Qinghai Zhang

A Fourth-Order Accurate Finite-Volume Method with Structured Adaptive Mesh Refinement for Solving the Advection-Diffusion Equation

We present a fourth-order accurate algorithm for solving Poissons equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditions; the resulting linear system is solved with a multigrid algorithm. For time integration, we couple the elliptic solver to a fourth-order accurate Runge-Kutta method, introduced by Kennedy and Carpenter [Appl. Numer. Math., 44 (2003), pp. 139-181], which enables us to treat the nonstiff advection term explicitly and the stiff diffusion term implicitly. We demonstrate the spatial and temporal accuracy by comparing results with analytical solutions. Because of the general formulation of the approach, the algorithm is easily extensible to more complex physical systems.

A new interface tracking method: The polygonal area mapping method

We present a new method, the polygonal area mapping (PAM) method, for tracking a non-diffusive, immiscible material interface between two materials in two-dimensional incompressible flows. This method represents material areas explicitly as piecewise polygons, traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon-clippings in a discrete manner. The new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately. In addition to high accuracy, the PAM method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications. The mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons. The results from a set of widely used benchmark tests show that the PAM method is superior to existing volume-of-fluid (VOF) methods.

Neural mechanism of optimal limb coordination in crustacean swimming

Significance Despite the general belief that neural circuits have evolved to optimize behavior, few studies have clearly identified the neural mechanisms underlying optimal behavior. The distinct limb coordination in crustacean swimming and the relative simplicity of the neural coordinating circuit have allowed us to show that the interlimb coordination in crustacean swimming is biomechanically optimal and how the structure of underlying neural circuit robustly gives rise to this coordination. Thus, we provide a concrete example of how an optimal behavior arises from the anatomical structure of a neural circuit. Furthermore, our results suggest that the connectivity of the neural circuit underlying limb coordination during crustacean swimming may be a consequence of natural selection in favor of more effective and efficient swimming. A fundamental challenge in neuroscience is to understand how biologically salient motor behaviors emerge from properties of the underlying neural circuits. Crayfish, krill, prawns, lobsters, and other long-tailed crustaceans swim by rhythmically moving limbs called swimmerets. Over the entire biological range of animal size and paddling frequency, movements of adjacent swimmerets maintain an approximate quarter-period phase difference with the more posterior limbs leading the cycle. We use a computational fluid dynamics model to show that this frequency-invariant stroke pattern is the most effective and mechanically efficient paddling rhythm across the full range of biologically relevant Reynolds numbers in crustacean swimming. We then show that the organization of the neural circuit underlying swimmeret coordination provides a robust mechanism for generating this stroke pattern. Specifically, the wave-like limb coordination emerges robustly from a combination of the half-center structure of the local central pattern generating circuits (CPGs) that drive the movements of each limb, the asymmetric network topology of the connections between local CPGs, and the phase response properties of the local CPGs, which we measure experimentally. Thus, the crustacean swimmeret system serves as a concrete example in which the architecture of a neural circuit leads to optimal behavior in a robust manner. Furthermore, we consider all possible connection topologies between local CPGs and show that the natural connectivity pattern generates the biomechanically optimal stroke pattern most robustly. Given the high metabolic cost of crustacean swimming, our results suggest that natural selection has pushed the swimmeret neural circuit toward a connection topology that produces optimal behavior.

On a Family of Unsplit Advection Algorithms for Volume-of-Fluid Methods

Volume-of-fluid (VOF) methods are widely-used for the interface tracking problem, yet rigorous analyses of them are rare. This paper presents such an analysis for incompressible flows by combining the theories of ordinary differential equations, differential geometry, and Boolean algebra. Based on the concept of donating region (DR) [Q. Zhang, SIAM Rev., 55 (2013), pp. 443--461], the author classifies the fluxing particles of a fixed control volume into four categories and derives three analytical solutions for the advection equation of the color function. One edgewise solution provides a unified view of DR-based advection algorithms for VOF methods while another cellwise solution serves as the theoretical foundation of the recent polygonal area mapping method. The well-known second-order convergence rates of streamline-based VOF advection algorithms are proved rigorously. Potential deterioration of this second-order convergence is also discussed for several subtle issues such as exact mass conservation a...

FOURTH-ORDER INTERFACE TRACKING IN TWO DIMENSIONS VIA AN IMPROVED POLYGONAL AREA MAPPING METHOD ∗

We present an improved PAM (iPAM) method as the first fourth-order interface tracking method whose convergence rates are independent of

Handling solid-fluid interfaces for viscous flows: Explicit jump approximation vs. ghost cell approaches

C^1

On Donating Regions: Lagrangian Flux through a Fixed Curve

(derivative) discontinuities of the interface. As an improved version of the polygonal area mapping (PAM) method [Q. Zhang and P. L.-F. Liu, J. Comput. Phys., 227 (2008), pp. 4063--4088], the accuracy of the iPAM method is achieved via (i) augmenting the abstract data structure of PAM to faithfully represent multiple components of material regions within a single cell, (ii) removing restrictive assumptions of PAM, (iii) adjusting the volume of represented cell material regions via polygon ear removal, and (iv) maintaining a relation

GePUP: Generic Projection and Unconstrained PPE for Fourth-Order Solutions of the Incompressible Navier---Stokes Equations with No-Slip Boundary Conditions

(h_L=r_hh^{\alpha})

Implicit Parallel FEM Analysis of Shallow Water Equations

between the Eulerian grid size

A numerical study of swash flows generated by bores

h