Time-domain stochastic finite element simulation of uncertain seismic wave propagation through uncertain heterogeneous solids

• Stochastic Galerkin method with uncertain material parameter and forcing function.
• Non-Gaussian and heterogeneous/nonstationary input random field and process.
• Time domain solution algorithm.
• Sensitivity analysis with respect to input uncertainty parameters.

This paper presents an efficient numerical methodology in probabilistically solving the governing partial differential equation of solid mechanics with uncertainties in both the material parameter and forcing function in the time domain using the stochastic Galerkin approach. The methodology hypothesizes the input forcing function and the elastic modulus of the solid to be a nonstationary random process and a heterogeneous random field, respectively, and efficiently represents them in terms of multidimensional Hermite polynomial chaos – orthogonal and uncorrelated polynomials of zero-mean, unit variance Gaussian random variables – by taking advantage of the optimality of the Kosambi-Karhunen-Loève theorem. The methodology allows for any non-Gaussian marginal distributions and any arbitrary correlation structures for the input process and field. The solution random processes (displacement, velocity, and acceleration) are also represented in terms of multidimensional Hermite polynomial chaos expansions whose coefficients at each time step are estimated by applying a stochastic Galerkin projection with the time integration performed via the Newmark's method. The methodology is illustrated, keeping the geotechnical site response analysis in mind, with fully probabilistic, time-domain propagations of bedrock motions through an elastic soil deposit in one-dimension, and is verified using the Monte Carlo method. The effects of input uncertainty parameters of the soil modulus and bedrock motion on the simulated surface motion are also quantified through a parametric sensitivity study.