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At least 253 records · Page 14

A unified hp-HDG framework for Friedrichs' PDE systems

Here, this work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

97 MATHEMATICS AND COMPUTING↗

Modeling of switching regulator power stages with and without zero-inductor-current dwell time

State space techniques are employed to derive accurate models for buck, boost, and buck/boost converter power stages operating with and without zero-inductor-current dwell time. A generalized procedure is developed which treats the continuous-inductor-current mode without the dwell time as a special case of the discontinuous-current mode, when the dwell time vanishes. An abrupt change of system behavior including a reduction of the system order when the dwell time appears is shown both analytically and experimentally.

Lee, F. C.↗

Luna 24 - Opaque mineral chemistry of gabbroic and basaltic fragments from Mare Crisium

Spinels and ilmenites are relatively sparse in the Luna 24 gabbro and basalts. Spinel compositions show some affinities to those of spinels in Apollo 12, Apollo 14 and Luna 16 basalts; a characteristic feature is high Al2O3, reaching a maximum of 19.8 wt%. A comparison of spinels in the Luna 24 gabbro with those in other deep-seated lunar intrusive rocks shows a characteristic trend for Fe/Mg. This trend is systematic from gabbro to anorthosite to troctolite and is interpreted to be P-T dependent. Compositions of spinels in the gabbro fall within the Cr/Al trend defined by the spinels of the basalts, but form a Fe/Mg trend parallel to that of the basalts; this relationship suggests that both the gabbro and the basalts are derived from a closely similar source region, with the basalts originating at a slightly greater depth than the gabbro. The spinels in both rock types are considered to have formed at high crustal levels, at low pressures. The Luna 24 data suggest that the compositional discontinuities which exist between chromian spinels and titanian spinels in a large proportion of mare basalts are the result of nucleation of chromian spinels at high crustal levels prior to eruption, and of titanian spinels during melt crystallization at the lunar surface.

Haggerty, S. E.↗

Chemostatic concentration–discharge behaviour observed in a headwater catchment underlain with discontinuous permafrost

Concentration–discharge dynamics were evaluated in a small (~ 2.25 km2) headwater catchment underlain with discontinuous permafrost on the Seward Peninsula of western Alaska. A large storm, during which 48 mm of rain fell over a 24-h period, enabled the evaluation of solute concentration–discharge response to a sizeable hydrological event, while water stable isotopes enabled an appraisal of the contributions of event water. Under normal catchment conditions, chemostatic behaviour was observed for solutes typically derived from mineral weathering (e.g. calcium, magnesium, sodium and silica). The chemostatic behaviour observed for most solutes under normal catchment conditions indicated that catchment storage and residence times are sufficiently long for many solute generating reactions to approach equilibrium. Following the storm however, most solutes exhibited dilutive and highly variable behaviour. This likely indicated the exceedance of a discharge threshold where chemostatic behaviour could no longer be maintained for most solutes. Dissolved organic carbon and silica were the only solutes monitored to exhibit chemostatic behaviour during all time periods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Quaternary slip rates and most recent surface rupture of the Bitterroot fault, western Montana

