arXiv subjects

John M. Stockie

Publications and source records attributed to John M. Stockie.

At least 19 recordsLinked to original sources

arXiv2025

A mathematical model for Nordic skiing

Nordic skiing provides fascinating opportunities for mathematical modelling studies that exploit methods and insights from physics, applied mathematics, data analysis, scientific computing and sports science. A typical ski course winds over varied terrain with frequent changes in elevation and direction, and so its geometry is naturally described by a three-dimensional space curve. The skier travels along a course under the influence of various forces, and their dynamics can be described using a nonlinear system of ordinary differential equations (ODEs) that are derived from Newton's laws of motion. We develop an algorithm for solving the governing equations that combines Hermite spline interpolation, numerical quadrature and a high-order ODE solver. Numerical simulations are compared with measurements of skiers on actual courses to demonstrate the effectiveness of the model. Throughout, we aim to illustrate how elementary concepts from undergraduate courses in calculus and scientific computing can be applied to study real problems in sport, which we hope will provide stimulating examples for both instructors and students. At the same time, we demonstrate how these concepts are capable of providing novel insights into skiing that should also be of interest to sport scientists.

physics.class-ph↗

arXiv2024

Experimental and computational comparison of freeze-thaw induced pressure generation in red and sugar maple

Sap exudation is the process whereby trees such as sugar (Acer saccharum) and red maple (Acer rubrum) generate unusually high positive stem pressure in response to repeated cycles of freeze and thaw. This elevated xylem pressure permits the sap to be harvested over a period of several weeks and hence is a major factor in the viability of the maple syrup industry. The extensive literature on sap exudation documents competing hypotheses regarding the physical and biological mechanisms that drive positive pressure generation in maple, but to date relatively little effort has been expended on devising mathematical models for the exudation process. In this paper, we utilize an existing model of Graf et al. [J. Roy. Soc. Interface 12:20150665, 2015] that describes heat and mass transport within the multiphase gas-liquid-ice mixture in the porous xylem tissue. The model captures the inherent multiscale nature of xylem transport by including phase change and osmotic transport in wood cells on the microscale, which is coupled to heat transport through the tree stem on the macroscale. A parametric study based on simulations with synthetic temperature data identifies the model parameters that have greatest impact on stem pressure build-up. Measured daily temperature fluctuations are then used as model inputs and the resulting simulated pressures are compared directly with experimental measurements taken from mature red and sugar maple stems during the sap harvest season. The results demonstrate that our multiscale freeze-thaw model reproduces realistic exudation behavior, thereby providing novel insights into the specific physical mechanisms that dominate positive pressure generation in maple trees.

math.NA↗

arXiv2022

Simulating surface height and terminus position for marine outlet glaciers using a level set method with data assimilation

We implement a data assimilation framework for integrating ice surface and terminus position observations into a numerical ice-flow model. The model uses the well-known shallow shelf approximation (SSA) coupled to a level set method to capture ice motion and changes in the glacier geometry. The level set method explicitly tracks the evolving ice-atmosphere and ice-ocean boundaries for a marine outlet glacier. We use an Ensemble Transform Kalman Filter to assimilate observations of ice surface elevation and lateral ice extent by updating the level set function that describes the ice interface. Numerical experiments on an idealized marine-terminating glacier demonstrate the effectiveness of our data assimilation approach for tracking seasonal and multi-year glacier advance and retreat cycles. The model is also applied to simulate Helheim Glacier, a major tidewater-terminating glacier of the Greenland Ice Sheet that has experienced a recent history of rapid retreat. By assimilating observations from remotely-sensed surface elevation profiles we are able to more accurately track the migrating glacier terminus and glacier surface changes. These results support the use of data assimilation methodologies for obtaining more accurate predictions of short-term ice sheet dynamics.

physics.ao-ph↗

arXiv2020

An open-access clicker question bank for numerical analysis

We present a question bank consisting of over 250 multiple-choice and true-false questions covering a broad range of material typically taught in an introductory undergraduate course in numerical analysis or scientific computing. The questions are ideal for polling students during lectures by means of a student response system that uses clicker remotes or smartphones running a suitable app. We describe our experiences implementing these clicker questions in a recent class and provide evidence of their effectiveness in terms of testing students' prior knowledge, gauging understanding of new material, increasing participation, and especially improving student satisfaction. Our conclusions are supported by a mid-semester student survey as well as anecdotal observations. The question bank has been released as an open-access educational resource under a Creative Commons license (BY-NC-SA) for free use by the mathematics community.

