Front-tracking finite element methods for a void electro-stress migration problem

Continued research in electronic engineering technology has led to a miniaturisation of integrated circuits. Further reduction in the dimensions of the interconnects is impeded by the presence of small cracks or voids. Subject to high current and elastic stress, voids tend to drift and change shape in the interconnect, leading to a potential mechanical failure of the system. This thesis investigates the temporal evolution of voids moving along conductors, in the presence of surface diffusion, electric loading and elastic stress. We simulate a bulk-interface coupled system, with a moving interface governed by a fourth-order geometric evolution equation and a bulk where the electric potential and the displacement field are computed. We first give a general overview about geometric evolution equations, which define the motion of a hypersurface by prescribing its normal velocity in terms of geometric quantities. We briefly describe the three main approaches that have been proposed in the literature to solve numerically this class of equations, namely parametric approach, level set approach and phase field approach. We then present in detail two methods from the parametric approach category for the void electro-stress migration problem. We first introduce an unfitted method, where bulk and interface grids are totally independent, i.e. no topological compatibility between the two grids has to be enforced over time. We then discuss a fitted method, where the interface grid is at all times part of the boundary of the bulk grid. A detailed analysis, in terms of existence and uniqueness of the finite element solutions, experimental order of convergence (when the exact solution to the free boundary problem is known) and coupling operations (e.g., smoothing/remeshing of the grids, intersection between elements of the two grids), is carried out for both approaches. Several numerical simulations, both two- and three-dimensional, are performed in order to test the accuracy of the methods.Open Acces

Simulations of diffusion driven phase evolution in heterogenous solids

With reduction in size, ever greater operational demands are placed on electronics components at all levels of the device, starting from the transistor level to the level of the package and the solder interconnects. Concurrently, there has been a move to more complicated materials systems in order to meet health and environmental guidelines. These trends of reducing size, increasing loads have increased the necessity to understand the mechanisms of the failure. ^ As the length scales are reduced, it becomes increasingly important to consider interfacial and micro-structural effects that can be safely ignored at larger length scales owing to the randomness. It has become important to model the effect of interfacial motion and micro-structural evolution due to diffusion on the reliability of micro-electronics components. Examples of interfacial motion phenomena in solids include crack propagation, grain boundary motion, diffusion driven void motion through sur- face and bulk diffusion. The presence and evolution of these over the life-cycle of electronics components such as metal lines and solder joints presents a significant reliability challenge. The mathematical models that describe the evolution of these interfaces are usually formulated as systems of non-linear equations and hence, numerical methods provide an important method to study and understand them. The primary challenge in the study of these moving boundary problems is the tracking of the moving boundary and the application of appropriate boundary conditions on the moving boundary. ^ The phase field method tracks through smooth approximations of the Heaviside step and Dirac δ functions, which are maintained through the solution of a system of nonlinear differential equations. In this work, phase field approaches are developed for the study of diffusion driven phase evolution problems. First a phase field model for the evolution of voids in solder joints owing to electromigration and stress-migration both at the interface due to the surface gradients of the electric potential, temperature, curvature and strain energy, as well as self diffusion in the bulk on account of the chemical potential gradients as well as the electromigration force. This is modeled using a vacancy diffusion mechanism, while the growth of the voids is assumed to be due to the absorption of voids at the interface of pre-existing voids. A formal asymptotic analysis is performed to show the equivalence of the diffuse interface model to its sharp interface equivalents. Several numerical examples are presented. ^ Finally, an n-phase system of Cahn-Hilliard equations is developed to allow for the simulation of void evolution and growth in a multi-phase system. This is derived through a micro-force balance in order to eliminate the use of Lagrange multipliers that are commonly seen in such methods. A limited formal asymptotic analysis is performed to show the equivalence of the model to the standard surface diffusion model in regions where only two phase are present. This is numerically implemented and various numerical examples of phase evolution under simple surface diffusion, as well as surface diffusion with electromigration are demonstrated

