arXiv subjects

P. E. Farrell

Publications and source records attributed to P. E. Farrell.

17 recordsLinked to original sources

arXiv2026

Kinetic derivation of thermal viscous models for nematic liquid crystal dynamics

We develop a macroscopic thermodynamic theory of nematic liquid crystals starting from a kinetic theory of ordered fluids with a collision operator of Bhatnagar-Gross-Krook (BGK) type. The kinetic description incorporates mean-field alignment interactions through a Vlasov potential and relies on a separation of time scales, with orientational relaxation occurring on a faster time scale than translational momentum relaxation. At the continuum level, we establish the balance equations for mass, linear and angular momentum, energy, and entropy. Using the zeroth and first order Chapman-Enskog expansions, we derive a constitutive equation for the Helmholtz free energy and identify the associated structural form of the entropy production rate. We then exploit additional information from the kinetic description to determine a constitutive relation for the entropy production rate itself. Finally, by applying the constrained maximisation procedure of Rajagopal and Srinivasa, we obtain constitutive equations for the Cauchy stress and couple-stress tensors, as well as for the energy and entropy fluxes. In this way we generalise the recent inviscid kinetic theory of Farrell, Russo, and Zerbinati to account for viscous, thermal, and spin-diffusive effects, using the simplest BGK-type approximation of the collision operator. Both compressible and incompressible variants of the theory are presented.

math-ph↗

arXiv2025

Multiple solutions to the static forward free-boundary Grad-Shafranov problem on MAST-U

The Grad-Shafranov (GS) equation is a nonlinear elliptic partial differential equation that governs the ideal magnetohydrodynamic equilibrium of a tokamak plasma. Previous studies have demonstrated the existence of multiple solutions to the GS equation when solved in idealistic geometries with simplified plasma current density profiles and boundary conditions. Until now, the question of whether multiple equilibria might exist in real-world tokamak geometries with more complex current density profiles and integral free-boundary conditions (commonly used in production-level equilibrium codes) has remained unanswered. In this work, we discover multiple solutions to the static forward free-boundary GS problem in the MAST-U tokamak geometry using the validated evolutive equilibrium solver FreeGSNKE and the deflated continuation algorithm. By varying the plasma current, current density profile coefficients, or coil currents in the GS equation, we identify and characterise distinct equilibrium solutions, including both deeply and more shallowly confined plasma states. We suggest that the existence of even more equilibria is likely prohibited by the restrictive nature of the integral free-boundary condition, which globally couples poloidal fluxes on the computational boundary with those on the interior. We conclude by discussing the implications of these findings for wider equilibrium modelling and emphasise the need to explore whether multiple solutions are present in other equilibrium codes and tokamaks, as well as their potential impact on downstream simulations that rely on GS equilibria.

physics.plasm-ph↗

arXiv2023

Discrete Breathers in Klein-Gordon Lattices: a Deflation-Based Approach

Deflation is an efficient numerical technique for identifying new branches of steady state solutions to nonlinear partial differential equations. Here, we demonstrate how to extend deflation to discover new periodic orbits in nonlinear dynamical lattices. We employ our extension to identify discrete breathers, which are generic exponentially localized, time-periodic solutions of such lattices. We compare different approaches to using deflation for periodic orbits, including ones based on a Fourier decomposition of the solution, as well as ones based on the solution's energy density profile. We demonstrate the ability of the method to obtain a wide variety of multibreather solutions without prior knowledge about their spatial profile.

nlin.PS↗

arXiv2023

Two-Component 3D Atomic Bose-Einstein Condensates Support Complex Stable Patterns

We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multi-component nonlinear wave systems of nonlinear Schr{ö}dinger type. While our computations relate to two-component atomic Bose-Einstein condensates in parabolic traps, our methods can be broadly applied to high-dimensional, nonlinear systems of partial differential equations. The combination of the so-called deflation technique with a careful selection of initial guesses enables the computation of an unprecedented breadth of patterns, including ones combining vortex lines, rings, stars, and ``vortex labyrinths''. Despite their complexity, they may be dynamically robust and amenable to experimental observation, as confirmed by Bogolyubov-de Gennes spectral analysis and numerical evolution simulations.

nlin.PS↗

arXiv2021

Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling

When solving partial differential equations with random fields as coefficients the efficient sampling of random field realisations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the SPDE approach of Lindgren et al.~combined with a new fast algorithm for white noise sampling which is taylored to (ML)QMC. We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

math.NA↗

arXiv2021

Phase-field modeling of multivariant martensitic transformation at finite-strain: computational aspects and large-scale finite-element simulations