The Bitterroot fault is a ~100-km-long Quaternary active normal fault that bounds the eastern margin of the north–south-trending Bitterroot Mountains and accommodates extension near the Intermountain Seismic Belt. Here, new detailed mapping using high-resolution topographic data derived from light detection and ranging (lidar) along the southern Bitterroot Range documents multiple generations of fault scarps in Holocene–Pleistocene deposits with vertical off sets that increase in magnitude with age. Fault mapping indicates a complex fault geometry characterized by an en echelon pattern of discontinuous segments of 41–78° east-dipping normal faults that appear to cut the older Eocene detachment fault, and locally 70–88° west-dipping antithetic normal faults. 10 Be cosmogenic exposure dating provides in situ age control for 32 surface boulders (>1 m) sampled in glacial deposits. Near Como Dam, two Pinedale-age glacial moraine sequences yield peak age distributions of 15.0 ± 0.4 ka and 16.4 ± 0.6 ka as apparent exposure ages (e = 0), and 15.4 ± 0.4 ka and 16.8 ± 0.6 ka based on the maximum allowed boulder surface erosion rate (e = 2 mm/ka). Vertical separation of 3.5 ± 0.2 m across the ~16–17 ka glacial moraine off set by the Bitterroot fault scarp yields a fault slip rate of 0.2–0.3 mm/yr. Glacial Lake Missoula highstand shorelines, inset into the ~15 ka glacial moraine and vertically off set 4.6 ± 1.6 m by an antithetic strand of the Bitterroot fault, yield fault slip rates of 0.2–0.5 mm/yr that overlap with fault slip rates on the main strand near Lake Como (0.2–0.3 mm/yr). At the Ward Creek fan located ~15 km to the north of Lake Como, two glacial debris fan sequences yield peak age distributions of 16.6 ± 0.4 ka and 62.8 ± 1.7 ka (e = 0), and 17.0 ± 0.4 ka and 69.9 ± 2.2 ka (e = 2 mm/ka). Vertical separations of 2.4 ± 0.3 m and 4.5 ± 0.2 m on the ~17 ka and ~63–70 ka fan surfaces off set by the Bitterroot fault yield fault slip rates of 0.2–0.3 mm/yr and 0.1 mm/yr, respectively. Our results indicate broadly consistent fault slip rates for the main fault segments at Lake Como (0.2–0.3 mm/yr) and the Ward Creek fan (0.1–0.3 mm/yr) with an along-strike range of 0.1–0.3 mm/yr for the southern Bitterroot fault. Fault scaling relations and evidence of multiple late Quaternary fault surface ruptures suggest the Bitterroot fault could produce a Mw ~7.2 earthquake. Structural model constraints and our slip rate results indicate both high-angle or low-angle fault geometries are possible at depth. A seismogenic low-angle fault model could generate a larger earthquake of Mw >7.2. Earthquake history is unknown for the Bitterroot fault, but fault scarps in young glacial deposits demonstrate its seismogenic potential. Data from this study suggest seismic hazards from the Bitterroot fault may pose a high level of risk to the Missoula metropolitan area, the State’s second most populous region, and major infrastructures across the Missoula and Bitterroot Valleys.

58 GEOSCIENCES↗

An extended classical solution of the droplet growth problem

Problems of applying the classical kinetic theory to the growth of small droplets from vapor are examined. A solution for the droplet growth equation is derived which is based on the assumption of a diffusive field extending to the drop surface. The method accounts for partial thermal and mass accommodation at the interface and the kinetic limit to the mass and heat fluxes, and it avoids introducing the artifact of a discontinuity in the thermal and vapor field near the droplet. Consideration of the environmental fields in spherical geometry utilizing directional fluxes yields boundary values in terms of known parameters and a new Laplace transform integral.

Anderson, B. J.↗

The Effects of a 1998 Observing System Change on MERRA-2-Based Ozone Profile Simulations

Model simulations of ozone (O3) driven by meteorological reanalyses are useful for filling observational gaps and interpreting observed O3 variability and trends. However, the transport circulation of reanalysis products is impacted by changes to the observing system (the data assimilated into the reanalyses). We examine the impacts of these changes on simulated O3 from two models, Global Modeling Initiative (GMI) Chemistry Transport Model (GMI CTM) and Modern‐Era Retrospective Analysis for Research Applications version 2 (MERRA‐2) GMI Replay (M2 GMI Replay) simulation, using observations from global ozonesondes (>50,000 profiles) and satellites from 1980 to 2016. Both models are constrained by meteorology from the NASA MERRA‐2 reanalysis, and both use versions of NASA's GMI chemical mechanism. We focus on an observing system change affecting simulated O3 after 1998, associated with the assimilation of temperature and humidity data from new microwave profiling satellites. A large post‐1998 O3 increase, mainly confined to 15–20‐km altitude, of ~10 Dobson units (DU) in midlatitudes occurs in the GMI CTM, worsening the bias compared to observations. In contrast, an increase in M2 GMI Replay simulation O3 of ~10 DU is observed only near −60° latitude, reducing the bias compared to observations. The GMI CTM O3 high biases display a Quasi‐Biennial Oscillation (QBO)‐like periodicity that result from excessive transport from the tropical stratosphere to the midlatitude lower stratosphere during the QBO westerly phase. We quantify O3 discontinuities caused by MERRA‐2 observing system changes and demonstrate how the MERRA‐2 Global Modeling Initiative Replay simulation dampens the effects of these changes and QBO‐driven artifacts on simulated lower stratospheric and total O3. We caution against using simulations driven by a reanalysis to derive multidecadal O3 trends, especially prior to 1998.

