Computing error bounds improves damage prediction in materials models
A posteriori error bounds for cumulative energy fields in irreversible gradient damage
Computational Engineering, Finance, and Science
Summary
This paper looks at a way to measure errors in models that track how materials get damaged over time based on how they are loaded or stressed. The authors develop a method to bound the errors in calculating cumulative energy fields, which matter because the damage evolves irreversibly and depends on the path taken during loading. They use mathematical tools to enclose the computed results within guaranteed error margins, separating different sources of error like spatial or algebraic ones. Their approach is tested on many simulated examples, showing it can drastically reduce the uncertainty in predictions and improve computational efficiency.
What this means in practice
- •For structural engineers: Validate and certify numerical damage simulations in materials under complex loading histories to improve safety assessments.
- •For finite element software developers: Incorporate certified error bounds into gradient damage models to ensure simulation reliability and optimize computational resources.
Authors
Yubo Huang, Xiongyu Hu, Xin Lai, Cheng Lyu
Abstract
Cumulative energy fields in irreversible evolution depend on the loading path, even when terminal states and global work agree. We derive a posteriori bounds for these fields in scalar gradient damage with a prescribed driving density. A perturbation estimate for the incremental obstacle map yields a continuous loading history, which is enclosed by nondecreasing residual supersolutions. A primitive identity transfers this enclosure to the cumulative field. An exact endpoint-rule defect and outward arithmetic certify the output delivered by the numerical method. The analysis separates spatial, loading, and algebraic errors and gives sufficient conditions for certificate decay. Computable realizations use splines and verified supersolutions in one dimension and on two-dimensional rectangles. The production bounds cover the independently enclosed errors of all 417 saved histories from 28 adaptive calculations and all 25 fixed-parameter controls. In a narrow-localization case, the verified relative error decreases from approximately 158.78\% to below 2.125\% at a 5\% certified target. Source-oriented marking reduces accumulated matrix--vector work by 35.9\% and 51.9\% in two comparisons. A mechanically generated two-dimensional benchmark shows how auxiliary supersolutions sharpen constant barriers and enable acceptance of both loading orders.