Large-scale 3D martensitic microstructure evolution problems are studied using a finite-element discretization of a finite-strain phase-field model. The model admits an arbitrary crystallography of transformation and arbitrary elastic anisotropy of the phases, and incorporates Hencky-type elasticity, a penalty-regularized double-obstacle potential, and viscous dissipation. The finite-element discretization of the model is performed in Firedrake and relies on the PETSc solver library. The large systems of linear equations arising are efficiently solved using GMRES and a geometric multigrid preconditioner with a carefully chosen relaxation. The modeling capabilities are illustrated through a 3D simulation of the microstructure evolution in a pseudoelastic CuAlNi single crystal during nano-indentation, with all six orthorhombic martensite variants taken into account. Robustness and a good parallel scaling performance have been demonstrated, with the problem size reaching 150 million degrees of freedom.

physics.comp-ph↗

arXiv2020

Deflation-based Identification of Nonlinear Excitations of the 3D Gross--Pitaevskii equation

We present previously unknown solutions to the 3D Gross--Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures exhibit striking topological features, involving combinations of vortex rings and multiple, possibly bent vortex lines. Although unstable, many of them persist for long times in dynamical simulations. These solutions were identified by a state-of-the-art numerical technique called deflation, which is expected to be applicable to many problems from other areas of physics.

nlin.PS↗

arXiv2020

An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow

We propose an augmented Lagrangian preconditioner for a three-field stress-velocity-pressure discretization of stationary non-Newtonian incompressible flow with an implicit constitutive relation of power-law type. The discretization employed makes use of the divergence-free Scott-Vogelius pair for the velocity and pressure. The preconditioner builds on the work [P. E. Farrell, L. Mitchell, and F. Wechsung, SIAM J. Sci. Comput., 41 (2019), pp. A3073-A3096], where a Reynolds-robust preconditioner for the three-dimensional Newtonian system was introduced. The preconditioner employs a specialized multigrid method for the stress-velocity block that involves a divergence-capturing space decomposition and a custom prolongation operator. The solver exhibits excellent robustness with respect to the parameters arising in the constitutive relation, allowing for the simulation of a wide range of materials.

math.NA↗

arXiv2020

Monolithic Multigrid for Magnetohydrodynamics

The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems.

math.NA↗

arXiv2019

Bifurcation analysis of stationary solutions of two-dimensional coupled Gross-Pitaevskii equations using deflated continuation

Recently, a novel bifurcation technique known as the deflated continuation method (DCM) was applied to the single-component nonlinear Schrödinger (NLS) equation with a parabolic trap in two spatial dimensions. The bifurcation analysis carried out by a subset of the present authors shed light on the configuration space of solutions of this fundamental problem in the physics of ultracold atoms. In the present work, we take this a step further by applying the DCM to two coupled NLS equations in order to elucidate the considerably more complex landscape of solutions of this system. Upon identifying branches of solutions, we construct the relevant bifurcation diagrams and perform spectral stability analysis to identify parametric regimes of stability and instability and to understand the mechanisms by which these branches emerge. The method reveals a remarkable wealth of solutions: these do not only include some of the well-known ones including, e.g., from the Cartesian or polar small amplitude limits of the underlying linear problem but also a significant number of branches that arise through (typically pitchfork) bifurcations. In addition to presenting a ``cartography'' of the landscape of solutions, we comment on the challenging task of identifying {\it all} solutions of such a high-dimensional, nonlinear problem.

nlin.PS↗

arXiv2019

Complexity bounds on supermesh construction for quasi-uniform meshes

Projecting fields between different meshes commonly arises in computational physics. This operation requires a supermesh construction and its computational cost is proportional to the number of cells of the supermesh $n$. Given any two quasi-uniform meshes of $n_A$ and $n_B$ cells respectively, we show under standard assumptions that n is proportional to $n_A + n_B$. This result substantially improves on the best currently available upper bound on $n$ and is fundamental for the analysis of algorithms that use supermeshes.

math.NA↗

arXiv2017

Higher-order moving mesh methods for PDE-constrained shape optimization

We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of PDE-constraints. This shape optimization method is based on discretized deformation diffeomorphisms and allows for arbitrarily high resolution of shapes with arbitrary smoothness. Numerical experiments show that it allows the solution of PDE-constrained shape optimization problems to high accuracy.

