# Enriched Isogeometric Analysis for Parametric Domain Decomposition and Fracture Analysis

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-suitable 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.

An enriched field approximation is then developed for characterizing stress singularities at junctions of general multi-material corners including crack tips. Using enriched isogeometric analysis, the developed method explicitly tracks the singular points and interfaces embedded in a non-conforming mesh. Solution convergence to those of linear elastic fracture mechanics is verified through several examples. More importantly, the proposed method enables direct extraction of generalized stress intensity factors upon solution of the problems without the need to use

*a posteriori*path-independent integral such as the J-integral. Next, the analysis of crack initiation and propagation is carried out using the alternative concept of configurational force. The configurational force is first shown to result from a configurational optimization problem, which yields a configurational derivative as a necessary condition. For specific velocities imposed on the heterogeneities corresponding to translation, rotation or scaling, the configurational derivative is shown to yield the configurational force. The use of configurational force to analyze crack propagation is demonstrated through examples.The developed methods are lastly applied to investigate the risk of ratcheting-induced fracture in the back end of line structure during thermal cycle test of a epoxy molded microelectronic package. The first principal stress and the opening mode stress intensity factor are proposed as the failure descriptors. A finite element analysis sub-modeling and load decomposition procedure is proposed to study the accumulation of plastic deformation in the metal line and to identify the critical loading mode. Enriched isogeometric analysis with singular stress enrichment is carried out to identify the interfacial corners most vulnerable to stress concentration and crack initiation. Correlation is made between the failure descriptors and the design parameters of the structure. Crack path from the identified critical corner is predicted using both linear elastic fracture mechanics criterion and configurational force criterion.