High‐dimensional uncertainty quantification for an electrothermal field problem using stochastic collocation on sparse grids and tensor train decompositions

Abstract

The temperature developed in bondwires of integrated circuits (ICs) is a possible source of malfunction, and has to be taken into account during the design phase of an IC. Due to manufacturing tolerances, a bondwire's geometrical characteristics are uncertain parameters, and as such their impact has to be examined with the use of uncertainty quantification (UQ) methods. Considering a stochastic electrothermal problem featuring twelve (12) bondwire-related uncertainties, we want to quantify the impact of the uncertain inputs onto the temperature developed during the duty cycle of an IC. For this reason we apply the stochastic collocation (SC) method on sparse grids (SGs), which is considered the current state-of-the-art. We also implement an approach based on the recently introduced low-rank tensor decompositions, in particular the tensor train (TT) decomposition, which in theory promises to break the curse of dimensionality. A comparison of both methods is presented, with respect to accuracy and computational effort.keywords-electrothermal field simulation, IC packaging, low-rank tensor decompositions, sparse grids, stochastic collocation, tensor trains, uncertainty quantification.

Determination of bond wire failure probabilities in microelectronic packages

“…Apart from the failure probability of a single wire, the system failure probability was defined in Section 3.5. With (6), it can be estimated for the here considered values of δ M. The results range between 0.0191 and 0.0200. However, we recall that the results for δ M = {2, 4, 6} have not converged as observed from Fig.…”

“…where L m are multivariate Lagrange polynomials. However, when a high number of uncertain input parameters is involved, more sophisticated methods such as sparse grids [7] and low-rank tensor approximations [6] are used. In the following, the failure region evaluated with g sur instead of g is called Γ sur F and the associated failure probability reads…”

“…It has been observed [6] that taking the maximum over all bond wires gives rise to a performance function that is not smooth and hence difficult to approximate with polynomials. For an efficient surrogate approximation, we thus calculate the failure probability P hyb F, j for every wire independently.…”

“…We can then focus on the iterative hybrid sampling algorithm which is the main topic of this work. For the high-dimensional case N = N bw = 12, we refer to [8] and [6]. Since there is not sufficient measurement data available to determine the probability density function, we choose y i to be uniformly distributed in the interval [µ − σ , µ + σ ] with µ = 0.17 and σ = 0.048.…”

Special issue: The 10th International Symposium on Electric and Magnetic Fields (EMF 2016)

“…The third major topic concerns fast solvers for electromagnetic applications with contributions on ultra weak variational formulations and the behavior of natural and finite element interpolation functions, domain decomposition methods for finite volume, finite element and boundary element schemes,() explicit time integration of eddy current problems, and the GPU acceleration of Maxwell solvers for differents applications. () The fourth major topic concerns the application of electromagnetic modelling, uncertainty quantification and model order reduction techniques to novel or challenging applications, with contributions to electrothermal field problems, electric machine modelling,() lightning‐produced electromagnetic fields, electronic circuits, and electrical capacitance tomography sensors …”