Replay↗

DORIS Satellite Phase Center Determination and Consequences on the Derived Scale of the Terrestrial Reference Frame

This slide presentation reports on the analysis of several years of Delft Object-oriented Radar Interferometric Software (Doris) data undertaken to estimate daily determination of satellite antenna phase center corrections. This was done on a satellite by satellite basis. For each DORIS satellite, we considered long-term time series of such individual estimations, looking for possible biases, discontinuities, trends or annual signals. The analysis compared DORIS to GPS estimates for common satellites (Jason and TOPEX/Poseidon).

terrestrial reference frame scale↗

Flux splitting algorithms for two-dimensional viscous flows with finite-rate chemistry

The Roe flux-difference splitting method has been extended to treat two-dimensional viscous flows with nonequilibrium chemistry. The derivations have avoided unnecessary assumptions or approximations. For spatial discretization, the second-order Roe upwind differencing is used for the convective terms and central differencing for the viscous terms. An upwind-based TVD scheme is applied to eliminate oscillations and obtain a sharp representation of discontinuities. A two-stage Runge-Kutta method is used to time integrate the discretized Navier-Stokes and species transport equations for the asymptotic steady solutions. The present method is then applied to two types of flows: the shock wave/boundary layer interaction problems and the jet in cross flows.

Shuen, Jian-Shun↗

Flux splitting algorithms for two-dimensional viscous flows with finite-rate chemistry

The Roe flux difference splitting method was extended to treat 2-D viscous flows with nonequilibrium chemistry. The derivations have avoided unnecessary assumptions or approximations. For spatial discretization, the second-order Roe upwind differencing is used for the convective terms and central differencing for the viscous terms. An upwind-based TVD scheme is applied to eliminate oscillations and obtain a sharp representation of discontinuities. A two-state Runge-Kutta method is used to time integrate the discretized Navier-Stokes and species transport equations for the asymptotic steady solutions. The present method is then applied to two types of flows: the shock wave/boundary layer interaction problems and the jet in cross flows.

Shuen, Jian-Shun↗

The Stress-strain Behavior of Polymer-Nanotube Composites from Molecular Dynamics Simulations

Stress-strain curves of polymer-carbon nanotube composites are derived from molecular dynamics simulations of a single-walled carbon nanotube embedded in polyethylene. A comparison is made between the response to mechanical loading of a composite with a long, continuous nanotube (replicated via periodic boundary conditions) and the response of a composite with a short, discontinuous nanotube. Both composites are mechanically loaded in the direction of and transverse to the NT axis. The long-nanotube composite shows an increase in the stiffness relative to the polymer and behaves anisotropically under the different loading conditions. The short-nanotube composite shows no enhancement relative to the polymer, most probably because of its low aspect ratio. The stress-strain curves are compared with rule-of-mixtures predictions.

Frankland, S. J. V.↗

Capillary surface discontinuities above reentrant corners

A particular configuration of a vertical capillary tube for which S is the equilibrium interface between two fluids in the presence of a downward pointing gravitational field was investigated. S is the graph a function u whose domain is the (horizontal) cross section gamma of the tube. The mean curvature of S is proportional to its height above a fixed reference plane and lambda is a prescribed constant and may be taken between zero and pi/2. Domains gamma for which us is a bounded function but does not extend continuously to d gamma are sought. Simple domains are found and the behavior of u in those domains is studied. An important comparison principle that has been used in the literature to derive many of the results in capillarity is reviewed. It allows one to deduce the approximate shape of a capillary surface by constructing comparison surfaces with mean curvature and contact angle close to those of the (unknown) solution surface. In the context of nonparametric problems the comparison principle leads to height estimates above and below for the function u. An example from the literature where these height estimates have been used successfully is described. The promised domains for which the bounded u does not extend continuously to the boundary are constructed. The point on the boundary at which u has a jump discontinuity will be the vertext of a re-entrant corner having any interior angle theta pi. Using the comparison principle the behavior of u near this point is studied.

