<div>Many Engineering structures, including electronic component assemblies, are inherently multi-scale in nature. These structures often experience complex local nonlinear behavior such as plasticity, damage or fracture. These local behaviors eventually lead to the failure at the macro length scale. Connecting the behavior across the length scales to develop an understanding of the failure mechanism is important for developing reliable products.</div><div><br></div><div>To solve multi-scale problems in which the critical region is much smaller than the entire structure, an iterative solution approach based on domain decomposition techniques is proposed. Two independent models are constructed to model the global and local substructures respectively. The unbalanced force at the interface is iteratively reduced to ensure force equilibrium of the overall structure in the final solution. The approach is non-intrusive since only nodal values on the interface are transferred between the global and local models. Solution acceleration using SR1 and BFGS updates is also demonstrated. Equally importantly, the two updates are applied in a non-intrusive manner, meaning that the technique is implemented without needing access to the codes using which the sub-domains are analyzed. Code- and mesh-agnostic solutions for problems with local nonlinear material behavior or local crack growth are demonstrated. Analysis in which the global and local models are solved using two different commercial codes is also demonstrated.</div><div><br></div><div>Engineering analysis using numerical models are helpful in providing insight into the connection between the structure, loading history, behavior and failure. Specifically, Isogeometric analysis (IGA) is advantageous for engineering problems with evolving geometry compared to the traditional finite element method (FEM). IGA carries out analysis by building behavioral approximations isoparametrically on the geometrical model (commonly NURBS) and is thus a promising approach to integrating Computer-Aided Design (CAD) with Computer-Aided Engineering (CAE).</div><div><br></div><div>In enriched isogeometric Analysis (EIGA), the solution is enriched with known behavior on lower dimensional geometrical features such as crack tips or interfaces. In the present research, enriched field approximation techniques are developed for the application of boundary conditions, coupling patches with non-matching discretizations and for modeling singular stresses in the structure.</div><div><br></div><div>The first problem solution discussed is to apply Dirichlet and Neumann boundary conditions on boundary representation (B-rep) CAD models immersed in an underlying domain of regular grid points. The boundary conditions are applied on the degrees of freedom of the lower dimensional B-rep part directly. The solution approach for the immersed analysis uses signed algebraic level sets constructed from the B-rep surfaces to blend the enriched</div><div>field with the underlying field. The algebraic level sets provide a surrogate for distance, are non-iteratively (or algebraically) computed and allow implicit Boolean compositions.</div><div><br></div><div>The methodology is also applied to couple solution approximations of decomposed patches by smoothly blending incompatible geometries to an arbitrary degree of smoothness. A parametrically described frame or interface is introduced to “stitch” the adjacent patches. A hierarchical blending procedure is then developed to stitch multiple unstructured patches including those with T-junctions or extraordinary vertices.</div><div><br></div><div>Finally, using the EIGA technique, a computational method for analyzing general multimaterial sharp corners that enables accurate estimations of the generalized stress intensity factors is proposed. Explicitly modeled geometries of material junctions, crack tips and deboned interfaces are isogeometrically and hierarchically enriched to construct approximations with the known local behavior. specifically, a vertex enrichment is used to approximate the asymptotic field near the re-entrant corner or crack tip, Heaviside function is used to approximate the discontinuous crack face and the parametric smooth stitching technique is used to approximate the behavior across material interface. The developed method allows direct extraction of generalized stress intensity factors without needing a posteriori evaluation of path independent integrals for decisions on crack propagation. The numerical implementation is validated through analysis of a bi-material corner, interface crack and growth of an inclined crack in a homogeneous solid. The developed procedure demonstrates rapid convergence to the solution stress intensity factors with relatively fewer degrees of freedom, even with uniformly coarse discretizations.</div>

<p>Moving boundary problems such as solidification, crack propagation, multi-body contact or shape optimal design represent an important class of engineering problems. Common to these problems are one or more moving interfaces or boundaries. One of the main challenges associated with boundary evolution is the difficulty that arises when the topology of the geometry changes. Other geometric issues such as distance to the boundary, projected point on the boundary and intersection between surfaces are also important and need to be efficiently solved. In general, the present thesis is concerned with the geometric arrangement and behavioral analysis of evolving parametric boundaries immersed in a domain. </p> <p>The first problem addressed in this thesis is the packing of ellipsoidal fillers in a regular domain and to estimate their effective physical behavior. Particle packing problem arises when one generates simulated microstructures of particulate composites. Such particulate composites used as thermal interface materials (TIMs) motivates this work. The collision detection and distance calculation between ellipsoids is much more difficult than other regular shapes such as spheres or polyhedra.  