About

Noel J. Walkington is a Professor in the Department of Mathematical Sciences at Carnegie Mellon University, located in Wean Hall, Pittsburgh. He holds degrees in mechanical engineering and mathematics, including an M.S. and Ph.D. in Mechanical Engineering from the University of Missouri-Rolla, and a Ph.D. in Mathematics from the University of Texas at Austin. His research involves the development and analysis of numerical algorithms required for the approximation of partial differential equations, with a particular interest in working at the interface of mechanical engineering and mathematics.

Research topics

• Computer Science
• Mathematics
• Mechanics
• Artificial Intelligence
• Physics
• Algorithm
• Mathematical analysis
• Mechanical engineering
• Geotechnical engineering
• Geology
• Classical mechanics
• Engineering
• Materials science
• Composite material
• Geometry
• Thermodynamics

Selected publications

• A Kinetic Phase-Field Model of Diffusion Bonding: A Nonlocal Approach to Interface Coalescence

  Journal of Applied Mechanics · 2026-02-18

  articleOpen access

  Abstract Conventional phase-field models often drive solid-solid interfaces to coalesce when in close proximity. This feature limits their use for processes like diffusion bonding, where the interfaces might need to remain distinct under certain thermodynamic conditions. We develop a kinetic phase-field model to address this problem, using an evolution equation based on a geometric conservation law for interfaces, rather than the gradient descent evolution that is typical in phase-field modeling. This formulation enables us to specify complex kinetic laws, and we use this to incorporate a physically motivated geometric criterion to control interface merging. This criterion, based on nonlocal higher-derivative curvature invariants of the phase field, can be temperature-dependent, allows for a range of behaviors from complete coalescence to the preservation of distinct boundaries. Simulations show controlled bonding kinetics, demonstrating capabilities that are not available with existing methods for modeling interfaces that must remain distinct under given thermodynamic conditions.

  PublisherOA PDFDOI

• Stress Asymmetry in Hard Magnetic Soft Materials

  Journal of Applied Mechanics · 2026-03-30

  articleOpen access

  Abstract Hard magnetic soft materials – soft polymers embedded with hard magnetic particles – are modeled using continuum magnetomechanical formulations in which the deformation and the magnetization field are the primary kinematic variables. A recent question in such formulations is whether the Cauchy stress is symmetric, which is directly related to frame invariance and angular momentum balance. This note discusses energetically equivalent formulations, related by a change of variables between referential and current descriptions of the magnetization, and shows that they generally yield different Cauchy stresses, including a change in their symmetry. Specifically, the formulation based on a referential magnetization produces a symmetric Cauchy stress, while that based on a current magnetization generally yields an asymmetric Cauchy stress. We highlight that when the internal variable (magnetization field) is at the energy-minimizing equilibrium configuration, the divergences of these stresses are the same, and both stresses are symmetric.

  PublisherOA PDFDOI

• Aeroacoustic signatures reveal fast transient dynamics of vapor-jet-driven cavity oscillations in metallic additive manufacturing

  ArXiv.org · 2026-02-28

  articleOpen access

  Aeroacoustic emissions from intense evaporation are widely measured yet often treated as noisy byproducts and used mainly in empirical monitoring. Here, we show that airborne sound encodes physics-governed sub-millisecond fingerprints of vapor-jet dynamics in excessive vaporization, exemplified by vapor keyholes in laser metal processing. From first principles, we develop a vapor-jet-cavity oscillation framework and incorporate it into an aeroacoustic formulation, thereby coupling measured sound to transient cavity depth and oscillation frequency. Reconciled with synchronized multimodal in-situ data, airborne acoustics enable accurate tracking of vapor-cavity properties within tens to hundreds of microseconds. Combined with newly discovered correlations, cavity-jet-acoustic theory recasts the transition from steady, pore-free to pore-shedding vaporizations as a critical-frequency event. Aeroacoustic emissions thus become scalable, physics-guided, and cost-efficient probes of rapidly evolving liquid-vapor systems.

  PublisherOA PDF

• Stress Asymmetry in Hard Magnetic Soft Materials

  arXiv (Cornell University) · 2026-03-31

  preprintOpen access

  Hard magnetic soft materials -- soft polymers embedded with hard magnetic particles -- are modeled using continuum magnetomechanical formulations in which the deformation and the magnetization field are the primary kinematic variables. A recent question in such formulations is whether the Cauchy stress is symmetric, which is directly related to frame invariance and angular momentum balance. This note discusses energetically equivalent formulations, related by a change of variables between referential and current descriptions of the magnetization, and shows that they generally yield different Cauchy stresses, including a change in their symmetry. Specifically, the formulation based on a referential magnetization produces a symmetric Cauchy stress, while that based on a current magnetization generally yields an asymmetric Cauchy stress. We highlight that when the internal variable (magnetization field) is at the energy-minimizing equilibrium configuration, the divergences of these stresses are the same, and both stresses are symmetric.

