Computational Physics Group

Karel Matous










Asynchronous Space-Time Domain Decomposition Method with Localized Uncertainty Quantification

W. Subber1 and  K. Matous1,2

1Center for Shock Wave-processing of Advanced Reactive Materials,
2Department of Aerospace and Mechanical Engineering,
University of Notre Dame, Notre Dame, IN, 46556, USA.


  The computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest. In this case, a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response. We present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions. In particular, we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions. As a result, computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required. To verify the numerical implementation, we consider elastic wave propagation in an axially loaded beam. Moreover, we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain. A projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method.


   We have developed an asynchronous space and time computational algorithm with localized uncertainty quantification. The framework is based on the recently proposed PASTA-DDM algorithm in conjunction with the intrusive polynomial chaos expansion for the stochastic representation that can be efficiently localized to selected subdomains. The proposed algorithm is customized to reduce the computational cost of uncertainty quantification in problems featuring localized phenomenon and described by stochastic PDE's (i.e., PDE's with random coefficients). In the region of interest, a rich stochastic model with high mesh and time resolutions is used, while a low dimensional stochastic representation with coarse spatial and temporal resolutions are allowed apart from the region of interest. We verify our algorithm with the traditional Monte-Carlo sampling technique and study different scenarios to reduce the computational cost of uncertainty quantification. For the elastic wave propagation problem, the algorithm shows second order convergence in the mean and standard deviation of both the displacement and velocity with respect to mesh size and time increment. Furthermore, the first order convergence rate is achieved with respect to the localization length. The application of PASTA-DDM-UQ to an impact problem shows that localizing the random variability of the material parameters under the impact zone gives the closest results to the case when uncertainty is considered in the entire domain.


    This work has been supported by the Department of Energy, National Nuclear Security Administration, under the Award No. DE-NA0002377 as part of the Predictive Science Academic Alliance Program II.

Download the paper here.

(c) 2017 Notre  Dame  and Dr. Karel Matous