Assessing the robustness and scalability of the accelerated pseudo-transient method towards exascale computing

The development of highly efficient, robust and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. We address this challenge with the accelerated pseudo-transient iterative method and present here a physically motivated derivation. We analytically determine optimal iteration parameters for a variety of basic physical processes and confirm the validity of theoretical predictions with numerical experiments. We provide an efficient numerical implementation of pseudo-transient solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency over 96 % on 2197 GPUs in distributed memory parallelisation weak scaling benchmarks. 2197 GPUs allow for unprecedented terascale solutions of 3D variable viscosity Stokes flow on 49953 grid cells involving over 1.2 trillion degrees of freedom. We verify the robustness of the method by handling contrasts up to 9 orders of magnitude in material parameters such as viscosity, and arbitrary distribution of viscous inclusions for different flow configurations. Moreover, we show that this method is well suited to tackle strongly nonlinear problems such as shear-banding in a visco-elasto-plastic medium. A GPU-based implementation can outperform CPU-based direct-iterative solvers in terms of wall-time even at relatively low resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy has thus a great potential for future high-performance computing applications, and for paving the road to exascale in the geosciences and beyond.

Investigation of Shear Flow in a Planar Cylindrical Hybrid Geometry for Rheo-NMR Applications

<p>Materials which exhibit peculiar behaviour due to applied mechanical deformations are abundant in everyday life. Rheo-NMR is an established technique which has been used to study these responses for the past three decades by combining methodologies from rheometry and nuclear magnetic resonance (NMR). The technique enhances standard rheological studies of bulk properties, such as viscosity and elasticity, by applying the tools of NMR (e.g. spectroscopy, diffusion, relaxometry, imaging, and velocimetry) to matter under deformation. This allows for the exploration of molecular origins and / or local responses within the material which lead to the macroscopic behaviour. These materials are deformed (most commonly sheared) inside geometric housings with a NMR experiment running in parallel. For complex material studies it is desirable for these geometries to provide a simple homogeneous deformation. In reality, all standard rheometry geometries have inhomogeneity characteristics. In fact there is evidence to suggest that some material responses may be influenced by a small degree of deviation from pure homogeneity. This makes it harder to isolate any inherent material behaviour due to a magnitude or rate of deformation from the specific characteristics of how the deformation was applied. This contribution reports on the continued design and method development of a novel geometry for rheo-NMR - a planar cylindrical hybrid (PCH) shear geometry. The geometry includes planar sections with the aim to provide planar Couette flow, a simple truly homogeneous shear profile. It comprises of two parallel sections of planar flow connected by two semi-circular sections of circular flow to give a closed flow path in the shape of a racetrack. Shear is applied by rotating a band around the inner section like a conveyor belt. The purpose of the PCH geometry is to study the complex responses of materials under shear in this atypical shear environment. A paragon of a model system for exploring the novel geometry is a shear banding wormlike micelle (WLM) solution. It has a well documented nonlinear response to steady shear and previous work demonstrated that the curvature of a standard concentric cylinder geometric housing influenced the observed WLM’s rheological response. Strikingly, what was discovered by this thesis research was that there was no visible appearance of this material separating into bands in the planar (or cylindrical) regions in the PCH geometry when probed with an NMR velocity encoded imaging experiment. The more Newtonian-like response of the complex material differs from the intriguing curved flow profile seen for an actual Newtonian sample (which additionally evolves over the planar region) meaning the WLM’s response is still complex in nature. From these findings it is clear that geometry did not impart the homogeneous planar Couette flow for a Newtonian sample. However it has introduced a new deformation environment to study complex materials, acting completely differently to the geometries typically used in rheo-NMR and rheometry. Implications of this and motivation for work study are discussed.</p>

Investigation of Shear Flow in a Planar Cylindrical Hybrid Geometry for Rheo-NMR Applications

<p>Materials which exhibit peculiar behaviour due to applied mechanical deformations are abundant in everyday life. Rheo-NMR is an established technique which has been used to study these responses for the past three decades by combining methodologies from rheometry and nuclear magnetic resonance (NMR). The technique enhances standard rheological studies of bulk properties, such as viscosity and elasticity, by applying the tools of NMR (e.g. spectroscopy, diffusion, relaxometry, imaging, and velocimetry) to matter under deformation. This allows for the exploration of molecular origins and / or local responses within the material which lead to the macroscopic behaviour. These materials are deformed (most commonly sheared) inside geometric housings with a NMR experiment running in parallel. For complex material studies it is desirable for these geometries to provide a simple homogeneous deformation. In reality, all standard rheometry geometries have inhomogeneity characteristics. In fact there is evidence to suggest that some material responses may be influenced by a small degree of deviation from pure homogeneity. This makes it harder to isolate any inherent material behaviour due to a magnitude or rate of deformation from the specific characteristics of how the deformation was applied. This contribution reports on the continued design and method development of a novel geometry for rheo-NMR - a planar cylindrical hybrid (PCH) shear geometry. The geometry includes planar sections with the aim to provide planar Couette flow, a simple truly homogeneous shear profile. It comprises of two parallel sections of planar flow connected by two semi-circular sections of circular flow to give a closed flow path in the shape of a racetrack. Shear is applied by rotating a band around the inner section like a conveyor belt. The purpose of the PCH geometry is to study the complex responses of materials under shear in this atypical shear environment. A paragon of a model system for exploring the novel geometry is a shear banding wormlike micelle (WLM) solution. It has a well documented nonlinear response to steady shear and previous work demonstrated that the curvature of a standard concentric cylinder geometric housing influenced the observed WLM’s rheological response. Strikingly, what was discovered by this thesis research was that there was no visible appearance of this material separating into bands in the planar (or cylindrical) regions in the PCH geometry when probed with an NMR velocity encoded imaging experiment. The more Newtonian-like response of the complex material differs from the intriguing curved flow profile seen for an actual Newtonian sample (which additionally evolves over the planar region) meaning the WLM’s response is still complex in nature. From these findings it is clear that geometry did not impart the homogeneous planar Couette flow for a Newtonian sample. However it has introduced a new deformation environment to study complex materials, acting completely differently to the geometries typically used in rheo-NMR and rheometry. Implications of this and motivation for work study are discussed.</p>