physics.ed-ph↗

arXiv2018

A Bayesian approach for energy-based estimation of acoustic aberrations in high intensity focused ultrasound treatment

High intensity focused ultrasound is a non-invasive method for treatment of diseased tissue that uses a beam of ultrasound to generate heat within a small volume. A common challenge in application of this technique is that heterogeneity of the biological medium can defocus the ultrasound beam. Here we reduce the problem of refocusing the beam to the inverse problem of estimating the acoustic aberration due to the biological tissue from acoustic radiative force imaging data. We solve this inverse problem using a Bayesian framework with a hierarchical prior and solve the inverse problem using a Metropolis-within-Gibbs algorithm. The framework is tested using both synthetic and experimental datasets. We demonstrate that our approach has the ability to estimate the aberrations using small datasets, as little as 32 sonication tests, which can lead to significant speedup in the treatment process. Furthermore, our approach is compatible with a wide range of sonication tests and can be applied to other energy-based measurement techniques.

physics.med-ph↗

arXiv2017

A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.

math.AP↗

arXiv2017

Asymptotic and numerical analysis of a porous medium model for transpiration-driven sap flow in trees

We develop a 3D porous medium model for sap flow within a tree stem, which consists of a nonlinear parabolic partial differential equation with a suitable transpiration source term. Using an asymptotic analysis, we derive approximate series solutions for the liquid saturation and sap velocity for a general class of coefficient functions. Several important non-dimensional parameters are identified that can be used to characterize various flow regimes. We investigate the relative importance of stem aspect ratio versus anisotropy in the sapwood hydraulic conductivity, and how these two effects impact the radial and vertical components of sap velocity. The analytical results are validated by means of a second-order finite volume discretization of the governing equations, and comparisons are drawn to experimental results on Norway spruce trees.

physics.flu-dyn↗

arXiv2016

Airborne contaminant source estimation using a finite-volume forward solver coupled with a Bayesian inversion approach

We propose a numerical algorithm for solving the atmospheric dispersion problem with elevated point sources and ground-level deposition. The problem is modelled by the 3D advection-diffusion equation with delta-distribution source terms, as well as height-dependent advection speed and diffusion coefficients. We construct a finite volume scheme using a splitting approach in which the Clawpack software package is used as the advection solver and an implicit time discretization is proposed for the diffusion terms. The algorithm is then applied to an actual industrial scenario involving emissions of airborne particulates from a zinc smelter using actual wind measurements. We also address various practical considerations such as choosing appropriate methods for regularizing noisy wind data and quantifying sensitivity of the model to parameter uncertainty. Afterwards, we use the algorithm within a Bayesian framework for estimating emission rates of zinc from multiple sources over the industrial site. We compare our finite volume solver with a Gaussian plume solver within the Bayesian framework and demonstrate that the finite volume solver results in tighter uncertainty bounds on the estimated emission rates.

math.NA↗

arXiv2016

Bayesian estimation of airborne fugitive emissions using a Gaussian plume model

A new method is proposed for estimating the rate of fugitive emissions of particulate matter from multiple time-dependent sources via measurements of deposition and concentration. We cast this source inversion problem within the Bayesian framework, and use a forward model based on a Gaussian plume solution. We present three alternate models for constructing the prior distribution on the emission rates as functions of time. Next, we present an industrial case study in which our framework is applied to estimate the rate of fugitive emissions of lead particulates from a smelter in Trail, British Columbia, Canada. The Bayesian framework not only provides an approximate solution to the inverse problem, but also quantifies the uncertainty in the solution. Using this information we perform an uncertainty propagation study in order to assess the impact of the estimated sources on the area surrounding the industrial site.

stat.AP↗

arXiv2015

Multi-layer asymptotic solution for wetting fronts in porous media with exponential moisture diffusivity

We study the asymptotic behaviour of sharp front solutions arising from the nonlinear diffusion equation θ_t = (D(θ)θ_x)_x, where the diffusivity is an exponential function D(θ) = D_o exp(βθ). This problem arises for example in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is D(0) = D_o << 1 so that the diffusion problem is nearly degenerate. Such problems are characterised by wetting fronts that sharply delineate regions of saturated and unsaturated flow, and that propagate with a well-defined speed. Using matched asymptotic expansions in the limit of large β, we derive an analytical description of the solution that is uniformly valid throughout the wetting front. This is in contrast with most other related analyses that instead truncate the solution at some specific wetting front location, which is then calculated as part of the solution, and beyond that location the solution is undefined. Our asymptotic analysis demonstrates that the solution has a four-layer structure, and by matching through the adjacent layers we obtain an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original nonlinear diffusion equation, we demonstrate that the first few terms in our series solution provide approximations of physical quantities such as wetting front location and speed of propagation that are more accurate (over a wide range of admissible β values) than other asymptotic approximations reported in the literature.

