-
Accelerating frequency-domain numerical methods for weakly nonlinear focused ultrasound using nested meshes
Authors:
Samuel P. Groth,
Pierre Gélat,
Seyyed R. Haqshenas,
Nader Saffari,
Elwin van 't Wout,
Timo Betcke,
Garth N. Wells
Abstract:
The numerical simulation of weakly nonlinear ultrasound is important in treatment planning for focused ultrasound (FUS) therapies. However, the large domain sizes and generation of higher harmonics at the focus make these problems extremely computationally demanding. Numerical methods typically employ a uniform mesh fine enough to resolve the highest harmonic present in the problem, leading to a v…
▽ More
The numerical simulation of weakly nonlinear ultrasound is important in treatment planning for focused ultrasound (FUS) therapies. However, the large domain sizes and generation of higher harmonics at the focus make these problems extremely computationally demanding. Numerical methods typically employ a uniform mesh fine enough to resolve the highest harmonic present in the problem, leading to a very large number of degrees of freedom. This paper proposes a more efficient strategy in which each harmonic is approximated on a separate mesh, the size of which is proportional to the wavelength of the harmonic. The increase in resolution required to resolve a smaller wavelength is balanced by a reduction in the domain size. This nested meshing is feasible owing to the increasingly localised nature of higher harmonics near the focus.
Numerical experiments are performed for FUS transducers in homogeneous media in order to determine the size of the meshes required to accurately represent the harmonics. In particular, a fast \emph{volume potential} approach is proposed and employed to perform convergence experiments as the computation domain size is modified. This approach allows each harmonic to be computed via the evaluation of an integral over the domain. Discretising this integral using the midpoint rule allows the computations to be performed rapidly with the FFT. It is shown that at least an order of magnitude reduction in memory consumption and computation time can be achieved with nested meshing. Finally, it is demonstrated how to generalise this approach to inhomogeneous propagation domains.
△ Less
Submitted 22 July, 2021; v1 submitted 5 November, 2020;
originally announced November 2020.
-
A hybridizable discontinuous Galerkin method for the Navier--Stokes equations with pointwise divergence-free velocity field
Authors:
Sander Rhebergen,
Garth N. Wells
Abstract:
We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We show that with modifications of the function spaces in the method of Labeur and Wells it is possible t…
▽ More
We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.
△ Less
Submitted 31 January, 2018; v1 submitted 25 April, 2017;
originally announced April 2017.
-
Torsion of a cylinder of partially molten rock with a spherical inclusion: theory and simulation
Authors:
Laura Alisic,
Sander Rhebergen,
John F. Rudge,
Richard F. Katz,
Garth N. Wells
Abstract:
The processes that are involved in migration and extraction of melt from the mantle are not yet fully understood. Gaining a better understanding of material properties of partially molten rock could help shed light on the behavior of melt on larger scales in the mantle. In this study, we simulate three-dimensional torsional deformation of a partially molten rock that contains a rigid, spherical in…
▽ More
The processes that are involved in migration and extraction of melt from the mantle are not yet fully understood. Gaining a better understanding of material properties of partially molten rock could help shed light on the behavior of melt on larger scales in the mantle. In this study, we simulate three-dimensional torsional deformation of a partially molten rock that contains a rigid, spherical inclusion. We compare the computed porosity patterns to those found in recent laboratory experiments. The laboratory experiments show emergence of melt-rich bands throughout the rock sample, and pressure shadows around the inclusion. The numerical model displays similar melt-rich bands only for a small bulk-to-shear-viscosity ratio (five or less). The results are consistent with earlier two-dimensional numerical simulations; however, we show that it is easier to form melt-rich bands in three dimensions compared to two. The addition of strain-rate dependence of the viscosity causes a distinct change in the shape of pressure shadows around the inclusion. This change in shape presents an opportunity for experimentalists to identify the strain-rate dependence and therefore the dominant deformation mechanism in torsion experiments with inclusions.
△ Less
Submitted 16 October, 2015; v1 submitted 13 August, 2015;
originally announced August 2015.
