Magister Scientiae - MSc / Many chemical and physical problems are mathematically described by partial differential equations (PDEs). These PDEs are often highly nonlinear and therefore have no closed form solutions. Thus, it is necessary to recourse to numerical approaches to determine suitable approximations to the solution of such equations. For solutions possessing sharp spatial transitions (such as boundary or interior layers), standard numerical methods have shown limitations as they fail to capture large gradients. The method of lines (MOL) is one of the numerical methods used to solve PDEs. It proceeds by the discretization of all but one dimension leading to systems of ordinary di erential equations. In the case of time-dependent PDEs, the MOL consists of discretizing the spatial derivatives only leaving the time variable continuous. The process results in a system to which a numerical method for initial value problems can be applied. In this project we consider various types of singularly perturbed time-dependent PDEs. For each type, using the MOL, the spatial dimensions will be discretized in many different ways following fitted numerical approaches. Each discretisation will be analysed for stability and convergence. Extensive experiments will be conducted to confirm the analyses.

<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>