Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods

dc.contributor.advisorDemkowicz, Leszek
dc.contributor.committeeMemberBabuska, Ivo M.
dc.contributor.committeeMemberCaffarelli, Luis A.
dc.contributor.committeeMemberHughes, Thomas J. R.
dc.contributor.committeeMemberOden, J. Tinsley
dc.contributor.committeeMemberWilder, Aleta
dc.creatorFuentes, Federico, Ph. D.
dc.creator.orcid0000-0002-4039-082X
dc.date.accessioned2018-07-20T15:34:43Z
dc.date.available2018-07-20T15:34:43Z
dc.date.created2018-05
dc.date.issued2018-06-25
dc.date.submittedMay 2018
dc.date.updated2018-07-20T15:34:43Z
dc.description.abstractDiscontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems. In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.
dc.description.departmentComputational Science, Engineering, and Mathematics
dc.format.mimetypeapplication/pdf
dc.identifierdoi:10.15781/T2D795V1C
dc.identifier.urihttp://hdl.handle.net/2152/65710
dc.language.isoen
dc.subjectFinite element methods
dc.subjectNumerical analysis
dc.subjectComputational mathematics
dc.subjectPDEs
dc.subjectDPG methods
dc.subjectDLS methods
dc.subjectPolyDPG methods
dc.subjectLinear elasticity
dc.subjectViscoelasticity
dc.subjectThermoviscoelasticity
dc.subjectDMA experiments
dc.subjectForm-wound coils
dc.titleVarious applications of discontinuous Petrov-Galerkin (DPG) finite element methods
dc.typeThesis
dc.type.materialtext
thesis.degree.departmentComputational Science, Engineering, and Mathematics
thesis.degree.disciplineComputational Science, Engineering, and Mathematics
thesis.degree.grantorThe University of Texas at Austin
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

Access full-text files

Original bundle

Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
FUENTES-DISSERTATION-2018.pdf
Size:
18.53 MB
Format:
Adobe Portable Document Format

License bundle

Now showing 1 - 2 of 2
No Thumbnail Available
Name:
PROQUEST_LICENSE.txt
Size:
4.46 KB
Format:
Plain Text
Description:
No Thumbnail Available
Name:
LICENSE.txt
Size:
1.85 KB
Format:
Plain Text
Description: