Brian T. Helenbrook

Analysis of ``p''-Multigrid for Continuous and Discontinuous Finite Element Discretizations

Analysis and numerical experiments examining the behavior and performance of p-multigrid (p = polynomial degree) for solving hp-finite element (FEM) discretizations are presented. We begin by demonstrating the mesh and order independent properties of p-multigrid when used to solve a C0 continuous FEM discretization of the Laplace equation. We then apply pmultigrid to both continuous and discontinuous FEM discretizations of the convection equation. Although 1D Fourier analysis predicts that mesh independent results should be possible for both discretizations, in 2D the results are sensitive to both the mesh resolution and the degree of polynomial approximation. Examining the solutions, we find that for both discretizations, the slowest converging mode is long-wavelength along the streamwise direction and short wavelength normal to this direction. Because of the isotropic coarsening of p-multigrid, this mode is not damped on coarse levels.

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method. 1 https://ntrs.nasa.gov/search.jsp?R=20080014258 2020-06-28T15:20:56+00:00Z

A two-fluid spectral-element method

Abstract A spectral-element formulation for two-fluid flows is presented which uses arbitrary-Lagrangian–Eulerian mesh movement with an unstructured mesh. The main advantage of this approach is that the spectral elements provide a high order-of-accuracy while the unstructured mesh allows geometric flexibility. The accuracy of this approach is demonstrated for several relatively simple configurations with and without moving interfaces. Another unique feature of the formulation as compared to previous (single-fluid) spectral-element methods is that a fully implicit time advancement scheme is used. This is made possible by the development of a spectral-multigrid iteration which solves the implicit governing equations in a fixed number of iterations insensitive to both the polynomial degree of the spectral elements and the number of elements in the mesh. By using this approach, solutions for both stiff unsteady problems and steady-state problems can be obtained efficiently.

Inertialess instability of a two-layer liquid film flow

The physical mechanism of instability in a superposed two-layer liquid film flow down an incline plane is analyzed. If the layer adjacent to the wall is sufficiently thin and less viscous in certain two-layer parallel Newtonian liquid flows of the same density with an interface but without a free surface, the flows are stable with respect to long waves. This is the so-called “thin lubrication layer effect.” However, when a free surface exists in the two-layered flow, the flow becomes unstable even when the Reynolds number approaches zero. Thus the thin-layer lubrication effect is lost due to the presence of the free surface, and inertialess instability occurs. The reason for the loss of the lubrication effect and the mechanism of inertialess instability are explained by use of the energy budget in the mechanical energy equation. Contrary to the case of two-layer flows without a free surface, the flow with a free surface is stable if the layer adjacent to the solid wall is more viscous. The stabilization i...

Transience to instability in a liquid sheet

Series solutions are found which describe the evolution to absolute and convective instability in an inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. These solutions are used to validate asymptotic stability predictions for sinuous and varicose disturbances. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate or reduce the order of singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow like t 1/3 .I f only the position is perturbed, we find that the sheet is stable and decays like t −2/3 at the origin. Furthermore, if both the position and velocity of a sheet are perturbed in the absence of ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and varicose waves grow like t 1/2 with a convective instability. A brief history of the fluid dynamics of liquid sheets and its applications is given in Lin & Jiang (2003). Here, we mention only the work that is directly relevant to this study. As shown in the analysis of Rayleigh (1896), there are two linearly independent wave modes of a liquid sheet. The sinuous mode moves the two free surfaces of a sheet in phase. The varicose mode symmetrically moves the free surfaces in opposite directions. These modes were later confirmed in the experiments of Taylor (1959). The onset of wave instability was analysed by Squire (1953) through the use of classical temporal stability theory. The classical theory predicts, for finite Q = ρg/ρl (ρg and ρl being, respectively, the gas and liquid densities), that the sinuous wave is only unstable if the Weber number is greater than one. The Weber number is defined as We = ρlU 2 h0/S, where U is the liquid velocity in the sheet, h0 is the half-sheet thickness and S is the interfacial tension. The experiments of Brown (1961) indicated instability for We <1, which seemed to contradict the classical theory. Around this time, a new stability theory was being developed to study the complex roots of a dispersion relationship and to take into consideration the possibility of both spatial and temporal growth (Sturrock 1958). Sturrock introduced a method for determining the nature of spatio-temporal growth, and established that the mapping between frequency and wavenumber in their corresponding complex