math.AP↗

arXiv2015

Multiscale model of a freeze-thaw process for tree sap exudation

Sap transport in trees has long fascinated scientists, and a vast literature exists on experimental and modelling studies of trees during the growing season when large negative stem pressures are generated by transpiration from leaves. Much less attention has been paid to winter months when trees are largely dormant but nonetheless continue to exhibit interesting flow behaviour. A prime example is sap exudation, which refers to the peculiar ability of sugar maple (Acer saccharum) and related species to generate positive stem pressure while in a leafless state. Experiments demonstrate that ambient temperatures must oscillate about the freezing point before significantly heightened stem pressures are observed, but the precise causes of exudation remain unresolved. The prevailing hypothesis attributes exudation to a physical process combining freeze-thaw and osmosis, which has some support from experimental studies but remains a subject of active debate. We address this knowledge gap by developing the first mathematical model for exudation, while also introducing several essential modifications to this hypothesis. We derive a multiscale model consisting of a nonlinear system of differential equations governing phase change and transport within wood cells, coupled to a suitably homogenized equation for temperature on the macroscale. Numerical simulations yield stem pressures that are consistent with experiments and provide convincing evidence that a purely physical mechanism is capable of capturing exudation.

physics.bio-ph↗

arXiv2015

Mathematics For Industry: A Personal Perspective

"I am an industrial mathematician." When asked to identify my profession or academic field of study, this is the most concise answer I can provide. However, this seemingly straightforward statement is commonly greeted by a blank stare or an uncomfortable silence, regardless of whether I am speaking to a fellow mathematician or a non-mathematician. I usually follow up with the clarification: "I am an applied mathematician who derives much of my inspiration from the study of industrial problems that I encounter through collaborations with companies." This dispels some confusion, but unfortunately still leaves a great deal open to interpretation owing to the vagueness of the words mathematics, industry and company, each of which covers an extremely broad range of scientific or socio-economic activity. To those academics who actually work in the field of industrial mathematics (and whose "perspective" referred to in the title is the focus of this article) this ambiguity is familiar and untroubling. However, for anyone less acquainted with the work of industrial mathematicians, some clarification is desirable especially for anyone who might be considering entering the field. This essay therefore aims to shed light upon the nature of research being done at the interface between mathematics and industry, paying particular attention to the following questions: What is industrial mathematics? Where is industrial mathematics? How does one do industrial mathematics? Why (or more precisely, what value is there in doing) industrial mathematics? I will attempt to answer these questions by means of several case studies drawn from my own experience in tackling mathematical problems from industry.

math.HO↗

arXiv2015

On regularizations of the delta distribution

In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the convergence of a sequence of distributions $\mathcal{S}_H$ to a singular term $\mathcal{S}$ as a parameter $H$ (associated with the {support size} of $\mathcal{S}_H$) shrinks to zero. We characterize this convergence in both the weak-$\ast$ topology of distributions, as well as in a weighted Sobolev norm. These notions motivate a framework for constructing regularizations of the delta distribution that includes a large class of existing methods in the literature. This framework allows different regularizations to be compared. The convergence of solutions of PDEs with these regularized source terms is then studied in various topologies such as pointwise convergence on a deleted neighborhood and weighted Sobolev norms. We also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems.

math.NA↗

arXiv2015

An immersed boundary model of the cochlea with parametric forcing

The cochlea or inner ear has a remarkable ability to amplify sound signals. This is understood to derive at least in part from some active process that magnifies vibrations of the basilar membrane (BM) and the cochlear partition in which it is embedded, to the extent that it overcomes the effect of viscous damping from the surrounding cochlear fluid. Many authors have associated this amplification ability to some type of mechanical resonance within the cochlea, however there is still no consensus regarding the precise cause of amplification. Our work is inspired by experiments showing that the outer hair cells within the cochlear partition change their lengths when stimulated, which can in turn cause periodic distortions of the BM and other structures in the cochlea. This paper investigates a novel fluid-mechanical resonance mechanism that derives from hydrodynamic interactions between an oscillating BM and the surrounding cochlear fluid. We present a model of the cochlea based on the immersed boundary method, in which a small-amplitude periodic internal forcing due to outer hair cells can induce parametric resonance. A Floquet stability analysis of the linearized equations demonstrates the existence of resonant (unstable) solutions within the range of physical parameters corresponding to the human auditory system. Numerical simulations of the immersed boundary equations support the analytical results and clearly demonstrate the existence of resonant solution modes. These results are then used to illustrate the influence of parametric resonance on wave propagation along the BM and explicit comparisons are drawn with results from another two-dimensional cochlea model.