RARE Velocimetry of Shear Banded Flow in Cylindrical Couette Geometry

<p>A flow phenomena called ‘shear banding’ is often observed for a certain class of complex fluids, namely wormlike micellar solutions. Wormlike micelles are elongated flexible self-assembly structures formed by the aggregation of amphiphiles, which may entangle into a dynamic network above a certain concentration threshold. The entanglement results in the sample having both solid-like (elastic) and liquid-like (viscous) properties, an ambiguity commonly found in complex fluids. Under certain shear conditions, the flow couples with the structure of the micellar network, leading to the formation of (shear) bands with differing viscosity. The principle goal of this work is to address open questions regarding the temporal and spatial stability of shear banded flow. Shear banding is often studied in cylindrical Couette cells, where the fluid is sheared in a gap between differentially rotating concentric cylinders. For the sake of an accurate description of the flow in such a shear cell, the methodology for a 2D Nuclear Magnetic Resonance (NMR) velocimetry technique (known as PGSE-RARE), which offers high temporal and spatial resolution, is improved and refined. Two main challenges are identified and overcome. The first concerns the fact that the velocity imaging process operates on a Cartesian grid, whereas the flow in the Couette cell is of cylindrical symmetry. Numerical calculations and NMR simulations based on the Bloch equations, as well as experimental evidence, give insight on the appropriate selection of the fluid volume over which velocity information is accumulated and the preferred scheme through which the NMR image is acquired in the so-called k-space. The small extent of the fluid gap for the cells in use is the second challenge. In this respect, a variant of the velocimetry technique is developed, which offers ultra high resolution in the gap direction, necessary for a detailed description of the flow profile in the banded state. The refined methodology is applied in a thorough study of a certain wormlike micellar solution (‘10% CPCl’), which is known to exhibit spatiotemporal fluctuations and has been subject of numerous studies over the past 20 years. NMR results are supported by a recently developed 2D Rheo-USV (Ultrasonic Speckle Velocimetry) method, which offers an even higher temporal resolution. The two complementary methods show good agreement for averaged velocity profiles. In line with previous studies the fluid is found to follow a standard anomalous lever rule, which is characterized by a constant shear rate in the high viscosity band and a varying shear rate and proportion of the high shear rate band. In particular, the high resolution NMR variant allows a refined picture on the dynamics of the interface between the two bands. Furthermore, slip is observed for all investigated shear rates. The amount of slip, however, is found to strongly depend on the specifities of the Couette cells in use. Spatially and temporally resolved flow maps reveal various flow instabilities. Ultrasound measurements show vorticity structures in the order of the gap width. In the NMR case no such structures are observed due to the lower resolution in the axial direction. For higher shear rates the occurrence of turbulent bursts is detected for USV. No direct evidence of similar flow instabilities is found in the NMR case. Finally, broad distributions dominate the high shear rate band in temporally and spatially resolved velocity profiles, showing the fluctuative nature of the flow.</p>

New Methods for Studying Materials Under Shear with Nuclear Magnetic Resonance

<p>For over 30 years, nuclear magnetic resonance (NMR) techniques have been used to study materials under shear. Collectively referred to as Rheo-NMR, these methods measure material behaviour due to external stimuli and provide spatially and temporally resolved maps of NMR spectra, intrinsic NMR parameters (e.g. relaxation times) or motion (e.g. diffusion or flow). As a consequence, Rheo-NMR has been established as a complementary technique to conventional rheological measurements. In this thesis, new hardware and experimental methods are presented with the goal of advancing this exciting field through further integration of traditional rheometry techniques with NMR experiments. Three key areas of hardware development have been addressed, including: 1) integrating torque sensing into the Rheo-NMR experiment for simultaneous bulk shear stress measurements, 2) constructing shear devices with geometric parameters closer to those used on commercial rheometers and 3) implementing an advanced drive system which allows for new shear profiles including oscillatory shear. In addition to presenting the design and construction of various prototype instruments, results from validation and proof of concept studies are discussed. This information demonstrates that the hardware operates as expected and establishes an experimental parameter space for these new techniques. Furthermore, these methods have been applied to open questions in various physical systems. This includes exploring the influence of shear geometry curvature on the onset of shear banding in a wormlike micelle surfactant system, observing shear induced structural changes in a lyotropic nonionic surfactant simultaneously via deuterium spectroscopy and bulk viscosity as well as studying interactions of flowing granular materials. The interpretation and implication of these observations are discussed in addition to motivating further studies.</p>

