Coupled Simulation of Transient Heat Flow and Electric Currents in Thin Wires: Application to Bond Wires in Microelectronic Chip Packaging

This work addresses the simulation of heat flow and electric currents in thin wires. An important application is the use of bond wires in microelectronic chip packaging. The heat distribution is modeled by an electrothermal coupled problem, which poses numerical challenges due to the presence of different geometric scales. The necessity of very fine grids is relaxed by solving and embedding a 1D sub-problem along the wire into the surrounding 3D geometry. The arising singularities are described using de Rham currents. It is shown that the problem is related to fluid flow in porous 3D media with 1D fractures [C. D'Angelo, SIAM Journal on Numerical Analysis 50.1, pp. 194-215, 2012]. A careful formulation of the 1D-3D coupling condition is essential to obtain a stable scheme that yields a physical solution. Elliptic model problems are used to investigate the numerical errors and the corresponding convergence rates. Additionally, the transient electrothermal simulation of a simplified microelectronic chip package as used in industrial applications is presented.Comment: all numerical results can be reproduced by the Matlab code openly available at https://github.com/tc88/ETwireSi

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Multiscale Method for Elastic Wave Propagation in the Heterogeneous, Anisotropic Media

Seismic wave simulation in realistic Earth media with full wavefield methods is a fundamental task in geophysical studies. Conventional approaches such as the finite-difference method and the finite-element method solve the wave equation in geological models represented with discrete grids and elements. When the Earth model includes complex heterogeneities at multiple spatial scales, the simulation requires fine discretization and therefore a system with many degrees of freedom, which often exceeds current computational abilities. In this dissertation, I address this problem by proposing new multiscale methods for simulating elastic wave propagation based on previously developed algorithms for solving the elliptic partial differential equations and the acoustic wave equation. The fundamental motivation for developing the multiscale method is that it can solve the wave equation on a coarsely discretized mesh by incorporating the effects of fine-scale medium properties using so-called multiscale basis functions. This can greatly reduce computation time and degrees of freedom compared with conventional methods. I first derive a numerical homogenization method for arbitrarily heterogeneous, anisotropic media that utilizes the multiscale basis functions determined from a local linear elasticity equation to compute effective, anisotropic properties, and these equivalent elastic medium parameters can be used directly in existing elastic modeling algorithms. Then I extend the approach by constructing multiple basis functions using two types of appropriately defined local spectral linear elasticity problems. Given the eigenfunctions determined from local spectral problems, I develop a generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media in both continuous Galerkin (CG) and discontinuous Galerkin (DG) formulations. The advantage of the multiscale basis functions is they are model-dependent, unlike the predefined polynomial basis functions applied in conventional finite-element methods. For this reason, the GMsFEM can effectively capture the influence of fine-scale variation of the media. I present results for several numerical experiments to verify the effectiveness of both the numerical homogenization method and GMsFEM. These tests show that the effectiveness of the multiscale method relies on the appropriate choice of boundary conditions that are applied for the local problem in numerical homogenization method and on the selection of basis functions from a large set of eigenfunctions contained in local spectral problems in GMsFEM. I develop methods for solving both these problems, and the results confirm that the multiscale method can be powerful tool for providing accurate full wavefield solutions in heterogeneous, anisotropic media, yet with reduced computation time and degrees of freedom compared with conventional full wavefield modeling methods. Specially, I applied the DG-GMsFEM to the Marmousi-2 elastic model, and find that DG-GMsFEM can greatly reduce the computation time compared with continuous Galerkin (CG) FEM

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Multiscale aeroelastic modelling in porous composite structures

Driven by economic, environmental and ergonomic concerns, porous composites are increasingly being adopted by the aeronautical and structural engineering communities for their improved physical and mechanical properties. Such materials often possess highly heterogeneous material descriptions and tessellated/complex geometries. Deploying commercially viable porous composite structures necessitates numerical methods that are capable of accurately and efficiently handling these complexities within the prescribed design iterations. Classical numerical methods, such as the Finite Element Method (FEM), while extremely versatile, incur large computational costs when accounting for heterogeneous inclusions and high frequency waves. This often renders the problem prohibitively expensive, even with the advent of modern high performance computing facilities. Multiscale Finite Element Methods (MsFEM) is an order reduction strategy specifically developed to address such issues. This is done by introducing meshes at different scales. All underlying physics and material descriptions are explicitly resolved at the fine scale. This information is then mapped onto the coarse scale through a set of numerically evaluated multiscale basis functions. The problems are then solved at the coarse scale at a significantly reduced cost and mapped back to the fine scale using the same multiscale shape functions. To this point, the MsFEM has been developed exclusively with quadrilateral/hexahedral coarse and fine elements. This proves highly inefficient when encountering complex coarse scale geometries and fine scale inclusions. A more flexible meshing scheme at all scales is essential for ensuring optimal simulation runtimes. The Virtual Element Method (VEM) is a relatively recent development within the computational mechanics community aimed at handling arbitrary polygonal (potentially non-convex) elements. In this thesis, novel VEM formulations for poromechanical problems (consolidation and vibroacoustics) are developed. This is then integrated at the fine scale into the multiscale procedure to enable versatile meshing possibilities. Further, this enhanced capability is also extended to the coarse scale to allow for efficient macroscale discretizations of complex structures. The resulting Multiscale Virtual Element Method (MsVEM) is originally applied to problems in elastostatics, consolidation and vibroacoustics in porous media to successfully drive down computational run times without significantly affecting accuracy. Following this, a parametric Model Order Reduction scheme for coupled problems is introduced for the first time at the fine scale to obtain a Reduced Basis Multiscale Virtual Element Method. This is used to augment the rate of multiscale basis function evaluation in spectral acoustics problems. The accuracy of all the above novel contributions are investigated in relation to standard numerical methods, i.e., the FEM and MsFEM, analytical solutions and experimental data. The associated efficiency is quantified in terms of computational run-times, complexity analyses and speed-up metrics. Several extended applications of the VEM and the MsVEM are briefly visited, e.g., VEM phase field Methods for brittle fracture, structural and acoustical topology optimization, random vibrations and stochastic dynamics, and structural vibroacoustics

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described