physics.bio-ph↗

arXiv2015

Simulating biofilm deformation and detachment with the immersed boundary method

We apply the immersed boundary (or IB) method to simulate deformation and detachment of a periodic array of wall-bounded biofilm colonies in response to a linear shear flow. The biofilm material is represented as a network of Hookean springs that are placed along the edges of a triangulation of the biofilm region. The interfacial shear stress, lift and drag forces acting on the biofilm colony are computed by using fluid stress jump method developed by Williams, Fauci and Gaver [Disc. Contin. Dyn. Sys. B 11(2):519-540, 2009], with a modified version of their exclusion filter. Our detachment criterion is based on the novel concept of an averaged equivalent continuum stress tensor defined at each IB point in the biofilm which is then used to determine a corresponding von Mises yield stress; wherever this yield stress exceeds a given critical threshold the connections to that node are severed, thereby signalling the onset of a detachment event. In order to capture the deformation and detachment behaviour of a biofilm colony at different stages of growth, we consider a family of four biofilm shapes with varying aspect ratio. Our numerical simulations focus on the behaviour of weak biofilms (with relatively low yield stress threshold) and investigate features of the fluid-structure interaction such as locations of maximum shear and increased drag. The most important conclusion of this work is that the commonly employed detachment strategy in biofilm models based only on interfacial shear stress can lead to incorrect or inaccurate results when applied to the study of shear induced detachment of weak biofilms. Our detachment strategy based on equivalent continuum stresses provides a unified and consistent IB framework that handles both sloughing and erosion modes of biofilm detachment, and is consistent with strategies employed in many other continuum based biofilm models.

physics.flu-dyn↗

arXiv2014

Homogenization of the Stefan problem, with application to maple sap exudation

The technique of periodic homogenization with two-scale convergence is applied to the analysis of a two-phase Stefan-type problem that arises in the study of a periodic array of melting ice bars. For this "reduced model" we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, which requires solving a macroscale problem for the global temperature field and a reference cell problem at each point in space which captures the underlying phase change process occurring on the microscale. We state a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. The same homogenized temperature equations are then applied to solve a much more complicated problem involving multi-phase flow and heat transport in trees, where the sap is present in both frozen and liquid forms and a third gas phase is also present. Our homogenization approach has the advantage that the global temperature field is a solution of the same reduced model equations, while all the remaining physics are relegated to the reference cell problem. Numerical simulations are performed to validate our results and draw conclusions regarding the phenomenon known as sap exudation, which is of great importance in sugar maple trees and few other related species.

math.AP↗

arXiv2014

Correction to "Parametric Resonance in Immersed Elastic Boundaries"

This note is a correction to a paper of Cortez, Peskin, Stockie & Varela [SIAM J. Appl. Math., 65(2):494-520, 2004], who studied the stability of a parametrically-forced, circular, elastic fiber immersed in an incompressible fluid in 2D, and showed the existence of parametric resonance. The results were represented as plots that separate parameter space into regions where the solution is either stable or unstable. We uncovered two errors in the paper: the first was in the derivation of the eigenvalue problem, and the second was in the code to used to calculate the stability contours.

math.AP↗

arXiv2014

A three-phase free boundary problem with melting ice and dissolving gas

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves the interactions between gas, water and ice. The dynamics are driven by melting of the ice layer, while the pressurized gas also dissolves within the meltwater. The model incorporates a Stefan condition at the water-ice interface along with Henry's law for dissolution of gas at the gas-water interface. We employ a quasi-steady approximation for the phase temperatures and then derive a series solution for the interface positions. A non-standard feature of the model is an integral free boundary condition that arises from mass conservation owing to changes in gas density at the gas-water interface, which makes the problem non-self-adjoint. We derive a two-scale asymptotic series solution for the dissolved gas concentration, which because of the non-self-adjointness gives rise to a Fourier series expansion in eigenfunctions that do not satisfy the usual orthogonality conditions. Numerical simulations of the original governing equations are used to validate the series approximations.

math-ph↗