RARE Velocimetry of Shear Banded Flow in Cylindrical Couette Geometry

<p>A flow phenomena called ‘shear banding’ is often observed for a certain class of complex fluids, namely wormlike micellar solutions. Wormlike micelles are elongated flexible self-assembly structures formed by the aggregation of amphiphiles, which may entangle into a dynamic network above a certain concentration threshold. The entanglement results in the sample having both solid-like (elastic) and liquid-like (viscous) properties, an ambiguity commonly found in complex fluids. Under certain shear conditions, the flow couples with the structure of the micellar network, leading to the formation of (shear) bands with differing viscosity. The principle goal of this work is to address open questions regarding the temporal and spatial stability of shear banded flow. Shear banding is often studied in cylindrical Couette cells, where the fluid is sheared in a gap between differentially rotating concentric cylinders. For the sake of an accurate description of the flow in such a shear cell, the methodology for a 2D Nuclear Magnetic Resonance (NMR) velocimetry technique (known as PGSE-RARE), which offers high temporal and spatial resolution, is improved and refined. Two main challenges are identified and overcome. The first concerns the fact that the velocity imaging process operates on a Cartesian grid, whereas the flow in the Couette cell is of cylindrical symmetry. Numerical calculations and NMR simulations based on the Bloch equations, as well as experimental evidence, give insight on the appropriate selection of the fluid volume over which velocity information is accumulated and the preferred scheme through which the NMR image is acquired in the so-called k-space. The small extent of the fluid gap for the cells in use is the second challenge. In this respect, a variant of the velocimetry technique is developed, which offers ultra high resolution in the gap direction, necessary for a detailed description of the flow profile in the banded state. The refined methodology is applied in a thorough study of a certain wormlike micellar solution (‘10% CPCl’), which is known to exhibit spatiotemporal fluctuations and has been subject of numerous studies over the past 20 years. NMR results are supported by a recently developed 2D Rheo-USV (Ultrasonic Speckle Velocimetry) method, which offers an even higher temporal resolution. The two complementary methods show good agreement for averaged velocity profiles. In line with previous studies the fluid is found to follow a standard anomalous lever rule, which is characterized by a constant shear rate in the high viscosity band and a varying shear rate and proportion of the high shear rate band. In particular, the high resolution NMR variant allows a refined picture on the dynamics of the interface between the two bands. Furthermore, slip is observed for all investigated shear rates. The amount of slip, however, is found to strongly depend on the specifities of the Couette cells in use. Spatially and temporally resolved flow maps reveal various flow instabilities. Ultrasound measurements show vorticity structures in the order of the gap width. In the NMR case no such structures are observed due to the lower resolution in the axial direction. For higher shear rates the occurrence of turbulent bursts is detected for USV. No direct evidence of similar flow instabilities is found in the NMR case. Finally, broad distributions dominate the high shear rate band in temporally and spatially resolved velocity profiles, showing the fluctuative nature of the flow.</p>

New Methods for Studying Materials Under Shear with Nuclear Magnetic Resonance

<p>For over 30 years, nuclear magnetic resonance (NMR) techniques have been used to study materials under shear. Collectively referred to as Rheo-NMR, these methods measure material behaviour due to external stimuli and provide spatially and temporally resolved maps of NMR spectra, intrinsic NMR parameters (e.g. relaxation times) or motion (e.g. diffusion or flow). As a consequence, Rheo-NMR has been established as a complementary technique to conventional rheological measurements. In this thesis, new hardware and experimental methods are presented with the goal of advancing this exciting field through further integration of traditional rheometry techniques with NMR experiments. Three key areas of hardware development have been addressed, including: 1) integrating torque sensing into the Rheo-NMR experiment for simultaneous bulk shear stress measurements, 2) constructing shear devices with geometric parameters closer to those used on commercial rheometers and 3) implementing an advanced drive system which allows for new shear profiles including oscillatory shear. In addition to presenting the design and construction of various prototype instruments, results from validation and proof of concept studies are discussed. This information demonstrates that the hardware operates as expected and establishes an experimental parameter space for these new techniques. Furthermore, these methods have been applied to open questions in various physical systems. This includes exploring the influence of shear geometry curvature on the onset of shear banding in a wormlike micelle surfactant system, observing shear induced structural changes in a lyotropic nonionic surfactant simultaneously via deuterium spectroscopy and bulk viscosity as well as studying interactions of flowing granular materials. The interpretation and implication of these observations are discussed in addition to motivating further studies.</p>