| Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations |
Nov 2012 |
63 pages |
| Authors:
Ju Liu; Hector Gomez; John A Evans; Thomas J Hughes; Chad M Landis; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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 | We propose a new methodology for the numerical solution of the isothermal Navier-Stokes- Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the ... |
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| Discrete Spectrum Analyses for Various Mixed Discretizations of the Stokes Eigenproblem |
Aug 2012 |
11 pages |
| Authors:
John A Evans; Thomas J Hughes; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We conduct discrete spectrum analyses for a selection of mixed discretization schemes for the Stokes eigenproblem. In particular, we consider the MINI element, the Crouzeix-Raviart element, the Marker-and- Cell scheme, the Taylor-Hood element, the Q(sub k)/P(sub k-1) element, the divergence-conforming discontinuous Galerkin method, and divergence-conforming B-splines. For each of these schemes, we compare the spectrum for the continuous Stokes problem with the spectrum for the discrete Stokes problem, and we ... |
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| Isogeometric Divergence-conforming B-splines for the Steady Navier-Stokes Equations |
Apr 2012 |
72 pages |
| Authors:
John A Evans; Thomas J Hughes; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier-Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart-Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier-Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity ... |
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| Isogeometric Divergence-conforming B-splines for the Unsteady Navier-Stokes Equations |
Apr 2012 |
49 pages |
| Authors:
John A Evans; Thomas J Hughes; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| Divergence-conforming B-splines are developed for application to the incompressible Navier-Stokes equations on geometrically mapped domains. These enable smooth pointwise divergence-free solutions and thus satisfy mass conservation in the strongest possible sense. Semi-discrete methods based on divergence-conforming B-splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity, enstrophy, and helicity. These are geometric structure-preserving quantities and numerical simulations that are sensitive to them are shown ... |
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| An Isogeometric Design-through-analysis Methodology based on Adaptive Hierarchical Refinement of NURBS, Immersed Boundary Methods, and T-spline CAD Surfaces |
22 Jan 2012 |
61 pages |
| Authors:
Dominik Schillinger; Luca Dede; Michael A Scott; John A Evans; Michael J Borden; Ernst Rank; Thomas J Hughes; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We explore hierarchical refinement of NURBS as a basis for adaptive isogeometric and immersed boundary analysis. We use the principle of B-spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions. We test hierarchical refinement of NURBS for some elementary fluid and structural analysis problems in two and three dimensions and ... |
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| Isogeometric Divergence-conforming B-splines for the Darcy-Stokes-Brinkman Equations |
Jan 2012 |
88 pages |
| Authors:
John A Evans; Thomas J Hughes; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy-Stokes-Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart-Thomas elements. The new discretizations are (at least) patch-wise C(0) and can be directly utilized in the Galerkin solution of Darcy- Stokes-Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free ... |
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| Explicit Trace Inequalities for Isogeometric Analysis and Parametric Hexahedral Finite Elements |
May 2011 |
25 pages |
| Authors:
John A Evans; Thomas J Hughes; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We derive new trace inequalities for NURBS-mapped domains. In addition to Sobolev-type inequalities, we derive discrete trace inequalities for use in NURBS-based isogeometric analysis. All dependencies on shape, size, polynomial order, and the NURBS weighting function are precisely specified in our analysis, and explicit values are provided for all bounding constants appearing in our estimates. As hexahedral finite elements are special cases of NURBS our results specialize to parametric hexahedral ... |
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