Kerstin Weinberg

A strategy for damage assessment of thermally stressed copper vias in microelectronic printed circuit boards

Abstract The thermal fatigue of plated-through vias remains a subject of concern, particularly when exposed to the high operating temperatures associated with automotive applications. In this paper the performance of different types of copper vias in different positions of a printed circuit board is analyzed. To this end a two-scale finite element analysis under the loading conditions of thermal cycling is employed. A new material model for the electrolytically deposited copper accounts for large elastic and plastic deformations and, additionally, for the growth of pores within the material. It is common practice to extrapolate the plastic straining computed within few steps of thermal cycling by means of a Coffin–Manson-Equation. We critical examine this strategy and point out, that a certain number of about 20 cycling steps is necessary to obtain meaningful extrapolations. Furthermore, an extrapolation of the computed porosity up to critical values allows similar conclusions. The presented strategy can serve as a predictive tool for plated through holes and vias and can reduce the need of repetitive experimental failure tests.

Condensation and Growth of Kirkendall Voids in Intermetallic Compounds

A model for the simulation of Kirkendall voiding in metallic materials is presented based on vacancy diffusion, elastic-plastic and rate-dependent deformation of the material. Starting with a phenomenological explanation of the Kirkendall effect we briefly discuss the consequences on the reliability of microelectronic components. Then, a constitutive model for void nucleation and growth is introduced, which can be used to predict the temporal development of voids in solder joints during thermal cycling. We present numerical studies and discuss the potential of the results for the failure analysis of joining connections.

Adaptive mixed finite element method for Reissner-Mindlin plate

Abstract Mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of Reissner–Mindlin plate models. Automatic adaptive mesh-refining algorithms are an important tool to improve the approximation behavior of the finite element discretization. In this paper, a reliable and robust residual-based a posteriori error estimate is derived, which evaluates a t -depending residual norm based on results in [D. Arnold, R. Falk, R. Winther, Math. Modell. Numer. Anal. 31 (1997) 517–557]. The localized error indicators suggest an adaptive algorithm for automatic mesh refinement. Numerical examples prove that the new scheme is efficient.

Kidney damage in extracorporeal shock wave lithotripsy: a numerical approach for different shock profiles.

In shock-wave lithotripsy—a medical procedure to fragment kidney stones—the patient is subjected to hypersonic waves focused at the kidney stone. Although this procedure is widely applied, the physics behind this medical treatment, in particular the question of how the injuries to the surrounding kidney tissue arise, is still under investigation. To contribute to the solution of this problem, two- and three-dimensional numerical simulations of a human kidney under shock-wave loading are presented. For this purpose a constitutive model of the bio-mechanical system kidney is introduced, which is able to map large visco-elastic deformations and, in particular, material damage. The specific phenomena of cavitation induced oscillating bubbles is modeled here as an evolution of spherical pores within the soft kidney tissue. By means of large scale finite element simulations, we study the shock-wave propagation into the kidney tissue, adapt unknown material parameters and analyze the resulting stress states. The simulations predict localized damage in the human kidney in the same regions as observed in animal experiments. Furthermore, the numerical results suggest that in first instance the pressure amplitude of the shock wave impulse (and not so much its exact time-pressure profile) is responsible for damaging the kidney tissue.

Application of operator-scaling anisotropic random fields to binary mixtures

In modern technical applications various multiphase mixtures are used to meet demanding mechanical, chemical and electrical requirements. To understand their structural properties as continuous macroscopic materials, it is important to capture the microstructure of these mixtures. Due to their vast range of applications multicomponent systems are subjected to microstructural changes such as phase separation and coarsening. Therefore the ultimate microstructural arrangement depends on the systems configuration and on exterior driving forces. In addition to this, random physical imperfections within the material and random noise in the exterior thermodynamic fields influence in essence the microstructural evolution. Since all physical processes are subjected to a certain degree of random inhomogeneity under realistic conditions, the influence of random phenomena cannot be neglected in modern physical models. An advanced mathematical description and an implementation of these stochastic processes are required to adapt simulation results based on deterministic mathematical models to experimental observations. In our contribution we will present an operator-scaling anisotropic random field embedded in the Cahn–Hilliard phase-field model to describe the phase evolution in a binary mixture. The arising nonlinear diffusion equation will be solved numerically in the innovative framework of the isogeometric finite element method. To illustrate the flexibility and versatility of our approach, numerical and experimental results for a eutectic Sn-Pb alloy are contraposed. This is the first time that the microstructural evolution in a multicomponent system has been associated with operator-scaling anisotropic random fields. Due to its enormous potential as an essential ingredient in stochastic mathematical and physical modeling it is only a matter of time until these processes will become prevalent in engineering applications.

Mesoscopic Modeling for Continua with Pores: Application in Biological Soft Tissue

Abstract In this work, the damage in biological soft tissue induced by bubble cavitation is investigated. A typical medical procedure with such damaging side effects is the kidney stone fragmentation by shock-wave lithotripsy. We start with a mesoscopic continuum model that allows the consideration of microstructural information within the macroscopic balance equations. An evolution equation for the temporal development of the bubble distribution function is derived. Furthermore, the constitutive relations of bubble expansion are deduced by means of a spherical shell model. Numerical simulations are presented for a typical soft tissue material and different definitions for a damage parameter are discussed.

An adaptive finite element approach for a mixed Reissner–Mindlin plate formulation

Abstract We use a recently introduced mixed finite element method for the Reissner–Mindlin plate model and study its numerical properties in practical use. After introducing a technique for stabilizing the discretisation, we derive an a posteriori error estimate and study the question of reliability. Numerical examples prove that it enables an efficient adaptive scheme.

Numerical investigation of diffusion induced coarsening processes in binary alloys

Microscopic small solder joints are part of most microelectronic packages. They show a variety of micro-morphological changes such as decomposition into phases and phase coarsening. In order to analyze the evolution of the alloy by means of a diffusion theory of heterogeneous solid mixtures we apply an extended Cahn-Hilliard phase-field model. The Cahn-Hilliard equation involves a bipotential operator and, accordingly, fourth-order spatial derivatives. In our contribution we discretize the diffusion equation spatially by means of rational B-spline finite element basis functions. To illustrate the versatility of this approach numerical simulations of phase decomposition and coarsening controlled by diffusion and by mechanical loading are discussed and compared with experimental results.

Mesoscopic Modeling for Continua with Pores: Dynamic Void Growth in Viscoplastic Materials

Abstract The temporal development of arbitrarily distributed voids in a viscoplastic material under different loading regimes is investigated. For this reason, we make use of a mesoscopic continuum model extending the classical space–time domain of continuum mechanics. This extended domain requires a reformulation of the classical balance equations as well as the consideration of additional constitutive quantities. Furthermore, a mesoscopic distribution function is formulated to describe the temporal evolution of different void regimes. Here, we assume a spherical shell model for the porous composites and elaborate all required steps in order to describe load-induced void growth in a metal-like matrix. We conclude with some exemplary results that confirm experimental observations of dynamical fracture.

An adaptive pNh‐technique for global‐local finite element analysis

The paper presents a new technique for a compatible transition from a h‐refined to a p‐refined finite element mesh. At one or more faces of particularly designed pNh‐transition elements a low order h‐discretization may be combined with a usual p‐mesh in the other parts of the elements. The pNh‐elements are conform finite elements which can be applied in an adaptive scheme controlled by a residue based error estimate. Typical applications which require strongly a local mesh refinement within a p‐finite element mesh are, e.g. the approximation of high gradients and the determination of contact areas. Numerical examples demonstrate the efficiency of the pNh‐element technique for such problems.