While many existing methods address the spherical packing problems, few appear to achieve volume loading exceeding 60%. The packing of ellipsoidal particles is even more difficult than that of spherical particles due to the need to detect contact between the particles. In this thesis, an efficient and robust ultra-packing algorithm termed Modified Drop-Fall-Shake is developed. The algorithm is used to simulate the real mixing process when manufacturing TIMs with hundreds of thousands ellipsoidal particles. The effective thermal conductivity of the particulate system is evaluated using an algorithm based on Random Network Model. </p> <p><br></p> <p>In problems where general free-form parametric surfaces (as opposed to the ellipsoidal fillers) need to be evolved inside a regular domain, the geometric distance from a point in the domain to the boundary is necessary to determine the influence of the moving boundary on the underlying domain approximation. Furthermore, during analysis, since the driving force behind interface evolution depends on locally computed curvatures and normals, it is ideal if the parametric entity is not approximated as piecewise-linear. To address this challenge,  an algebraic procedure is presented here to find the level sets of rational parametric surfaces commonly utilized by commercial CAD systems. The developed technique utilizes the resultant theory to construct implicit forms of parametric Bezier patches, level sets of which are termed algebraic level sets (ALS). Boolean compositions of the algebraic level sets are carried out using the theory of R-functions. The algebraic level sets and their gradients at a given point on the domain can also be used to project the point onto the immersed boundary. Beginning with a first-order algorithm, sequentially refined procedures culminating in a second-order projection algorithm are described for NURBS curves and surfaces. Examples are presented to illustrate the efficiency and robustness of the developed method. More importantly, the method is shown to be robust and able to generate valid solutions even for curves and surfaces with high local curvature or G₀ continuity---problems where the Newton--Raphson method fails due to discontinuity in the projected points or because the numerical iterations fail to converge to a solution, respectively. </p> <p><br></p> <p>Next, ALS is also extended for boundary representation (B-rep) models that are popularly used in CAD systems for modeling solids. B-rep model generally contains multiple NURBS patches due to the trimming feature used to construct such models, and as a result are not ``watertight" or mathematically compatible at patch edges. A time consuming geometry clean-up procedure is needed to preprocess geometry prior to finite element mesh generation using a B-rep model, which can take up to 70% of total analysis time according to literature. To avoid the need to clean up geometry and directly provide link between CAD and CAE integration,  signed algebraic level sets using novel inner/outer bounding box strategy is proposed for point classification of B-rep model. Several geometric examples are demonstrated, showing that this technique naturally models single patch NURBS geometry as well, and can deal with multiple patches involving planar trimming feature and Boolean operation. During the investigation of algebraic level sets, a complex self-intersection problem is also reported, especially for three-dimensional surface. The self-intersection may occur within an interval of interest during implicitization of a curve or surface since the implicitized curve or surface is not trimmed and extends to infinity. Although there is no robust and universal solution the problem, two potential solutions are provided and discussed in this thesis.</p> <p><br></p> <p>In order to improve the computational efficiency of analysis in immersed boundary problems, an efficient local refinement technique for both mesh and quadrature  using the kd-tree data structure is further proposed. The kd-tree sub-division is theoretically proved to be more efficient against traditional quad-/oct-tree subdivision methods. In addition, an efficient local refinement strategy based on signed algebraic level sets is proposed to divide the cells. The efficiency of kd-tree based mesh refinement and adaptive quadrature is later shown through numerical examples comparing with oct-tree subdivision, revealing significant reduction of degrees of freedom and quadrature points.</p> <p><br></p> <p>Towards analysis of moving boundaries problems, an explicit interface tracking method termed enriched isogeometric analysis (EIGA) is adopted in this thesis. EIGA utilizes NURBS shape function for both geometry representation and field approximation. The behavior field is modeled by a weighted blending of the underlying domain approximation and enriching field, allowing high order continuity naturally. Since interface is explicitly represented, EIGA provides direct geometric information such as normals and curvatures. In addition, the blending procedure ensures strong enforced boundary conditions. An important moving boundary problem, namely, reaction-diffusion problem, is investigated using EIGA. In reaction-diffusion problems, the phase interfaces evolve due to chemical reaction and diffusion under multi-physics driven forces, such as mechanical, electrical, thermal, etc. Typical failure phenomenon due to reaction-diffusion problems include void formation and intermetallic compound (IMC) growth. EIGA is applied to study factors and behavior patterns in these failure phenomenon, including void size, current direction, current density, etc. A full joint simulation is also conducted to study the degradation of solder joint under thermal aging and electromigration. </p>