  PublisherDOI

• Aeroacoustic signatures reveal fast transient dynamics of vapor-jet-driven cavity oscillations in metallic additive manufacturing

  Open MIND · 2026-02-28

  preprint

  Aeroacoustic emissions from intense evaporation are widely measured yet often treated as noisy byproducts and used mainly in empirical monitoring. Here, we show that airborne sound encodes physics-governed sub-millisecond fingerprints of vapor-jet dynamics in excessive vaporization, exemplified by vapor keyholes in laser metal processing. From first principles, we develop a vapor-jet-cavity oscillation framework and incorporate it into an aeroacoustic formulation, thereby coupling measured sound to transient cavity depth and oscillation frequency. Reconciled with synchronized multimodal in-situ data, airborne acoustics enable accurate tracking of vapor-cavity properties within tens to hundreds of microseconds. Combined with newly discovered correlations, cavity-jet-acoustic theory recasts the transition from steady, pore-free to pore-shedding vaporizations as a critical-frequency event. Aeroacoustic emissions thus become scalable, physics-guided, and cost-efficient probes of rapidly evolving liquid-vapor systems.

  DOI

• Impact of Gas/Liquid Phase Change of CO2 during Injection for Sequestration

  2025-06-25

  preprintOpen access

  CO 2 sequestration in deep saline formations is an effective and important process to control the rapid rise in CO 2 emissions.The process of injecting CO 2 requires reliable predictions of the stress in the formation and the fluid pressure distributions -particularly since monitoring of the CO 2 migration is difficult -to mitigate leakage, prevent induced seismicity, and analyze wellbore stability.A key aspect of CO 2 is the gas-liquid phase transition at the temperatures and pressures of relevance to leakage and sequestration, which has been recognized as being critical for accurate predictions but has been challenging to model without ad hoc empiricisms.This paper presents a robust multiphase thermodynamics-based poromechanics model to capture the complex phase transition behavior of CO 2 and predict the stress and pressure distribution under super-and sub-critical conditions during the injection process.A finite element implementation of the model is applied to analyze the behavior of a multiphase porous system with CO 2 as it displaces the fluid brine phase.We find that if CO 2 undergoes a phase transition in the geologic reservoir, the spatial variation of the density is significantly affected, and the migration mobility of CO 2 decreases in the reservoir.A key feature of our approach is that we do not a priori assume the location of the CO 2 gas/liquid interface -or even if it occurs at all -but rather, this is a prediction of the model, along with the spatial variation of the phase of CO 2 and the change of the saturation profile due to the phase change.

  PublisherOA PDFDOI

• Accretion and ablation in deformable solids using an Eulerian formulation: A finite deformation numerical method

  Journal of the Mechanics and Physics of Solids · 2025-02-21 · 3 citations

  articleOpen access

  Surface growth, i.e., the addition or removal of mass from the boundary of a solid body, occurs in a wide range of processes, including the growth of biological tissues, solidification and melting, and additive manufacturing. To understand nonlinear phenomena such as failure and morphological instabilities in these systems, accurate numerical models are required to study the interaction between mass addition and stress in complex geometrical and physical settings. Despite recent progress in the formulation of models of surface growth of deformable solids, current numerical approaches require several simplifying assumptions. This work formulates a method that couples an Eulerian surface growth description to a phase-field approach. It further develops a finite element implementation to solve the model numerically using a fixed computational domain with a fixed discretization. This approach bypasses the challenges that arise in a Lagrangian approach, such as having to construct a four-dimensional reference configuration, remeshing, and/or changing the computational domain over the course of the numerical solution. It also enables the modeling of several settings – such as non-normal growth of biological tissues and stress-induced growth – which can be challenging for available methods. The numerical approach is demonstrated on a model problem that shows non-normal growth, wherein growth occurs by the motion of the surface in a direction that is not parallel to the normal of the surface, that can occur in hard biological tissues such as nails, horns, etc. Next, a thermomechanical model is formulated and used to investigate the kinetics of freezing and melting in ice under complex stress states, particularly to capture regelation which is a key process in frost heave and basal slip in glaciers.

