High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints

Responses of the Pine Island and Thwaites Glaciers to Melt and Sliding Parameterizations

The Pine Island and Thwaites glaciers are the two largest contributors to sea level rise from Antarctica. Here we examine the influence of basal friction and ice shelf basal melt in determining projected losses. We examine both Weertman and Coulomb friction laws with explicit weakening as the ice thins to flotation, which many friction laws include implicitly via the effective pressure. We find relatively small differences with the choice of friction law (Weertman or Coulomb) but find losses to be highly sensitive to the rate at which the basal traction is reduced as the area upstream of the grounding line thins. Consistent with earlier work on Pine Island Glacier, we find sea level contributions from both glaciers to vary linearly with the melt volume averaged over time and space, with little influence from the spatial or temporal distribution of melt. Based on recent estimates of melt from other studies, our simulations suggest that the combined melt-driven and sea level rise contribution from both glaciers may not exceed 10 cm by 2200, although the uncertainty in model parameters allows for larger increases. We do not include other factors, such as ice shelf breakup, that might increase loss, or factors such as increased accumulation and isostatic uplift that may mitigate loss

Consistent Point Data Assimilation in Firedrake and Icepack

We present methods and tools that significantly improve the ability to estimate quantities and fields which are difficult to directly measure, such as the fluidity of ice, using point data sources, such as satellite altimetry. These work with both sparse and dense point data with estimated quantities and fields becoming more accurate as the number of measurements are increased. Such quantities and fields are often used as inputs to mathematical models that are used to make predictions so improving their accuracy is of vital importance. We demonstrate how our methods and tools can increase the accuracy of results, ensure posterior consistency, and aid discourse between modellers and experimenters. To do this, we bring point data into the finite element method ecosystem as discontinuous fields on meshes of disconnected vertices. Point evaluation can then be formulated as a finite element interpolation operation (dual-evaluation). Our new abstractions are well-suited to automation. We demonstrate this by implementing them in Firedrake, which generates highly optimised code for solving PDEs with the finite element method. Our solution integrates with dolfin-adjoint/pyadjoint which allows PDE-constrained optimisation problems, such as data assimilation, to be solved through forward and adjoint mode automatic differentiation. We demonstrate our new functionality through examples in the fields of groundwater hydrology and glaciology

Ice-shelf retreat drives recent Pine Island Glacier speedup

Speedup of Pine Island Glacier over the past several decades has made it Antarctica’s largest contributor to sea-level rise. The past speedup is largely due to grounding-line retreat in response to ocean-induced thinning that reduced ice-shelf buttressing. While speeds remained fairly steady from 2009 to late 2017, our Copernicus Sentinel 1A/B–derived velocity data show a >12% speedup over the past 3 years, coincident with a 19-km retreat of the ice shelf. We use an ice-flow model to simulate this loss, finding that accelerated calving can explain the recent speedup, independent of the grounding-line, melt-driven processes responsible for past speedups. If the ice shelf’s rapid retreat continues, it could further destabilize the glacier far sooner than would be expected due to surface- or ocean-melting processes

Entrepreneurship Assessment in Higher Education: A Research Review for Engineering Education Researchers

BackgroundDespite the wide adoption of entrepreneurship by United States engineering programs, there have been few advances in how to measure the influences of entrepreneurial education on engineering students. We believe the inadequate growth in engineering entrepreneurship assessment research is due to the limited use of research emerging from the broader entrepreneurship education assessment community.PurposeThis paper explores entrepreneurship education assessment by documenting the current state of the research and identifying the theories, variables, and research designs most commonly used by the broader community. We then examine if and how these theories and constructs are used in engineering entrepreneurship education.Scope/MethodTwo literature databases, Scopus® and Proquest, were searched systematically for entrepreneurship education assessment research literature. This search yielded 2,841 unique papers. Once inclusion and exclusion criteria were applied, 359 empirical research papers were coded for study design, theory, variables measured, instruments, and validity and reliability.ConclusionsWhile there has been growth in entrepreneurship education assessment research, little exchange of ideas across the disciplines of business, engineering, and education is occurring. Nonempirical descriptions of programs outweigh empirical research, and these empirical studies focus on affective, rather than cognitive or behavioral, outcomes. This pattern within the larger entrepreneurship community is mirrored in engineering where the use of theoryâ based, validated entrepreneurship education assessment instruments generally focuses on the context of intent to start a new company. Given the engineering community’s goals to support engineering entrepreneurship beyond business creation, the engineering education community should consider developing assessment instruments based in theory and focused on engineeringâ specific entrepreneurship outcomes.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145556/1/jee20197.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/145556/2/jee20197_am.pd

