Efficient damage simulations under material uncertainties in a weakly-intrusive implementation

TL;DR

Uncertainty quantification for structures with inelastic damage is computationally expensive due to history-dependent behavior. The authors develop a weakly-intrusive implementation of time-separated stochastic mechanics (TSM) in Abaqus, exploiting a separated representation $y(t,\\boldsymbol{x},\\xi)=y^{(0)}(t,\\boldsymbol{x})+\\sum_{i=1}^{p}\\mathbf{I}^{(i)}(t,\\boldsymbol{x})\\xi^{i}$ (often $p=1$) to decouple stochastic from deterministic dynamics. For homogeneous material fluctuations, this reduces the required simulations to two deterministic FE runs plus post-processing to estimate $\\langle \\cdot \\rangle$ and $\\mathrm{Std}(\\cdot)$ of quantities like $d$, $\\boldsymbol{\\sigma}$, and $\\boldsymbol{F}$, with results matching a reference Monte Carlo solution while achieving speed-ups on the order of $10^2$–$10^3$. The method is demonstrated on viscous damage models in two 3D benchmarks (double-notched specimen and plate with a hole), showing accurate mean and standard deviation predictions and enabling industrial-scale uncertainty quantification in standard FE software.

Abstract

Uncertainty quantification is not yet widely adapted in the design process of engineering components despite its importance for achieving sustainable and resource-efficient structures. This is mainly due to two reasons: 1) Tracing the effect of uncertainty in engineering simulations is a computationally challenging task. This is especially true for inelastic simulations as the whole loading history influences the results. 2) Implementations of efficient schemes in standard finite element software are lacking. In this paper, we are tackling both problems. We are proposing a \rev{weakly}-intrusive implementation of the time-separated stochastic mechanics in the finite element software Abaqus. The time-separated stochastic mechanics is an efficient and accurate method for the uncertainty quantification of structures with inelastic material behavior. The method effectivly separates the stochastic but time-independent from the deterministic but time-dependent behavior. The resulting scheme consists only two deterministic finite element simulations for homogeneous material fluctuations in order to approximate the stochastic behavior. This brings down the computational cost compared to standard Monte Carlo simulations by at least two orders of magnitude while ensuring accurate solutions. In this paper, the implementation details in Abaqus and numerical comparisons are presented for the example of damage simulations.

Figures (8)

• Figure 1: Literature overview of methods for uncertainty quantification of structures with inelastic material behavior. Grey boxes represent existing techniques.
• Figure 2: Graphical overview of the time-separated stochastic mechanics approach.
• Figure 3: Implementation scheme of the time-separated stochastic mechanics in Abaqus. The user-defined subroutines vexternaldb, vusdfld, vumat enable the coupling of the simulations.
• Figure 4: Geometry of the double-notched specimen with the boundary conditions. Additionally, the orientation of a line for closer investigation of the results is indicated. All lengths are given in the unit $m$.
• Figure 5: Results for the damage function $f$ along the line as indicated in Figure \ref{['fig:DNSGeometry']} at three time points. The expectation $\langle f \rangle$ is given by a solid line, the results for $\langle f \rangle \pm \textrm{Std}(f)$ by dashed lines. The results of the TSM are given in blue and the results of MC in red.