Jon Wilkening

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which in the case of the Benjamin–Ono equation, are the mean, a spatial phase, a temporal phase, and the real part of one of the Fourier modes at t=0.We use our method to study global paths of nontrivial time-periodic solutions connecting stationary and traveling waves of the Benjamin–Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached. By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

Breakdown of self-similarity at the crests of large-amplitude standing water waves.

We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokess sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.

Overdetermined shooting methods for computing standing water waves with spectral accuracy

A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wiltons ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counter-propagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss the existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth. In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Efficiency is achieved in the trust-region approach by parallelizing the Jacobian computation, so the setup cost of computing the Dirichlet-to-Neumann operator in the variational equation is not repeated for each column. Updates of the Jacobian are also delayed until the previous Jacobian ceases to be useful. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high-order Runge–Kutta or spectral deferred correction method in time and quadruple precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for transferring much of the computation to a graphic processing units are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.

Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension

A numerical method is introduced for the computation of time-periodic vortex sheets with surface tension separating two immiscible, irrotational, two-dimensional ideal fluids of equal density. The approach is based on minimizing a nonlinear functional of the initial conditions and supposed period that is positive unless the solution is periodic, in which case it is zero. An adjoint-based optimal control technique is used to efficiently compute the gradient of this functional. Special care is required to handle singular integrals in the adjoint formulation. Starting with a solution of the linearized problem about the flat rest state, a family of smooth, symmetric breathers is found that, at quarter-period time intervals, alternately pass through a flat state of maximal kinetic energy, and a rest state in which all the energy is stored as potential energy in the interface. In some cases, the interface overturns before returning to the initial, flat configuration. It is found that the bifurcation diagram describing these solutions contains several disjoint curves separated by near-bifurcation events.

Parameter estimation by implicit sampling

Author(s): Morzfeld, M; Tu, X; Wilkening, J; Chorin, AJ | Abstract:

A local construction of the Smith normal form of a matrix polynomial

Author(s): Yu, Jia | Advisor(s): Wilkening, Jon A | Abstract: This dissertation consists of two separate chapters. In the first chapter, we present an algorithm for computing a Smith normal form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column) to obtain the Smith normal form. The performance of the algorithm in exact arithmetic is reported for several test cases.The second chapter is devoted to a numerical method for computing nontrivial time-periodic, gravity-driven water waves with or without surface tension. This method is essentially a shooting method formulated as a minimization problem. The objective function depends on the initial conditions and the proposed period, and measures deviation from time-periodicity. We adapt an adjoint-based optimal control method to rapidly compute the gradient of the functional. The main technical challenge involves handling the nonlocal Dirichlet to Neumann operator of the water wave equations in the adjoint formulation. Several families of traveling waves and symmetric breathers are simulated. In the latter case, we observe disconnections in the bifurcation curves due to nonlinear resonances at critical bifurcation parameters.

Stress concentrations, diffusionally accommodated grain boundary sliding and the viscoelasticity of polycrystals

Using analytical and numerical methods, we analyse the Raj–Ashby bicrystal model of diffusionally accommodated grain-boundary sliding for finite interface slopes. Two perfectly elastic layers of finite thickness are separated by a given fixed spatially periodic interface. Dissipation occurs by time-periodic shearing of the viscous interfacial region, and by time-periodic grain-boundary diffusion. Although two time scales govern these processes, of particular interest is the characteristic time tD for grain-boundary diffusion to occur over distances of order of the grain size. For seismic frequencies ωtD≫1, we find that the spectrum of mechanical loss Q−1 is controlled by the local stress field near corners. For a simple piecewise linear interface having identical corners, this localization leads to a simple asymptotic form for the loss spectrum: for ωtD≫1, Q−1∼const.ω−α. The positive exponent α is determined by the structure of the stress field near the corners, but depends both on the angle subtended by the corner and on the orientation of the interface; the value of α for a sawtooth interface having 120° angles differs from that for a truncated sawtooth interface whose corners subtend the same 120° angle. When corners on an interface are not all identical, the behaviour is even more complex. Our analysis suggests that the loss spectrum of a finely grained solid results from volume averaging of the dissipation occurring in the neighbourhood of a randomly oriented three-dimensional network of grain boundaries and edges.