Data assimilation problems in glaciology

Thesis (Ph.D.)--University of Washington, 2017-03Rising sea levels due to mass loss from Greenland and Antarctica threaten to inun- date coastal areas the world over. For the purposes of urban planning and hazard mitigation, policy makers would like to know how much sea-level rise can be antici- pated in the next century. To make these predictions, glaciologists use mathematical models of ice sheet flow, together with remotely-sensed observations of the current state of the ice sheets. The quantities that are observable over large spatial scales are the ice surface elevation and speed, and the elevation of the underlying bedrock. There are other quantities, such as the viscosity within the ice and the friction co- efficient for sliding over the bed, that are just as important in dictating how fast the glacier flows, but that are not observable at large scales using current meth- ods. These quantities can be inferred from observations by using data assimilation methods, applied to a model of glacier flow. In this dissertation, I will describe my work on data assimilation problems in glaciology. My main contributions so far have been: computing the bed stress underneath the three biggest Greenland outlet glaciers; developing additional tools for glacier modelling and data assimi- lation in the form of the open-source library iceack; and improving the statistical methodology through the user of total variation priors

Ice surface elevation and change at Jakobshavn Isbrae, West Greenland, 2008-2014

Contains: * initial surface elevation map of Jakobshavn Isbrae in January 2008 * differences every 91 days from the initially recorded elevation map * geoid correction * error field

Data associated with "Ice-Shelf Retreat Drives Recent Pine Island Glacier Speedup” and "Ocean-Induced Melt Volume Directly Paces Ice Loss from Pine Island Glacier "

Speedup of Pine Island Glacier over the last several decades has made it Antarctica’s largest contributor to sea-level rise. The past speedup is largely due to grounding-line retreat in response to ocean-induced thinning that reduced ice-shelf buttressing. These data are the model inputs and other observations associated with two papers that use an ice-flow model to investigate recent speedup of Pine Island Glacier, along with how continued melting may affect the glacier over the next 200 year

Photodissociation Dynamics of the Cyclohexyl Radical from the 3p Rydberg State at 248 nm.

The photodissociation of jet-cooled cyclohexyl was studied by exciting the radicals to their 3p Rydberg state by using 248 nm laser light and detecting photoproducts by photofragment translational spectroscopy. Both H atom loss and dissociation to heavy fragment pairs are observed. The H atom loss channel exhibits a two-component translational energy distribution. The fast photoproduct component is attributed to impulsive cleavage directly from an excited state, likely the Rydberg 3s state, forming cyclohexene. The slow component is due to statistical decomposition of hot cyclohexyl radicals that internally convert to the ground electronic state prior to H atom loss. The fast and slow components are present in an ∼0.7:1 ratio, similar to findings in other alkyl radicals. Internal conversion to the ground state also leads to ring-opening followed by dissociation to 1-buten-4-yl + ethene in comparable yield to H-loss, with the C4H7 fragment containing enough internal energy to dissociate further to butadiene via H atom loss. A very minor ground-state C5H8 + CH3 channel is observed, attributed predominantly to 1,3-pentadiene formation. The ground-state branching ratios agree well with RRKM calculations, which also predict C4H6 + C2H5 and C3H6 + C3H5 channels with similar yield to C5H8 + CH3. If these channels were active, it was at levels too low to be observed

Photodissociation of the Cyclopentadienyl Radical at 248 nm

The photodissociation of jet-cooled cyclopentadienyl radicals, <i>c</i>-C₅H₅, at 248 nm was studied using photofragment translational spectroscopy. Two dissociation channels were observed: H + C₅H₄ and C₃H₃ + C₂H₂. The C₅H₄ fragment was identified as ethynylallene by its ionization energy. The translational energy distribution determined for each channel suggests that both dissociation mechanisms occur via internal conversion to the ground electronic state followed by intramolecular vibrational redistribution and dissociation. The experimental branching ratio and RRKM (Rice–Ramsperger–Kassel–Marcus) calculations favor the formation of C₃H₃ + C₂H₂ over the H-atom loss channel. The RRKM calculations also support the observation of ethynylallene as the dominant C₅H₄ product isomer