Enriched Isogeometric Analysis for Parametric Domain Decomposition and Fracture Analysis
Abstract
As physical testing does not always yield insight into the mechanistic cause of failures, computational modeling is often used to develop an understanding of the goodness of a design and to shorten the product development time. One common, and widely used analysis technique is the Finite Element Method. A significant difficulty with the finite element method is the effort required to generate an analysis-ble mesh due to the difference in the mathematical representation of geometry CAD and CAE systems. CAD systems commonly use Non-Uniform Rational B-Splines (NURBS) while the CAE tools rely on the finite element mesh. Efforts to unify CAD and CAE by carrying out analysis directly using NURBS models termed Isogeometric Analysis reduces the gap between CAD and CAE phases of product development. However, several challenges still remain in the field of isogeometric analysis. A critical challenge relates to the output of commercial CAD systems. B-rep CAD models generated by commercial CAD systems contain uncoupled NURBS patches and are therefore not suitable for analysis directly. Existing literature is largely missing methods to smoothly couple NURBS patches. This is the first topic of research in this thesis. Fracture-caused failures are a critical concern for the reliability of engineered structures in general and semiconductor chips in particular. The back-end of the line structures in modern semiconductor chips contain multi-material junctions that are sites of singular stress, and locations where cracks originate during fabrication or testing. Techniques to accurately model the singular stress fields at interfacial corners are relatively limited. This is the second topic addressed in this thesis. Thus, the overall objective of this dissertation is to develop an isogeometric framework for parametric domain decomposition and analysis of singular stresses using enriched isogeometric analysis. Geometrically speaking, multi-material junctions, sub-domain interfaces and crack surfaces are lower-dimensional features relative to the two- or three-dimensional domain. The enriched isogeometric analysis described in this research builds enriching approximations directly on the lower-dimensional geometric features that then couple sub-domains or describe cracks. Since the interface or crack geometry is explicitly represented, it is easy to apply boundary conditions in a strong sense and to directly calculate geometric quantities such as normals or curvatures at any point on the geometry. These advantages contrast against those of implicit geometry methods including level set or phase-field methods. In the enriched isogeometric analysis, the base approximations in the domain/subdomains are enriched by the interfacial fields constructed as a function of distance from the interfaces. To circumvent the challenges of measuring distance and point of influence from the interface using iterative operations, algebraic level sets and algebraic point projection are utilized. The developed techniques are implemented as a program in the MATLAB environment named as Hierarchical Design and Analysis Code. The code is carefully designed to ensure simplicity and maintainability, to facilitate geometry creation, pre-processing, analysis and post-processing with optimal efficiency. To couple NURBS patches, a parametric stitching strategy that assures arbitrary smoothness across subdomains with non-matching discretization is developed. The key concept used to accomplish the coupling is the insertion of a “parametric stitching” or p-stitching interface between the incompatible patches. In the present work, NURBS is chosen for discretizing the parametric subdomains. The developed procedure though is valid for other representations of subdomains whose basis functions obey partition of unity. The proposed method is validated through patch tests from which near-optimal rate of convergence is demonstrated. Several two- and three-dimensional elastostatic as well as heat conduction numerical examples are presented.
Degree
Ph.D.
Advisors
Subbarayan, Purdue University.
Subject Area
Mathematics
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