes the pollution effects from corner singularities, obviating the need for excessive mesh refinement near boundary corners. In fact, the singularities in the stress variables for these systems are very severe. We believe that, left unaddressed, the singlarities may be a large factor in the difficulty that standard methods encounter. That is, these errors may be enough to cause some nonlinear iterations to "jump out of" the basin of attraction. Using the strength of singularities for Stokes equations as a guide, our approach employs an appropriate weighted norm for the problem. (3) Least-squares formulations lead to symmetric positive definite linear systems. Standard multigrid methods generally solve these systems with optimal complexity, providing a robust, scalable solver. Adaptive methods, including adaptive smoothed aggregation (aSA), remain of interest as well. Preliminary results indicate that these may also provide a fast solver.

In our current work we have both theoretical and numerical results that are very encouraging. We have two formulations: a div/curl approach and a mixed H(div) approach. In our numerical tests, both formulations converge at optimal rates and accurately predict large-scale flow features in agreement with benchmark solutions. These tests were done, in part, by Wabash undergraduate student Haris Amin in a summer research internship. We have theoretical error bounds for each formulation as well. Details can be found in forthcoming papers.

Collaborators:

Zhiqiang Cai, Purdue University

Haris Amin, Wabash College Undergraduate

Weighted-Norm Least-Squares Methods

Many second-order boundary value problems fail to have full regularity at locations of changing boundary condition type and at corners of polygonal/polyhedral domains. While these boundary singularities have purely local character, many methods suffer from a global loss of accuracy. The least-squares method can be particularly sensitive to a such a loss of regularity.

The weighted-norm method we've developed requires only a rough lower bound on the power of the singularity and can be applied to a wide range of elliptic equations.

Standard div-curl functionals minimized over polynomial finite element spaces converge to the closest H^1 approximation to the solution. When the solution is not in H^1 the method simply fails to converge. Our approach for div/curl functionals uses an agressive weighting to restore optimal order discretization accuracy in weighted L^2, H^1, and functional norms, while also recovering L^2 convergence. Our analysis is set in weighted Sobolev spaces, an area for which there is some beautiful theory in the analysis community. Our theoretical and numerical results can be found in our 2-d paper or a more recent 3-d paper.

For H(div)-conforming functionals using the appropriate Raviart-Thomas finite element spaces, the least-squares method suffers from slow convergence and errors with global support. A more subtle weighting procedure can increase discretization accuracy to optimal levels. In current work we show this and also establish new interpolation results in for RT approximations in weighted norms. As part of a summer research internship, Grinnell College undergraduate student Sam Calisch performed extensive parameter studies for this approach.

We've employed this idea with good success in equations for nonlinear elasticity, Newtonian fluids, and viscoelastic fluids.

Collaborators:

Tom Manteuffel, University of Colorado

Eunjung Lee, University of Colorado

Zhiqiang Cai, Purdue University

Sam Calisch, Grinnell College Undergraduate

Nonlinear Elasticity

In my Ph.D. thesis I developed a least-squares minimization strategy for solving the equations of geometrically nonlinear elasticity. One of the strengths of the formulation is that in the presence of full regularity the linearized functionals are equivalent to the product H^1 norm, allowing the use of simple polynomial finite element spaces. When full H^2 regularity is lost (which occurs at almost any corner in a polygonal domain) the weighted-norm procedure can be used to recover optimal-order discretization accuracy. When using an efficient nonlinear nested iteration strategy, the full nonlinear problem can generally be solved with work equivalent to the linear problem.

Nested Iteration

At the heart of the multigrid/multilevel philosophy is the idea that work done on coarse scales is cheap in comparison to the work required on the finest scale. On the other hand, completely resolving all of one type of error in an inner iteration is inefficient. We are interested in what can be considered efficient simultaneous continuation methods.

The iteration always consists of an innermost iteration of some number of multigrid cycles on an algebraic system of equations. We generally think of the outermost iteration as the refinement level of the mesh, either locally or globally refined. As intermediate parameters we consider linearization steps, material parameters (Reynolds number, Lame constants, etc.), polynomial order of basis, weighting parameter, and forcing functions.

The difficulty is, of course, using computable measures to predict the most accurate next step within the iteration. The eventual goal is the design of general algorithms that learn the behavior of the specific iteration while solving the coarse scale problems to choose a cost-optimal strategy for the finer and finest scale problems. These choices must be made on-the-fly and with as little input from the user as possible.

In computation we continually develop intuition regarding these choices. We have developed theory regarding the simple two parameter model of meshsize and multigrid cycle. In practice we employ this approach to other problems of interest, especially those with a nonlinear iteration.