Bryan Quaife

Bryan Quaife, George Biros

We construct a high-order adaptive time stepping scheme for vesicle suspensions with viscosity contrast. The high-order accuracy is achieved using a spectral deferred correction (SDC) method, and adaptivity is achieved by estimating the local truncation error with the numerical error of physically constant values. Numerical examples demonstrate that our method can handle suspensions with vesicles that are tumbling, tank-treading, or both. Moreover, we demonstrate that a user-prescribed tolerance can be automatically achieved for simulations with long time horizons.

Mary-Catherine Kropinski, Bryan Quaife

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, $u(\x) - α^2 Δu(\x) = 0$, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.

Nilima Nigam, Mary-Catherine Kropinski, Bryan Quaife

An integral equation method for solving the Yukawa-Beltrami equation on a multiply-connected sub-manifold of the unit sphere is presented. A fundamental solution for the Yukawa-Beltrami operator is constructed. This fundamental solution can be represented by conical functions. Using a suitable representation formula, a Fredholm equation of the second kind with a compact integral operator needs to be solved. The discretization of this integral equation leads to a linear system whose condition number is bounded independent of the size of the system. Several numerical examples exploring the properties of this integral equation are presented.

Alan E. Lindsay, Bryan Quaife, Laura Wendelberger

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green's functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.

Daryn Sagel, Bryan Quaife

The increasing frequency and severity of wildfires highlight the need for accurate fire and plume spread models. We introduce an approach that effectively isolates and tracks fire and plume behavior across various spatial and temporal scales and image types, identifying physical phenomena in the system and providing insights useful for developing and validating models. Our method combines image segmentation and graph theory to delineate fire fronts and plume boundaries. We demonstrate that the method effectively distinguishes fires and plumes from visually similar objects. Results demonstrate the successful isolation and tracking of fire and plume dynamics across various image sources, ranging from synoptic-scale ($10^4$-$10^5$ m) satellite images to sub-microscale ($10^0$-$10^1$ m) images captured close to the fire environment. Furthermore, the methodology leverages image inpainting and spatio-temporal dataset generation for use in statistical and machine learning models.

Mary-Catherine Kropinski, Bryan Quaife

We present an efficient integral equation approach to solve the heat equation, $u_t (\x) - Δu(\x) = F(\x,t)$, in a two-dimensional, multiply connected domain, and with Dirichlet boundary conditions. Instead of using integral equations based on the heat kernel, we take the approach of discretizing in time, first. This leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. The solution to this equation is formulated as a volume potential plus a double layer potential.The volume potential is evaluated using a fast multipole-accelerated solver. The boundary conditions are then satisfied by solving an integral equation for the homogeneous modified Helmholtz equation. The integral equation solver is also accelerated by the fast multipole method (FMM). For a total of $N$ points in the discretization of the boundary and the domain, the total computational cost per time step is $O(N)$ or $O(N\log N)$.

Xin Tong, Bryan Quaife

Data-driven techniques are being increasingly applied to complement physics-based models in fire science. However, the lack of sufficiently large datasets continues to hinder the application of certain machine learning techniques. In this paper, we use simulated data to investigate the ability of neural networks to parameterize dynamics in fire science. In particular, we investigate neural networks that map five key parameters in fire spread to the first arrival time, and the corresponding inverse problem. By using simulated data, we are able to characterize the error, the required dataset size, and the convergence properties of these neural networks. For the inverse problem, we quantify the network's sensitivity in estimating each of the key parameters. The findings demonstrate the potential of machine learning in fire science, highlight the challenges associated with limited dataset sizes, and quantify the sensitivity of neural networks to estimate key parameters governing fire spread dynamics.

Benjamin Ong, Andrew Christlieb, Bryan Quaife

In this paper, a new two-parameter family of regularized kernels is introduced, suitable for applying high-order time stepping to N-body systems. These high-order kernels are derived by truncating a Taylor expansion of the non-regularized kernel about $(r^2+ε^2)$, generating a sequence of increasingly more accurate kernels. This paper proves the validity of this two-parameter family of regularized kernels, constructs error estimates, and illustrates the benefits of using high-order kernels through numerical experiments.

Pieter Coulier, Bryan Quaife, Eric Darve

We consider an efficient preconditioner for boundary integral equation (BIE) formulations of the two-dimensional Stokes equations in porous media. While BIEs are well-suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as GMRES. In this paper, we apply a fast inexact direct solver, the inverse fast multipole method (IFMM), as an efficient preconditioner for GMRES. This solver is based on the framework of $\mathcal{H}^{2}$-matrices and uses low-rank compressions to approximate certain matrix blocks. It has a tunable accuracy $\varepsilon$ and a computational cost that scales as $\mathcal{O} (N \log^2 1/\varepsilon)$. We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block-diagonal preconditioner, especially for pipe flow problems involving many pores.