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2016-2017 CETL Annual Report
Get PDFThe Center for Excellence in Teaching & Learning description of events, attendance at events, teaching workshop
Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models
Get PDFIn this work, we develop an implicit real space method in 1D and 2D for the Cahn--Hilliard (CH) and vector Cahn--Hilliard (VCH) equations, based on the method of lines transpose (MOL) formulation. This formulation results in a semidiscrete time stepping algorithm, which we prove is gradient stable in the norm. The spatial discretization follows from dimensional splitting and an matrix-free solver, which applies fast convolution to the modified Helmholtz equation. We propose a novel factorization technique, in which fourth-order spatial derivatives are incorporated into the solver. The splitting error is included in the nonlinear fixed point iteration, resulting in a high-order, logically Cartesian (line-by-line) update. Our method is fast but not restricted to periodic boundaries like the fast Fourier transform (FFT). The basic solver is implemented using the backward Euler formulation, and we extend this to both backward difference formula (BDF) stencils, singly diagonal implicit Runge--Kutta (SDIRK), and spectral deferred correction (SDC) frameworks to achieve high orders of temporal accuracy. We demonstrate with numerical results that the CH and VCH equations maintain gradient stability in one and two spatial dimensions. We also explore time-adaptivity, so that meta-stable states and ripening events can be simulated both quickly and efficiently
Sequences which converge to e: New insights from an old formula
No full textOne of the most fundamental results in calculus was the discovery of the mathematical constant e = 2.718... by Jacob Bernoulli. Remarkably, new definitions of e are still being discovered, in part due to renewed interest at the advent of modern computing and the quest for more digits. In this work we review recent discoveries of sequences which tend to e, and propose a systematic approach for producing such sequences. In doing so, we establish several classes of sequences, and their generalizations. Our methods use only basic tools of calculus and numerical analysis, such as series expansions and Padé approximants. Numerical results demonstrate that our new sequences rapidly converge to e
Drivers of research topic selection for engineering doctoral students
Get PDFIn this study, we explored engineering doctoral students’ motivations for selecting their research topic. The extent to whichindividuals are intrinsically or extrinsically motivated and the autonomy they have to make their own decisions hasimplications for their enjoyment of and success at a particular task. Given the importance of motivation, we sought toaddress a gap in the understanding of how doctoral students in engineering decide on a particular problem to study. Ourfindings are based on interviews with students with varying past educational and professional experiences that enable us tocapture a wide range of motivations for engineering PhD students’ research subject decisions. We found that the majorityof students interviewed reported some form of extrinsic motivation guiding their decision, though these students varied intheir autonomy to select their own topic. Of the students who reported intrinsic motivations for their research topicselection, many had extensive prior work experience that informed their topic choice. Funding played a major role inshaping students’ project decisions, which is reflective of the scale and expense of much of engineering work. However, ourfindings suggest there are a number of opportunities for students to identify research topics in which they personallyperceive as important and interesting
