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Numerical models are an essential tool in predicting and monitoring the behavior of civil structures. Inferring the model parameters is a challenging tasks as they are often measured indirectly and are affected by uncertainties. Digital twins couple those models with real-world data and can introduce additional, systematic sensor uncertainties related to the sensor calibration, i.e. uncertain offsets and calibration factors.
In this work, the challenges of data processing, parameter identification, model selection and damage detection are explored using a lab-scale cable stayed bridge demonstrator. By combining force measurements in the cables with displacement measurements from both laser and stereo-photogrammetry systems, the elastic parameters of a three-dimensional finite element beam model are inferred.
Depending on the number of sensors and the number of datasets used, parametrizing the sensor offsets and factors, leads to model with over 100 parameters. With a real-time solution of the problem in mind, a highly efficient analytical variational Bayesian approach is used to solve it within seconds. An analysis of the required assumptions and limitations of the approach, especially w.r.t. to the computed evidence, is provided by a comparison with dynamic nested sampling in a simplified problem.
Finally, by inferring the value of additional damage parameters along the bridge, the method is successfully used to detect the location of an artificially introduced weak spot in the demonstrator bridge.
Numerical models built as virtual-twins of a real structure (digital-twins) are considered the future ofmonitoring systems. Their setup requires the estimation of unknown parameters, which are not directly measurable. Stochastic model identification is then essential, which can be computationally costly and even unfeasible when it comes to real applications. Efficient surrogate models, such as reduced-order method, can be used to overcome this limitation and provide real time model identification. Since their numerical accuracy influences the identification process, the optimal surrogate not only has to be computationally efficient, but also accurate with respect to the identified parameters. This work aims at automatically controlling the Proper Generalized Decomposition (PGD) surrogate’s numerical accuracy for parameter identification. For this purpose, a sequence of Bayesian model identification problems, in which the surrogate’s accuracy is iteratively increased, is solved with a variational Bayesian inference procedure. The effect of the numerical accuracy on the resulting posteriors probability density functions is analyzed through two metrics, the Bayes Factor (BF) and a criterion based on the Kullback-Leibler (KL) divergence. The approach is demonstrated by a simple test example and by two structural problems. The latter aims to identify spatially distributed damage, modeled with a PGD surrogate extended for log-normal random fields, in two different structures: a truss with synthetic data and a small, reinforced bridge with real measurement data. For all examples, the evolution of the KL-based and BF criteria for increased accuracy is shown and their convergence indicates when model refinement no longer affects the identification results.
Numerical simulators, such as finite element models, have become increasingly capable of predicting the behaviour of structures and components owing to more sophisticated underlying mathematical models and advanced computing power. A common challenge lies, however, in calibrating these models in terms of their unknown/uncertain parameters. When measurements exist, this can be achieved by comparing the model response against measured data. Besides uncertain model parameters, phenomena like damage can give rise to further uncertainties; in particular, quasi-brittle materials, like concrete, experience damage in a heterogeneous manner due to various imperfections, e.g. in geometry and boundary conditions. This hardens an accurate prediction of the damaged behaviour of real structures that comprise such materials.
In this study, which draws from a data-driven approach, we use the force-version of the finite element model updating method (FEMU-F) to incorporate measured displacements into the identification of the damage parameters, in order to cope with heterogeneity. In this method, instead of conducting a forward evaluation of the model and comparing the model response (displacements) against the data, we impose displacements to the model and compare the resulting force residuals with measured reaction forces. To account for uncertainties in the measurement of displacements, we endow this approach with a penalty term, which reflects the discrepancy between measured and imposed displacements, where the latter is assumed as unknown random variables to be identified as well. A Variational Bayesian approach is used as an approximating tool for computing posterior parameters. The underlying damage model considered in this work is a gradient-enhanced damage model.
We first establish the identification procedure through two virtual examples, where synthetic data (displacements) are generated over a certain spatially-dense set of points over the domain. The procedure is then validated on an experimental case-study; namely a 3-point bending experiment with displacement measurements resulting from a digital image correlation (DIC) analysis.
The problem of polydisperse sphere packings is applied to concrete mesoscale geometries in finite sized specimens. Realistic sphere diameter distributions are derived from concrete grading curves. An event-driven molecular dynamics simulation using growing particles is introduced. Compared to the widely used random sequential addition algorithm, it reaches denser aggregate packings and saves computation time at high volume fractions.
A minimal distance between particles strongly influences the maximum aggregate content. It is essential to obtain undistorted elements when meshing the geometry for finite element simulations. The algorithm maximizes this value and produces meshable concrete mesostructures with more than 70% aggregate content.
