Explicit geometry based enriched field approximations

Abhishek Tambat, Purdue University

Abstract

Boundaries with specified behavior, phase boundaries, crack surfaces or singular points are, geometrically speaking, lower-dimensional features relative to two- or three-dimensional geometrical domains. Often, the distinguishing characteristics of the behavior at these features are known a priori and may be exploited to enrich isogeometric models. Explicit geometrical representations possess parametrically computable tangents, normals and curvature, while in implicit strategies, the geometric "exactness" of enriching lower-dimensional features is not exploited or retrieved only in the limit of mesh refinement. In the present work, CAD-inspired hierarchical partition of unity field compositions are extended to modeling explicitly defined enrichments within the isogeometric framework. The base approximations are "enriched" isogeometrically on parametrically defined lower-dimensional geometrical features of the base entity and by constructing distance fields from them. The efficiency and robustness of distance computations is significantly improved by composing monotonic distance measures, defined piecewise on the enriching geometric entity, using R-Functions. The procedure allows both the behavioral approximation as well as the material description to be enriched enabling the modeling of material damage (or, alternatively, local stiffening). Further, the enrichments may ensure known function value or its derivative. Function value enrichments are demonstrated to model Dirichlet boundary conditions and propagating cracks. The derivative enrichments are used to model Neumann boundary conditions as well as strain jumps across material interfaces. The material enrichments are demonstrated through the use of a cohesive damage law to model arbitrary crack initiation and propagation within the domain. The developed behavioral as well as material enrichment strategies for modeling fracture are used to study fracture behavior in multilayered structures. A propagating crack impinging an interface between dissimilar materials could either deflect along the interface or penetrate the interface. The competition between crack deflection along the interface or penetration across the interface is numerically investigated. The energy release rate at the crack tip along with the fracture toughness of the interface as well as those of materials forming the interface play an important role in the competition. The numerically evaluated energy release rate for the deflected crack is compared with the maximum energy release rate for the crack penetrating the interface under Mode I and mixed mode conditions. Under Mode I conditions, the ratio of interfacial fracture toughness to the bulk fracture toughness of the material is numerically evaluated for the case of singly and doubly deflected cracks. For cracks under mixed mode conditions, the angle of crack penetration is numerically investigated. Based on the above studies, an automatic crack propagation algorithm is developed for simulating fracture in layered structures, which is demonstrated on an example problem. As a second application, the hierarchical enrichments are used to model a critical failure mechanism in semiconductor chip dielectric stacks. At the present time, fracture in the ILD (interlayer dielectric) stacks induced by assembly to either an organic substrate or a die stack (3-D) is an important reliability consideration. These interactions include what is popularly referred to as the chip-package interactions (CPI). In this work, insights are developed on the potential crack initiation site within the ILD, die-substrate geometrical parameters that cause most damage, as well as insights on the manufacturing process that is critical to failure. Towards this end analytical models based on classical elasticity theory as well as the enrichment technique that is capable of nucleating and propagating cracks at arbitrary locations within the structure without remeshing are utilized. Specifically, the strength of singularities at all the possible multimaterial corners in the ILD stack are analytically estimated to provide insight on the likely damage nucleation sites for various material configurations in the ILD stack. Both of the enrichment approaches developed in this thesis are used for fracture simulation. In the first, cracks are modeled as discontinuous enrichments over an underlying continuous behavioral approximation. In the second approach, the underlying material description is enriched with a cohesive damage description whose stiffness is evolved according to a prescribed damage law. Multi-level finite element models are used to determine the load imposed on the ILD structure by the substrate. Maximum damage induced in the ILD stack by the above load is used as an indicator of the reliability risk. Parametric simulations are conducted by varying ILD material, die size, die thickness, as well as the solder material. Through analytical models of bonded assemblies, groups of relevant dimensionless parameters are identified to relate the numerically estimated damage in ILD stacks to the die/substrate material and geometrical parameters. It is demonstrated that the damage in the ILD stack is least when the flexural rigidity of the die is matched to that of the assembled substrate. It is also demonstrated that ILD damage is only weakly correlated to shear deformation on the die surface due to assembly. The above observations are generalized into mathematical fits (for use as design rules) relating damage in ILD stacks to ILD material choice, relative substrate flexural rigidity, and die size.

Degree

Ph.D.

Advisors

Subbarayan-Shastri, Purdue University.

Subject Area

Engineering|Mechanical engineering

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