Design of a foldable sunshade as an attachment to a sports bag

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, June 2011."June 2011." Cataloged from PDF version of thesis.The purpose of this project was to design a foldable, lightweight, inexpensive addition to an existing duffel bag that will provide enough shade to cover the user's head and upper torso. The intended user for this product is an athlete who plays the sport of Ultimate Frisbee, as this typically involves bye rounds which some use for napping on the sidelines. The existing products to fill this need are too expensive, heavy, bulky or uncomfortable. The process for designing this feature included an extensive ideation phase, potential user interviews, prototype mock-ups, and the production of a final working prototype. This final prototype is made with the roll-out mat that comes attached to the Under Armour Medium Team Duffel bag and the metal support hoops from a collapsible laundry hamper. This prototype meets all of the product specifications that were based on the requirements of the interviewed users.by Cody A. Rebholz.S.B

Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form

We consider {\it local} balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory

Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation

In this paper, we consider the recently introduced EMAC formulation for the incompressible Navier-Stokes (NS) equations, which is the only known NS formulation that conserves energy, momentum and angular momentum when the divergence constraint is only weakly enforced. Since its introduction, the EMAC formulation has been successfully used for a wide variety of fluid dynamics problems. We prove that discretizations using the EMAC formulation are potentially better than those built on the commonly used skew-symmetric formulation, by deriving a better longer time error estimate for EMAC: while the classical results for schemes using the skew-symmetric formulation have Gronwall constants dependent on $\exp(C\cdot Re\cdot T)$ with $Re$ the Reynolds number, it turns out that the EMAC error estimate is free from this explicit exponential dependence on the Reynolds number. Additionally, it is demonstrated how EMAC admits smaller lower bounds on its velocity error, since {incorrect treatment of linear momentum, angular momentum and energy induces} lower bounds for $L^2$ velocity error, and EMAC treats these quantities more accurately. Results of numerical tests for channel flow past a cylinder and 2D Kelvin-Helmholtz instability are also given, both of which show that the advantages of EMAC over the skew-symmetric formulation increase as the Reynolds number gets larger and for longer simulation times.Comment: 21 pages, 5 figure

Improving mass conservation in FE approximations of the Navier Stokes equations using continuous velocity fields: a connection between grad-div stabilization and Scott-Vogelius elements

This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. Under mild restrictions, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that, again under these mild restrictions, the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, in this setting, we provide theoretical justification that choosing the parameter large does not destroy the solution. A limiting result is also proven for the general case. Numerical tests are provided which verify the theory, and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations

Stable computing with an enhanced physics based scheme for the 3d Navier-Stokes equations

We study extensions of an earlier developed energy and helicity preserving scheme for the 3D Navier-Stokes equations and apply them to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the schem