Isotropic damage models are widely used for the finite element simulation of softening materials, e.g. in mesoscale simulations of concrete. Regularization techniques must be employed to obtain a physically meaningful fracture energy upon mesh refinement.
In regularized local damage models the strains localize in single elements allowing them to represent weak or strong discontinuities. In implicit integration schemes, these models can exhibit convergence Problems caused by an ill-conditioned tangent stiffness. This corresponds to the loss of ellipticity of the local rate equilibrium equations.
Oliver et al. developed the implicit/explicit (IMPL-EX) integration scheme which overcomes These problems in local damage models. The internal damage driving variable is extrapolated based on previous implicitly determined values. This provides two main benefits: First, it always results in a symmetric positive semi-definite algorithmic stiffness matrix which precludes ill-posedness. Second, the system becomes incrementally linear and converges in one Newton-Raphson iteration. Even though the IMPL-EX algorithm, like explicit algorithms in general, requires smaller time steps than implicit schemes to obtain the same accuracy, it leads to a computational speedup.
The gradient enhanced damage model by Peerlings is a nonlocal damage model that provides the regularization by limiting the curvature of the damage-driving strains. These models do not lose their ellipticity. However, structural instabilities often require tiny time steps and many iterations to obtain convergence. Here, the second aspect of the IMPL-EX scheme reduces the computational costs. This is shown in simulations of the complex geometry of concrete mesostructures, where only the gradient enhanced matrix material and linear elastic aggregates are considered.
With regard to future mesoscale simulations, the remaining component of the mesoscopic structure, the interfacial transition zone and its degradation, has to be included. This adds a local damage model to the nonlocal problem. Thus, an IMPL-EX implementation has to be provided for both models to benefit from the increase of robustness and performance.
Concrete is a complex material. Its properties evolve over time, especially at early age, and are dependent on environmental conditions, i.e. temperature and moisture conditions, as well as the composition of the material.
This leads to a variety of macroscopic phenomena such as hydration/solidification/hardening, creep and shrinkage, thermal strains, damage and inelastic deformations. Most of these phenomena are characterized by specific set of model assumptions and often an additive decomposition of strains into elastic, plastic, shrinkage and creep components is performed. Each of these phenomena are investigated separately and a number of respective independent models have been designed. The interactions are then accounted for by adding appropriate correction factors or additional models for the particular interaction. This paper discusses the importance of reconsider even in the experimental phase the model assumptions required to generalize the experimental data into models used in design codes. It is especially underlined that the complex macroscopic behaviour of concrete is strongly influenced by its multiscale and multiphyscis nature and two examples (shrinkage and fatigue) of interacting phenomena are discussed.
A key limitation of the most constitutive models that reproduce a Degradation of quasi-brittle materials is that they generally do not address issues related to fatigue. One reason is the huge computational costs to resolve each load cycle on the structural level. The goal of this paper is the development of a temporal Integration scheme, which significantly increases the computational efficiency of the finite element method in comparison to conventional temporal integrations.
The essential constituent of the fatigue model is an implicit gradient-enhanced formulation of the damage rate. The evolution of the field variables is computed as amultiscale Fourier series in time.On a microchronological scale attributed to single cycles, the initial boundary value problem is approximated by linear BVPs with respect to the Fourier coefficients. Using the adaptive cycle jump concept, the obtained damage rates are transferred to a coarsermacrochronological scale associated with the duration of material deterioration. The performance of the developedmethod is hence improved due to an efficient numerical treatment of the microchronological problem in combination with the cycle jump technique on the macrochronological scale. Validation examples demonstrate the convergence of the obtained solutions to the reference simulations while significantly reducing the computational costs.
Simulating high-cycle fatigue with continuum models offers the possibility to model stress-redistributions, consider 3Dstress states and simplifies extensions to multi-physics problems. The computational cost of conventional cycle-by-cycle time integrations is reduced by reformulating the fatigue problem as an ordinary differential equation for the material state and solving it with high-order adaptive time integration schemes. The computational cost of calculating the Change of the material state in one cycle is further reduced by a high-order fatigue-specific time integration. The approach is exemplarily demonstrated for a fatigue extension of the implicit gradient-enhanced damage model in 3D and compared to experimental Wöhler lines.
Concrete is a complex material and can be modeled on various spatial and temporal scales. While simulations on coarse scales are practical for engineering applications, a deeper understanding of the material is gained on finer scales. This is at the cost of an increased numerical effort that can be reduced by the three methods developed and used in this work, each corresponding to one publication.
The coarse spatial scale is related to fully homogenized models. The material is described in a phenomenological approach and the numerous parameters sometimes lack a physical meaning. Resolving the three-phase mesoscopic structure consisting of aggregates, the mortar matrix and the interfaces between them allow to describe similar effects with simpler models.