Korevaar, H. J.↗

Modeling of switching regulator power stages with and without zero-inductor-current dwell time

State-space techniques are employed to derive accurate models for the three basic switching converter power stages: buck, boost, and buck/boost operating with and without zero-inductor-current dwell time. A generalized procedure is developed which treats the continuous-inductor-current mode without dwell time as a special case of the discontinuous-current mode when the dwell time vanishes. Abrupt changes of system behavior, including a reduction of the system order when the dwell time appears, are shown both analytically and experimentally. Merits resulting from the present modeling technique in comparison with existing modeling techniques are illustrated.

Lee, F. C. Y.↗

A kinematic and proper-motion survey of the Cygnus Loop

H-alpha radial velocities, line widths, and line intensities based on 71 Fabry-Perot interferometer scans of 61 fields in the Cygnus Loop supernova remnant SNR are reported. Also, measurements of proper motions at 39 locations within the Cygnus Loop are summarized. These results show that the remnant is expanding asymmetrically into the ambient interstellar medium, with rest-frame velocities as high as 380 km/s in the near half and 150 km/s in the far half. This can plausibly be interpreted as the result of the progenitor's evolution near a density discontinuity, the consequent formation of a cavity and partial shell, and the SN's interaction with that shell. The remnant's optically observed filamentary structure results from line-of-sight effects as shock waves advance into the irregular inner surface of the shell. In this context, the derived expansion velocities and proper motions give a formal distance to the Cygnus Loop of roughly 600 pc, with 1-sigma limits of 300 and 1200 pc.

Shull, Peter, Jr.↗

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Gnoffo, Peter A.↗

Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nev- ertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuni- form data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.

97 MATHEMATICS AND COMPUTING↗

Dynamical Modeling of Surface Tension

In a recent review it is said that free-surface flows 'represent some of the difficult remaining challenges in computational fluid dynamics'. There has been progress with the development of new approaches to treating interfaces, such as the level-set method and the improvement of older methods such as the VOF method. A common theme of many of the new developments has been the regularization of discontinuities at the interface. One example of this approach is the continuum surface force (CSF) formulation for surface tension, which replaces the surface stress given by Laplace's equation by an equivalent volume force. Here, we describe how CSF formulation might be made more useful. Specifically, we consider a derivation of the CSF equations from a minimization of surface energy as outlined by Jacqmin (1996). This reformulation suggests that if one eliminates the computation of curvature in terms of a unit normal vector, parasitic currents may be eliminated. For this reformulation to work, it is necessary that transition region thickness be controlled. Various means for this, in addition to the one discussed by Jacqmin (1996), are discussed.

Brackbill, Jeremiah U.↗

Interlaminar Stresses by Refined Beam Theories and the Sinc Method Based on Interpolation of Highest Derivative

Computation of interlaminar stresses from the higher-order shear and normal deformable beam theory and the refined zigzag theory was performed using the Sinc method based on Interpolation of Highest Derivative. The Sinc method based on Interpolation of Highest Derivative was proposed as an efficient method for determining through-the-thickness variations of interlaminar stresses from one- and two-dimensional analysis by integration of the equilibrium equations of three-dimensional elasticity. However, the use of traditional equivalent single layer theories often results in inaccuracies near the boundaries and when the lamina have extremely large differences in material properties. Interlaminar stresses in symmetric cross-ply laminated beams were obtained by solving the higher-order shear and normal deformable beam theory and the refined zigzag theory with the Sinc method based on Interpolation of Highest Derivative. Interlaminar stresses and bending stresses from the present approach were compared with a detailed finite element solution obtained by ABAQUS/Standard. The results illustrate the ease with which the Sinc method based on Interpolation of Highest Derivative can be used to obtain the through-the-thickness distributions of interlaminar stresses from the beam theories. Moreover, the results indicate that the refined zigzag theory is a substantial improvement over the Timoshenko beam theory due to the piecewise continuous displacement field which more accurately represents interlaminar discontinuities in the strain field. The higher-order shear and normal deformable beam theory more accurately captures the interlaminar stresses at the ends of the beam because it allows transverse normal strain. However, the continuous nature of the displacement field requires a large number of monomial terms before the interlaminar stresses are computed as accurately as the refined zigzag theory.

Slemp, Wesley C. H.↗