Practical Error Estimates for Reynolds' Lubrication Approximation and its Higher Order Corrections

PRACTICAL ERROR ESTIMATES FOR REYNOLDS’ LUBRICATION APPROXIMATION AND ITS HIGHER ORDER CORRECTIONS JON WILKENING Abstract. Reynolds’ lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds’ equation may be thought of as the zeroth order term in an expansion of the solution of the Stokes equations in powers of the aspect ratio e of the domain. In this paper, we show how to compute the terms in this expansion to arbitrary order on a two-dimensional, x-periodic domain and derive rigorous, a-priori error bounds for the difference between the exact solution and the truncated expansion solution. Unlike previous studies of this sort, the constants in our error bounds are either independent of the function h(x) describing the geometry, or depend on h and its derivatives in an explicit, intuitive way. Specifically, if the expansion is truncated at order 2k, the error is O(e 2k+2 ) and h enters into R 1 the error bound only through its first and third inverse moments 0 h(x) −m dx, m = 1, 3 and via ∂ x h ‚ ∞ , 1 ≤ ≤ 2k + 2. We validate our estimates by comparing with finite the max norms ‚ ! h element solutions and present numerical evidence that suggests that even when h is real analytic and periodic, the expansion solution forms an asymptotic series rather than a convergent series. Key words. Incompressible flow, lubrication theory, asymptotic expansion, Stokes equations, thin domain, a-priori error estimates AMS subject classifications. 76D08, 35C20, 41A80 1. Introduction. Reynolds’ lubrication equation [22, 20, 16, 12] is used exten- sively in engineering applications to study flows between moving machine parts, e.g. in journal bearings or computer disk drives. It is also used in micro- and bio-fluid me- chanics to model creeping flows through narrow channels and in thin films. Although there is a vast literature (including several textbooks) on viscous flows in thin geome- tries, the equations are normally derived either directly from physical arguments [16], or using formal asymptotic arguments [12]. This is acceptable in most circumstances as the original equations (Stokes or Navier–Stokes) have also been derived from phys- ical considerations, and by now the lubrication equations have been used frequently enough that one can draw on experience and intuition to determine whether they will work well for a given problem. On the other hand, as soon as the geometry of interest develops (or approaches) a singularity, or if we wish to compute several terms in the asymptotic expansion of the solution in powers of the aspect ratio e, we rapidly leave the space of problems for which we can use experience as a guide; thus, it would be helpful to have a rigorous proof of convergence to serve as a guide to identify the features of the geometry that could potentially invalidate the approximation. For example, in [25], the author and A. E. Hosoi used lubrication theory to study the optimal wave shapes that an animal such as a gastropod should use as it propagates ripples along its muscular foot to crawl over a thin layer of viscous fluid. In certain limits of this constrained optimization problem, the optimal wave shape develops a kink or cusp in the vicinity of the region closest to the substrate, and there is a competing mechanism controlling the size of the modeling error (singularity formation vs. nearness to the substrate). We found that shape optimization within (zeroth order) lubrication theory drives the geometry ∗ Department of Mathematics and Lawrence Berkeley National Laboratory, University of Cali- fornia, Berkeley, CA 94720 (wilken@math.berkeley.edu). This work was supported in part by the Director, Office of Science, Advanced Scientific Computing Research, U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

Shape optimization of a sheet swimming over a thin liquid layer

Motivated by the propulsion mechanisms adopted by gastropods, annelids and other invertebrates, we consider shape optimization of a flexible sheet that moves by propagating deformation waves along its body. The self-propelled sheet is separated from a rigid substrate by a thin layer of viscous Newtonian fluid. We use a lubrication approximation to model the dynamics and derive the relevant Euler-Lagrange equations to simultaneously optimize swimming speed, efficiency and fluid loss. We find that as the parameters controlling these quantities approach critical values, the optimal solutions become singular in a self-similar fashion and sometimes leave the realm of validity of the lubrication model. We explore these singular limits by computing higher order corrections to the zeroth order theory and find that wave profiles that develop cusp-like singularities are appropriately penalized, yielding non-singular optimal solutions. These corrections are themselves validated by comparison with finite element solutions of the full Stokes equations, and, to the extent possible, using recent rigorous a-priori error bounds.

Computation of three-dimensional standing water waves

We develop a method for computing three-dimensional gravity-driven water waves, which we use to search for time-periodic standing wave solutions. We simulate an inviscid, irrotational, incompressible fluid bounded below by a flat wall, and above by an evolving free surface. The computations make use of spectral derivatives on the surface, but also require computing a velocity potential in the bulk, which we carry out using a finite element method with fourth-order elements that are curved to match the free surface. This computationally expensive step is solved using a parallel multigrid algorithm, which is discussed in detail. Time-periodic solutions are searched for using a previously developed overdetermined shooting method. Several families of large-amplitude three-dimensional standing waves are found in both shallow and deep regimes, and their physical characteristics are examined and compared to previously known two-dimensional solutions.