A practical phase field method for an elliptic surface PDE

We consider a diffuse interface approach for solving an elliptic PDE on a given closed hypersurface. The method is based on a (bulk) finite element scheme employing numerical quadrature for the phase field function and hence is very easy to implement compared to other approaches. We estimate the error in natural norms in terms of the spatial grid size, the interface width and the order of the underlying quadrature rule. Numerical test calculations are presented, which confirm the form of the error bounds

Finite-Element Approximation of One-Sided Stefan Problems with Anisotropic, Approximately Crystalline, Gibbs--Thomson Law

We present a finite-element approximation for the one-sided Stefan problem and the one-sided Mullins--Sekerka problem, respectively. The problems feature a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our approximation, which couples a parametric approximation of the moving boundary with a finite element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys very good mesh properties which means that no mesh smoothing is necessary in practice. In our numerical computations we concentrate on the simulation of snow crystal growth. On choosing realistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in approximately crystalline evolutions can be observed.Comment: 50 pages, 32 figures, 14 tables. Corrected typo

On stable parametric finite element methods for the Stefan problem and the Mullins-Sekerka problem with applications to dendritic growth

We introduce a parametric finite element approximation for the Stefan problem with the Gibbs–Thomson law and kinetic undercooling, which mimics the underlying energy structure of the problem. The proposed method is also applicable to certain quasi-stationary variants, such as the Mullins–Sekerka problem. In addition, fully anisotropic energies are easily handled. The approximation has good mesh properties, leading to a well-conditioned discretization, even in three space dimensions. Several numerical computations, including for dendritic growth and for snow crystal growth, are presented

Numerical analysis and simulations of a tractable model for tumour growth

This thesis begins with the description of a tractable model for tumour growth. The unique feature of this model is that we pass through the thin rim limit. We derive the sharp interface weak form and finite element scheme. We discuss the mesh smoothing techniques used in the implementation of the sharp interface finite element scheme. We then introduce an unfitted finite element scheme, and a sharp interface finite element scheme in R3. We also write the model in the diffuse interface paradigm, along with the associated weak form. We prove the existence and uniqueness of the solution to the di_use interface version of the model, and prove convergence of the diffuse interface finite element method. We conclude this thesis with a number of simulations in R2 and R3. Here, we present rates of convergence, and also investigate the effect of parameter spaces on the morphology of the tumour. A biologically motivated investigation is made, and a brief comparison with in vivo tumours is presented

Modelling, analysis and simulation of incompressible multi-fluid flows

Multi-fluid flows are omnipresent in our lives, from the fabrication of integrated circuit components in most electronics to the miniature laboratories inside medical tools, and even as a drop of rain splashes onto the wing of an aeroplane. In this thesis we use theoretical and numerical tools to investigate topics from the fascinating world of interfacial flows. The first part of this dissertation is dedicated to the study of multi-fluid systems in small scale channel geometries in the presence of electric fields. We develop the theoretical machinery to address the stabilisation (to the point of complete suppression) of the classical Rayleigh-Taylor instability under the action of an electric field acting in the plane of the liquid-liquid interface. In a related context, in many situations electric fields normal to the fluid-fluid interface may be employed in order to accurately drive instabilities towards beneficial goals. In particular, we discuss novel mechanisms to generate pumping and mixing in millimetre-sized geometries without requiring moving parts or an oncoming flow. In the second part of this thesis we turn our attention to the area of aerodynamics, thus investigating multi-fluid flows in a very different regime, dictated by high speed environments. We initially address one of the canonical problems in fluid mechanics, drop impact onto solid or liquid coated surfaces. This situation arises naturally on an aircraft in either rain or de-icing conditions. A new model for water catch on a surface is proposed, incorporating the violent splashing dynamics occurring in realistic conditions. The impingement of a large number of droplets ultimately leads to the formation of a liquid layer on the surface. We extend the powerful asymptotic framework of triple-deck theory to analyse changes in the flow separation process in the presence of an additional liquid. Flows past surface roughnesses and corners/flaps are discussed as practical examples.Open Acces