Low-Reynolds-number instabilities in three-layer flow down an inclined wall

The finite wavelength instability of viscosity-stratified three-layer flow down an inclined wall is examined for small but finite Reynolds numbers. It has previously been demonstrated using linear theory that three-layer zero-Reynolds-number instabilities can have growth rates that are orders of magnitude larger than those that arise in twolayer structures. Although the layer configurations yielding large growth instabilities have been well characterized, the physical origin of the three-layer inertialess instability remains unclear. Using analytic, numerical and experimental techniques, we investigate the origin and evolution of these instabilities. Results from an energy equation derived from linear theory reveal that interfacial shear and Reynolds stresses contribute to the energy growth of the instability at finite Reynolds numbers, and that this remains true in the limit of zero Reynolds number. This is thus a rare example that demonstrates how the Reynolds stress can play an important role in flow instability, even when the Reynolds number is vanishingly small. Numerical solutions of the Navier–Stokes equations are used to simulate the nonlinear evolution of the interfacial deformation, and for small amplitudes the predicted wave shapes are in excellent agreement with those obtained from linear theory. Further comparisons between simulated interfacial deformations and linear theory reveal that the linear evolution equations are surprisingly accurate even when the interfaces are highly deformed and nonlinear effects are important. Experimental results obtained using aqueous gelatin systems exhibit large wave growth and are in agreement with both the theoretical predictions of small-amplitude behaviour and the nonlinear simulations of the large-amplitude behaviour. Quantitative agreement is confounded owing to water diffusion driven by differences in gelatin concentration between the layers in experiments. However, the qualitative agreement is sufficient to confirm that the correct mechanism for the experimental instability has been determined.

High order finite element model for core loss assessment in a hysteresis magnetic lamination

A dynamic model for evaluating core losses in a hysteretic magnetic lamination is developed and then solved using a high-order finite element method that includes time-history effects. It is demonstrated that the dynamic hysteresis effect, previously used to explain the frequency dependence of B−H loops, is not a fundamental phenomenon of magnetic materials but originates from the skin effect. It arises because the measured flux density is an averaged value over the lamination thickness, and this value is influenced strongly by the skin effect. The study verifies that, unlike the observed dynamic hysteresis effect, the local B−H loop is in fact frequency independent. The developed dynamic core loss model is thus derived based on the frequency-independent B−H loop. It is shown that the developed model can accurately evaluate the losses for different frequencies and thicknesses based on only one set of inputs of an experimental B−H loop at one low frequency without a huge database of experimental losses.

Proper-Orthogonal-Decomposition Based Thermal Modeling of Semiconductor Structures

A thermal model of a semiconductor structure is developed using a hierarchical function space, rather than physical space. The thermal model is derived using the proper orthogonal decomposition (POD) and does not require any assumptions about the physical geometry, dimensions, or heat flow paths, as is usually necessary for compact/lumped thermal models. The approach can be applied to complex geometries and provides detailed thermal information at a computational cost comparable to that of lumped thermal models. The POD thermal model is applied to steady thermal simulations of a 2-D silicon-on-insulator (SOI) device structure and validated at various power levels against detailed numerical simulation (DNS) data. It is shown that a POD thermal model using only four POD modes can duplicate the temperature solution derived from DNS, including the hot spot and temperature gradients along the device island. In addition, an unsteady POD model of the SOI device structure is constructed. A POD model incorporating ten modes yielded a virtually identical solution when compared to corresponding unsteady DNS results.

Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection-diffusion equation

An improved p-multigrid algorithm for discontinuous Galerkin (DG) discretizations of convection-diffusion problems is presented. The general p-multigrid algorithm for DG discretizations involves a restriction from the p=1 to p=0 discontinuous polynomial solution spaces. This restriction is problematic and has limited the efficiency of the p-multigrid method. For purely diffusive problems, Helenbrook and Atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p=1. It is shown that this method is not directly applicable to the convection-diffusion (CD) equation because it results in a central-difference discretization for the convective term. To remedy this, ideas from the streamwise upwind Petrov-Galerkin (SUPG) formulation are used to devise a transition from the discontinuous to continuous space at p=1 that yields an upwind discretization. The results show that the new method converges rapidly for all Peclet numbers.

Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis.