We present a machine-learning strategy for finite element analysis of solid mechanics wherein we replace complex portions of a computational domain with a data-driven surrogate. In the proposed strategy, we decompose a computational domain into an “outer” coarse-scale domain that we resolve using a finite element method (FEM) and an “inner” fine-scale domain. We then develop a machine-learned (ML) model for the impact of the inner domain on the outer domain. In essence, for solid mechanics, our machine-learned surrogate performs static condensation of the inner domain degrees of freedom. This is achieved by learning the map from displacements on the inner-outer domain interface boundary to forces contributed by the inner domain to the outer domain on the same interface boundary. We consider two such mappings, one that directly maps from displacements to forces without constraints, and one that maps from displacements to forces by virtue of learning a symmetric positive semi-definite (SPSD) stiffness matrix. We demonstrate, in a simplified setting, that learning an SPSD stiffness matrix results in a coarse-scale problem that is well-posed with a unique solution. We present numerical experiments on several exemplars, ranging from finite deformations of a cube to finite deformations with contact of a fastener-bushing geometry. We demonstrate that enforcing an SPSD stiffness matrix drastically improves the robustness and accuracy of FEM–ML coupled simulations, and that the resulting methods can accurately characterize out-of-sample loading configurations with significant speedups over the standard FEM simulations.
Threaded fastener behavior can be an important aspect of complex component and system behavior, but there is no one-size-fits-all finite element analysis technique. Proper modeling of threaded fastener joints requires careful consideration of many details, from test setup and data acquisition to constitutive modeling and uncertainty quantification approaches. This report details analysis of a “mini-radax” bolted-joint exemplar where a Discrete-Direct uncertainty quantification approach is employed to evaluate margin of the component. The mini-radax geometry is tested to failure on a drop table, and single-coupon tests of individual fasteners serve as foundational data for the analysis. Analysis predictions complement the test data well and provide additional context for engineering decision-making.
A collaborative testing and analysis effort investigating the effects of threaded fastener size on load-displacement behavior and failure was conducted to inform the modeling of threaded connections. A series of quasistatic tension tests were performed on #00, #02, #04, #06 and #4 (1/4”) A286 stainless steel fasteners (NAS1351N00-4, NAS1352N02-6, NAS1352N04-8, NAS1352N06-10, and NAS1352N4-24, respectively) to provide calibration and validation data for the analysis portion of the study. The data obtained from the testing series reveals that the size of the fastener may influence the characteristic stress-strain response, as the failure strains and ultimate loads varied between the smaller (#00 and #02) and larger (#04, #06, and #4) fasteners. These results motivated the construction of high-fidelity finite element models to investigate the underlying mechanics of these responses. Two threaded fastener models, one with axisymmetric threads and the other with full 3D helical threads, were calibrated to subsets of the data to compare modeling approaches, analyze fastener material properties, and assess how well these calibrated properties extend to fasteners of varying sizes and if trends exist that can inform future best modeling practices. The modeling results are complemented with a microstructural analysis to further investigate the root cause of size effects observed in the experimentally obtained load-displacement curves. These analyses are intended to inform and guide reduced-order modeling approaches that can be incorporated in system level analyses of abnormal environments where modeling fidelity is limited and each component is not always testable, but models must still capture fastener behavior up to and including failure. This complimentary testing and analysis study identifies differences in the characteristic stress-strain response of varying sized fasteners, provides microstructural evidence to support these variations, evaluates our ability to extrapolate calibrated properties to different sized fasteners, and ultimately further educates the analysis community on the robustness of fastener modeling.
Multiple fastener reduced-order models and fitting strategies are used on a multiaxial dataset and these models are further evaluated using a high-fidelity analysis model to demonstrate how well these strategies predict load-displacement behavior and failure. Two common reduced-order modeling approaches, the plug and spot weld, are calibrated, assessed, and compared to a more intensive approach – a “two-block” plug calibrated to multiple datasets. An optimization analysis workflow leveraging a genetic algorithm was exercised on a set of quasistatic test data where fasteners were pulled at angles from 0° to 90° in 15° increments to obtain material parameters for a fastener model that best capture the load-displacement behavior of the chosen datasets. The one-block plug is calibrated just to the tension data, the spot weld is calibrated to the tension (0°) and shear (90°), and the two-block plug is calibrated to all data available (0°-90°). These calibrations are further assessed by incorporating these models and modeling approaches into a high-fidelity analysis model of the test setup and comparing the load-displacement predictions to the raw test data.
Multiple fastener reduced-order models and fitting strategies are used on a multiaxial dataset and these models are further evaluated using a high-fidelity analysis model to demonstrate how well these strategies predict load-displacement behavior and failure. Two common reduced-order modeling approaches, the plug and spot weld, are calibrated, assessed, and compared to a more intensive approach – a “two-block” plug calibrated to multiple datasets. An optimization analysis workflow leveraging a genetic algorithm was exercised on a set of quasistatic test data where fasteners were pulled at angles from 0° to 90° in 15° increments to obtain material parameters for a fastener model that best capture the load-displacement behavior of the chosen datasets. The one-block plug is calibrated just to the tension data, the spot weld is calibrated to the tension (0°) and shear (90°), and the two-block plug is calibrated to all data available (0°-90°). These calibrations are further assessed by incorporating these models and modeling approaches into a high-fidelity analysis model of the test setup and comparing the load-displacement predictions to the raw test data.
This study details a complimentary testing and finite element analysis effort to model threaded fasteners subjected to multiple loadings and loading rates while identifying modeling sensitivities that impact this process. NAS1352-06-6P fasteners were tested in tension at quasistatic loading rates and tension and shear at dynamic loading rates. The quasistatic tension tests provided calibration and validation data for constitutive model fitting, but this process was complicated by the difference in the conventional (global) and novel (local) displacement measurements. The consequences of these differences are investigated in detail by obtaining calibrated models from both displacement measurements and assessing their performance when extended to the dynamic tension and shear applications. Common quantities of interest are explored, including failure load, time-to-failure, and displacement-at-failure. Finally, the mesh sensitivities of both dynamic analysis models are investigated to assess robustness and inform modeling fidelity. This study is performed in the context of applying these fastener models into large-scale, full system finite element analyses of complex structures, and therefore the models chosen are relatively basic to accommodate this desire and reflect typical modeling approaches. The quasistatic tension results reveal the sensitivity and importance of displacement measurement techniques in the testing procedure, especially when performing experiments involving multiple components that inhibit local specimen measurements. Additional compliance from test fixturing and load frames have an increasingly significant effect on displacement data as the measurement becomes more global, and models must necessarily capture these effects to accurately reproduce the test data. Analysis difficulties were also discovered in the modeling of shear loadings, as the results were very sensitive to mesh discretization, further complicating the ability to analyze joints subjected to diverse loadings. These variables can significantly contribute to the error and uncertainty associated with the model, and this study begins to quantify this behavior and provide guidance on mitigating these effects. When attempting to capture multiple loadings and loading rates in fasteners through simulation, it becomes necessary to thoroughly exercise and explore test and analysis procedures to ensure the final model is appropriate for the desired application.