A stability framework for the Galerkin approximation of multifield saddle point principles with applications to irreversible problems
Abstract/Contents
- Abstract
- This dissertation addresses the development of a variational-based stability framework and finite element design for the approximative description of a broad class of materials which in their mathematical description are expressed as saddle point principles. The primary focus lies in the treatment of standard dissipative solids for which the energetic response can be expressed by two scalar functions, a free energy density and a dissipation function. As these materials show a time-dependence, the variational problem may be related to a potential function in incremental form. A fundamental challenge lies in the proper approximation of interpolated field variables due to a possible lack of the uniqueness of the approximated solution. The presented framework aims to generalize the so-called discrete inf-sup theory and its numerical verification in the context of general problems in an arbitrary number of fields for both linear and nonlinear problems and to provide a basis for multi-fold inf-sup conditions in the presence of vanishing parameters. Its novel character lies in the notion of Hessian matrices through which the requirement, that a perturbation of a functional possesses the desired properties, is met. This framework is then applied to a number of examples for which new discretization techniques are developed and tested. The first example represents a two-field displacement-pressure saddle point principle for porous materials that governs the interaction of a deforming porous solid matrix with an inter-penetrating solvent, a description that is of interest in the context of many geotechnical problems, the study of polymeric gels, and a large variety of biological materials. For the study of this problem a subdivision-technique is proposed in the context of isogeometric analysis and the satisfaction of stability requirements is shown. The second example represents a description of the same problem derived from a three-field principle with an additional incompatible field for the deformation gradient. It is shown that a reduced formulation of previously used incompatible modes provides the desired stability and the accompanied smoothness of the pore pressure in a much larger range of deformations due to its capability to avoiding hourglassing in the finite deformation range. Finally, a recent four-field variational formulation in the context of gradient plasticity, that overcomes drawbacks regarding mesh sensitivity, and the corresponding finite element design is investigated based on the proposed framework and a test is conducted for three different element types.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2019; ©2019 |
| Publication date | 2019; 2019 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Krischok, Andreas Ralf | |
|---|---|---|
| Degree supervisor | Linder, Christian, 1949- | |
| Thesis advisor | Linder, Christian, 1949- | |
| Thesis advisor | Borja, Ronaldo Israel | |
| Thesis advisor | Lew, Adrian | |
| Degree committee member | Borja, Ronaldo Israel | |
| Degree committee member | Lew, Adrian | |
| Associated with | Stanford University, Civil & Environmental Engineering Department. |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Andreas Krischok. |
|---|---|
| Note | Submitted to the Civil & Environmental Engineering Department. |
| Thesis | Thesis Ph.D. Stanford University 2019. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2019 by Andreas Ralf Krischok
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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