-
CO$_2$ dissolution in a background hydrological flow
Authors:
H. Juliette T. Unwin,
Garth N. Wells,
Andrew W. Woods
Abstract:
We investigate the long time steady-state dissolution of CO$_{2}$ in a deep saline aqquifer in the presence of a background hydrological flow. In steady-state, the distribution of CO$_2$ in the groundwater upstream of the aquifer involves a balance between three competing effects: (i) the buoyancy-driven flow of CO$_2$ saturated water; (ii) the diffusion of CO$_2$ from saturated to under-saturated…
▽ More
We investigate the long time steady-state dissolution of CO$_{2}$ in a deep saline aqquifer in the presence of a background hydrological flow. In steady-state, the distribution of CO$_2$ in the groundwater upstream of the aquifer involves a balance between three competing effects: (i) the buoyancy-driven flow of CO$_2$ saturated water; (ii) the diffusion of CO$_2$ from saturated to under-saturated water; and (iii) the advection associated with the oncoming background flow. This leads to three limiting regimes. In the limit of very slow diffusion, a nearly static intrusion of dense fluid may extend a finite distance upstream, balanced by the pressure gradient associated with the oncoming background flow. In the limit of fast diffusion relative to the flow, a gradient zone may become established in which the along aquifer diffusive flux balances the advection associated with the background flow. However, if the buoyancy-driven flow speed exceeds the background hydrological flow speed, then a third, intermediate regime may become established. In this regime, a convective recirculation develops upstream of the anticline involving the vertical diffusion of CO$_2$ from an upstream propagating flow of dense CO$_2$ saturated water into the downstream propagating flow of CO$_2$ unsaturated water. For each limiting case, we find analytical solutions for the distribution of CO$_2$ upstream of the anticline, and test our analysis with full numerical simulations. A key result is that, although there may be very different controls on the distribution and extent of CO$_2$ bearing water upstream of the anticline, in each case the dissolution rate is given by the product of the background volume flux and the difference in concentration between the CO$_2$ saturated water and the original aquifer water upstream.
△ Less
Submitted 16 October, 2015; v1 submitted 14 June, 2015;
originally announced June 2015.
-
Three-field block-preconditioners for models of coupled magma/mantle dynamics
Authors:
Sander Rhebergen,
Garth N. Wells,
Andrew J. Wathen,
Richard F. Katz
Abstract:
For a prescribed porosity, the coupled magma/mantle flow equations can be formulated as a two-field system of equations with velocity and pressure as unknowns. Previous work has shown that while optimal preconditioners for the two-field formulation can be obtained, the construction of preconditioners that are uniform with respect to model parameters is difficult. This limits the applicability of t…
▽ More
For a prescribed porosity, the coupled magma/mantle flow equations can be formulated as a two-field system of equations with velocity and pressure as unknowns. Previous work has shown that while optimal preconditioners for the two-field formulation can be obtained, the construction of preconditioners that are uniform with respect to model parameters is difficult. This limits the applicability of two-field preconditioners in certain regimes of practical interest. We address this issue by reformulating the governing equations as a three-field problem, which removes a term that was problematic in the two-field formulation in favour of an additional equation for a pressure-like field. For the three-field problem, we develop and analyse new preconditioners and we show numerically that they are optimal in terms of problem size and less sensitive to model parameters, compared to the two-field preconditioner. This extends the applicability of optimal preconditioners for coupled mantle/magma dynamics into parameter regimes of physical interest.
△ Less
Submitted 4 June, 2015; v1 submitted 19 November, 2014;
originally announced November 2014.
-
Analysis of block-preconditioners for models of coupled magma/mantle dynamics
Authors:
Sander Rhebergen,
Garth N. Wells,
Richard F. Katz,
Andrew J. Wathen
Abstract:
This article considers the iterative solution of a finite element discretisation of the magma dynamics equations. In simplified form, the magma dynamics equations share some features of the Stokes equations. We therefore formulate, analyse and numerically test a Elman, Silvester and Wathen-type block preconditioner for magma dynamics. We prove analytically and demonstrate numerically the optimalit…
▽ More
This article considers the iterative solution of a finite element discretisation of the magma dynamics equations. In simplified form, the magma dynamics equations share some features of the Stokes equations. We therefore formulate, analyse and numerically test a Elman, Silvester and Wathen-type block preconditioner for magma dynamics. We prove analytically and demonstrate numerically the optimality of the preconditioner. The presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers. The analysis is verified through a range of two- and three-dimensional numerical examples on unstructured grids, from simple illustrative problems through to large problems on subduction zone-like geometries. The computer code to reproduce all numerical examples is freely available as supporting material.
△ Less
Submitted 29 May, 2014; v1 submitted 25 November, 2013;
originally announced November 2013.
-
Measurement requirements for a near-Earth asteroid impact mitigation demonstration mission
Authors:
Stephen D. Wolters,
Andrew J. Ball,
Nigel Wells,
Christopher Saunders,
Neil McBride
Abstract:
A concept for an Impact Mitigation Preparation Mission, called Don Quijote, is to send two spacecraft to a Near-Earth Asteroid (NEA): an Orbiter and an Impactor. The Impactor collides with the asteroid while the Orbiter measures the resulting change in the asteroid's orbit, by means of a Radio Science Experiment (RSE) carried out before and after impact. Three parallel Phase A studies on Don Quijo…
▽ More
A concept for an Impact Mitigation Preparation Mission, called Don Quijote, is to send two spacecraft to a Near-Earth Asteroid (NEA): an Orbiter and an Impactor. The Impactor collides with the asteroid while the Orbiter measures the resulting change in the asteroid's orbit, by means of a Radio Science Experiment (RSE) carried out before and after impact. Three parallel Phase A studies on Don Quijote were carried out for the European Space Agency: the research presented here reflects outcomes of the study by QinetiQ. We discuss the mission objectives with regards to the prioritisation of payload instruments, with emphasis on the interpretation of the impact. The Radio Science Experiment is described and it is examined how solar radiation pressure may increase the uncertainty in measuring the orbit of the target asteroid. It is determined that to measure the change in orbit accurately a thermal IR spectrometer is mandatory, to measure the Yarkovsky effect. The advantages of having a laser altimeter are discussed. The advantages of a dedicated wide-angle impact camera are discussed and the field-of-view is initially sized through a simple model of the impact.
△ Less
Submitted 21 July, 2011;
originally announced July 2011.
-
Structure and interactions of ultracold Yb ions and Rb atoms
Authors:
H. D. L. Lamb,
J. F. McCann,
B. M. McLaughlin,
J. Goold,
N. Wells,
I. Lane
Abstract:
In order to study ultracold charge-transfer processes in hybrid atom-ion traps, we have mapped out the potential energy curves and molecular parameters for several low lying states of the Rb, Yb$^+$ system. We employ both a multi-reference configuration interaction (MRCI) and a full configuration interaction (FCI) approach. Turning points, crossing points, potential minima and spectroscopic molecu…
▽ More
In order to study ultracold charge-transfer processes in hybrid atom-ion traps, we have mapped out the potential energy curves and molecular parameters for several low lying states of the Rb, Yb$^+$ system. We employ both a multi-reference configuration interaction (MRCI) and a full configuration interaction (FCI) approach. Turning points, crossing points, potential minima and spectroscopic molecular constants are obtained for the lowest five molecular states. Long-range parameters, including the dispersion coefficients are estimated from our {\it ab initio} data. The separated-atom ionization potentials and atomic polarizability of the ytterbium atom ($α_d=128.4$ atomic units) are in good agreement with experiment and previous calculations. We present some dynamical calculations for (adiabatic) scattering lengths for the two lowest (Yb,Rb$^+$) channels that were carried out in our work. However, we find that the pseudo potential approximation is rather limited in validity, and only applies to nK temperatures. The adiabatic scattering lengths for both the triplet and singlet channels indicate that both are large and negative in the FCI approximation.
△ Less
Submitted 23 July, 2012; v1 submitted 6 July, 2011;
originally announced July 2011.
-
A FEniCS-Based Programming Framework for Modeling Turbulent Flow by the Reynolds-Averaged Navier-Stokes Equations
Authors:
Mikael Mortensen,
Hans Petter Langtangen,
Garth N. Wells
Abstract:
Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by fi…
▽ More
Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble closely the mathematical formulation of equations and algorithms. The presented ideas and programming techniques are also applicable to other fields that involve systems of nonlinear partial differential equations. We demonstrate the framework in two applications and investigate the impact of various linearizations on the convergence properties of nonlinear solvers for a Reynolds-averaged Navier-Stokes model.
△ Less
Submitted 31 March, 2011; v1 submitted 14 February, 2011;
originally announced February 2011.
-
A discontinuous Galerkin method for strain gradient-dependent damage: Study of interpolations, convergence and two dimensional problems
Authors:
L. Molari,
G. N. Wells,
K. Garikipati,
F. Ubertini
Abstract:
A discontinuous Galerkin method has been developed for strain gradient-dependent damage. The strength of this method lies in the fact that it allows the use of $C^0$ interpolation functions for continuum theories involving higher-order derivatives, while in a conventional framework at least $C^1$ interpolations are required. The discontinuous Galerkin formulation thereby offers significant poten…
▽ More
A discontinuous Galerkin method has been developed for strain gradient-dependent damage. The strength of this method lies in the fact that it allows the use of $C^0$ interpolation functions for continuum theories involving higher-order derivatives, while in a conventional framework at least $C^1$ interpolations are required. The discontinuous Galerkin formulation thereby offers significant potential for engineering computations with strain gradient-dependent models. When using basis functions with a low degree of continuity, jump conditions arise at element edges which are incorporated in the weak form. In addition to the formulation itself, a detailed study of the convergence properties of the method for various element types is presented, an error analysis is undertaken, and the method is also shown to work in two dimensions.
△ Less
Submitted 2 November, 2004;
originally announced November 2004.