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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.
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.
The quality of a model - and thus its predictive capabilities - is influenced by numerous uncertainties. They include possibly unknown boundary and initial conditions, noise in the data used for its calibration and uncertainties in the model itself. Here, the latter part is not only restricted to uncertain model parameters, but also refers to the choice of the model itself. Inferring these uncertainties in an automatic way allows for an adaption of the model to new data sets and for a reliable, reproducible model assessment. Note that similar concepts apply at the structural level, where a continuously updated digital twin allows virtual measurements at inaccessible positions of the structure and a simulation based lifetime prediction.
This work presents an inference workflow that describes the difference of measured data and simulated model responses with a generic interface that is independent from the specific model or even the geometry and can easily incorporate multiple data sources. A variational Bayesian inference algorithm is then used to a) calibrate a set of models to given data and to b) identify the best fitting one. The developed concepts are applied to a bridge Demonstrator equipped with displacement sensors, force sensors and a stereophotogrammetry system to perform a system identification of the material parameters as well as a real-time identification of a moving load.
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.
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.
Quasi-brittle materials exhibit strain softening. Their modeling requires regularized constitutive formulations to avoid instabilities on the material level. A commonly used model is the implicit gradient-enhanced damage model. For complex geometries, it still Shows structural instabilities when integrated with classical backward Euler schemes. An alternative is the implicit–explicit (IMPL-EX) Integration scheme. It consists of the extrapolation of internal variables followed by an implicit calculation of the solution fields. The solution procedure for the nonlinear gradient-enhanced damage model is thus transformed into a sequence of problems that are algorithmically linear in every time step. Therefore, they require one single Newton–Raphson iteration per time step to converge. This provides both additional robustness and computational acceleration. The introduced extrapolation error is controlled by adaptive time-stepping schemes. This paper introduced and assessed two novel classes of error control schemes that provide further Performance improvements. In a three-dimensional compression test for a mesoscale model of concrete, the presented scheme was about 40 times faster than an adaptive backward Euler time integration.
Accurate models for the long term behavior of concrete structures are important to ensure a durable and reliable design.
A variety of interacting phenomena, such as the loss of prestress, the degradation due to chemical reactions or creep and shrinkage, influence the fatigue resistance. Therefore, a reliable numerical model to predict the performance of concrete over its lifetime is required.
The presented fatigue model is an extension of a static damage model to allow easy coupling in a multiphysics context. The evolution equation of the damage driving variable is enhanced to allow damage growth below the static limit. The model is defined in the time domain and does not include the number of cycles as a parameter.
Thus, it can capture both static and cyclic failure.
Additionally, this allows calibrating the majority of the model parameters static experiments. The model is integrated by resolving each loading cycle, requiring about ten time steps per cycle. The high computational costs are handled via a time scale separation.
The short time scale describes one cycle with marginal changes in the internal variables. These changes are integrated along the large time scale of material deterioration. Various high-order time integration schemes are compared.
Wöhler curves relate loading amplitudes to the number of cycles that the material endures. They are used to validate the model against experimental data.
Fatigue models that accurately resolve the complex three-dimensional failure mechanisms of concrete are numerically expensive. Especially the calibration of fatigue parameters to existing Wöhler lines requires solving for thousands or millions of cycles and a naive cycle-by-cycle integration is not feasible.
The proposed adaptive cycle jump methods provide a remedy to this challenge.
They greatly reduce the numerical effort of fatigue simulations and provide the basis for a development of those models.
One of the main challenges regarding our civil infrastructure is the efficient operation over their complete design lifetime while complying with standards and safety regulations. Thus, costs for maintenance or replacements must be optimized while still ensuring specified safety levels. This requires an accurate estimate of the current state as well as a prognosis for the remaining useful life. Currently, this is often done by regular manual or visual inspections within constant intervals. However, the critical sections are often not directly accessible or impossible to be instrumented at all. Model‐based approaches can be used where a digital twin of the structure is set up. For these approaches, a key challenge is the calibration and validation of the numerical model based on uncertain measurement data. The aim of this contribution is to increase the efficiency of model updating by using the advantage of model reduction (Proper Generalized Decomposition, PGD) and applying the derived method for efficient model identification of a random stiffness field of a real bridge.”
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.