math.NA↗

arXiv2016

Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation

In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schr\"odinger equation with a parabolic trap and repulsive interactions. Despite the fact that this system has been studied extensively, we discover a wide variety of previously unknown branches of solutions. We analyze the stability of the newly discovered branches and discuss the bifurcations that relate them to known solutions both in the near linear (Cartesian, as well as polar) and in the highly nonlinear regimes. While deflated continuation is not guaranteed to compute the full bifurcation diagram, this analysis is a potent demonstration that the algorithm can discover new nonlinear states and provide insights into the energy landscape of complex high-dimensional Hamiltonian dynamical systems.

nlin.PS↗

arXiv2016

Analysis of Carrier's problem

A computational and asymptotic analysis of the solutions of Carrier's problem is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to construct asymptotic solutions to the problem. This asymptotic approach explains the bifurcation structure identified numerically, and its predictions of the bifurcation points are in excellent agreement with the numerical results. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag is incorrect.

math.CA↗

arXiv2016

A Deflation Technique for Detecting Multiple Liquid Crystal Equilibrium States

Multiple equilibrium states arise in many physical systems, including various types of liquid crystal structures. Having the ability to reliably compute such states enables more accurate physical analysis and understanding of experimental behavior. This paper adapts and extends a deflation technique for the computation of multiple distinct solutions arising in the context of modeling equilibrium configurations of nematic and cholesteric liquid crystals. The deflation method is applied as part of an overall free-energy variational approach and is modified to fit the framework of optimization of a functional with pointwise constraints. It is shown that multigrid methods designed for the undeflated systems may be applied to efficiently solve the linear systems arising in the application of deflation. For the numerical algorithm, the deflation approach is interwoven with nested iteration, creating a dynamic and efficient method that further enables the discovery of distinct solutions. Finally, four numerical experiments are performed demonstrating the efficacy and accuracy of the algorithm in detecting important physical phenomena, including bifurcation and disclination behaviors. The final numerical experiment expands the algorithm to model cholesteric liquid crystals and illustrates the full discovery power of the deflation process.

math.NA↗

arXiv2013

Tidal turbine array optimisation using the adjoint approach

Oceanic tides have the potential to yield a vast amount of renewable energy. Tidal stream generators are one of the key technologies for extracting and harnessing this potential. In order to extract an economically useful amount of power, hundreds of tidal turbines must typically be deployed in an array. This naturally leads to the question of how these turbines should be configured to extract the maximum possible power: the positioning and the individual tuning of the turbines could significantly influence the extracted power, and hence is of major economic interest. However, manual optimisation is difficult due to legal site constraints, nonlinear interactions of the turbine wakes, and the cubic dependence of the power on the flow speed. The novel contribution of this paper is the formulation of this problem as an optimisation problem constrained by a physical model, which is then solved using an efficient gradient-based optimisation algorithm. In each optimisation iteration, a two-dimensional finite element shallow water model predicts the flow and the performance of the current array configuration. The gradient of the power extracted with respect to the turbine positions and their tuning parameters is then computed in a fraction of the time taken for a flow solution by solving the associated adjoint equations. These equations propagate causality backwards through the computation, from the power extracted back to the turbine positions and the tuning parameters. This yields the gradient at a cost almost independent of the number of turbines, which is crucial for any practical application. The utility of the approach is demonstrated by optimising turbine arrays in four idealised scenarios and a more realistic case with up to 256 turbines in the Inner Sound of the Pentland Firth, Scotland.

math.OC↗

arXiv2013

A framework for automated PDE-constrained optimisation

A generic framework for the solution of PDE-constrained optimisation problems based on the FEniCS system is presented. Its main features are an intuitive mathematical interface, a high degree of automation, and an efficient implementation of the generated adjoint model. The framework is based upon the extension of a domain-specific language for variational problems to cleanly express complex optimisation problems in a compact, high-level syntax. For example, optimisation problems constrained by the time-dependent Navier-Stokes equations can be written in tens of lines of code. Based on this high-level representation, the framework derives the associated adjoint equations in the same domain-specific language, and uses the FEniCS code generation technology to emit parallel optimised low-level C++ code for the solution of the forward and adjoint systems. The functional and gradient information so computed is then passed to the optimisation algorithm to update the parameter values. This approach works both for steady-state as well as transient, and for linear as well as nonlinear governing PDEs and a wide range of functionals and control parameters. We demonstrate the applicability and efficiency of this approach on classical textbook optimisation problems and advanced examples.

cs.MS↗