  PublisherDOI

• Crack Face Contact Modeling is Essential to Predict Crack-Parallel Stresses

  Journal of Applied Mechanics · 2025-04-22 · 1 citations

  articleOpen access

  Abstract Phase-field fracture models provide a powerful approach to modeling fracture, potentially enabling the unguided prediction of the crack growth in complex patterns. To ensure that only tensile stresses and not compressive stresses drive crack growth, several models have been proposed that aim to distinguish between the compressive and tensile loads. However, these models have a critical shortcoming: they do not account for the crack direction, and hence they cannot distinguish between crack-normal tensile stresses that drive crack growth and crack-parallel stresses that do not. In this study, we apply a phase-field fracture model, developed in our earlier work, that uses the crack direction to distinguish crack-parallel stresses from crack-normal stresses. This provides a transparent energetic formulation that drives cracks to grow in when crack faces open or slide past each other, while the cracks respond like the intact solid when the crack faces contact under normal compressive loads. We compare our approach against two widely used approaches, Spectral splitting and the volumetric-deviatoric splitting, and find that these predict unphysical crack growth and unphysical stress concentrations under loading conditions in which these should not occur. Specifically, we show that the splitting models predict spurious crack growth and stress concentration under pure crack-parallel normal stresses. However, our formulation, which resolves the crack-parallel stresses from the crack-normal stresses, predicts these correctly.

  PublisherOA PDFDOI

• Stability and Convergence of HDG Schemes under Minimal Regularity

  SIAM Journal on Numerical Analysis · 2025-09-25

  article
  PublisherDOI

• Phase‐Field Modeling of Fracture Under Compression and Confinement in Anisotropic Geomaterials

  International Journal for Numerical and Analytical Methods in Geomechanics · 2025-10-22

  article

  ABSTRACT Strongly anisotropic geomaterials, such as layered shales, have been observed to undergo fracture under compressive loading. This paper applies a phase‐field fracture model to study this fracture process. While phase‐field fracture models have several advantages—primarily that the fracture path is not pre‐determined but arises naturally from the evolution of a smooth non‐singular damage field—they provide unphysical predictions when the stress state is complex and includes compression that can cause crack faces to contact. Building on a recently‐developed phase‐field model that accounts for compressive traction across the crack face, this paper extends the model to the setting of anisotropic fracture. The key features of the model include the following: (1) a homogenized anisotropic elastic response and strongly‐anisotropic model for the work to fracture; (2) an effective damage response that accounts consistently for compressive traction across the crack face, that is, derived from the anisotropic elastic response; (3) a regularized crack normal field that overcomes the shortcomings of the isotropic setting, and enables the correct crack response, both across and transverse to the crack face. To test the model, we first compare the predictions to phase‐field fracture evolution calculations in a fully‐resolved layered specimen with spatial inhomogeneity, and show that it captures the overall patterns of crack growth. We then apply the model to previously‐reported experimental observations of fracture evolution in laboratory specimens of shales under compression with confinement, and find that it predicts well the observed crack patterns in a broad range of loading conditions. We further apply the model to predict the growth of wing cracks under compression and confinement. Prior approaches to simulate wing cracks have treated the initial cracks as an external boundary, which makes them difficult to apply to general settings. Here, the effective crack response model enables us to treat the initial crack simply as a non‐singular damaged zone within the computational domain, thereby allowing for easy and general computations.

  PublisherDOI

Recent grants

• Development and Analysis of Algorithms to Simulate Multi-Component Multi-Phase Porous Flows

  NSF · $400k · 2020–2024

• DMREF: Collaborative Research: Materials Engineering of Columnar and Living Liquid Crystals via Experimental Characterization, Mathematical Modeling, and Simulation

  NSF · $233k · 2017–2020

• Analysis of Algorithms for Simulating Complex Materials

  NSF · $401k · 2008–2011

• Analysis of Algorithms for Simulating Macroscopic Material Response

  NSF · $330k · 2014–2017

• Analysis of Algorithms for Continuum Models of Complex Materials

  NSF · $300k · 2011–2015

Frequent coauthors

• Gary L. Miller

  18 shared

• Oleg D. Lavrentovich

  University of Warsaw

  9 shared

• Dmitry Golovaty

  University of Akron

  7 shared

• Walter Eversman

  Missouri University of Science and Technology

  7 shared

• O. M. Tovkach

  Georgetown University

  6 shared

• Alexander Rand

  Siemens (United States)

  5 shared

• R. E. Showalter

  Oregon State University

  5 shared

• M. Carme Calderer

  University of Minnesota

  5 shared

Education

• Ph.D., Mechanical Engineering

  University of Missouri-Rolla

• Ph.D., Mathematics

  University of Texas at Austin

• M.S., Mechanical Engineering

  University of Missouri-Rolla