Use Of Random Fields Research Articles

Advanced numerical modelling for predicting residual compressive strength of corroded stiffened plates

An advanced methodology for predicting the residual compressive strength of corroded stiffened plates is developed here using the non-linear finite element method. The non-uniform loss of a plate thickness is accounted for on a macro-scale. In contrast, mechanical properties are changed using the constitutive model to reflect the corrosion degradation impact on a micro-scale. Three different stiffened plate thicknesses are considered, and ultimate compressive capacity is analysed for different severity of corrosion degradation. First, the deterministic analysis is performed, and numerical results are validated against the experiment. Then, the corrosion fields are modelled with the use of random fields. Different statistical characteristics of the generated random fields are investigated to identify their impact on the resulting structural behaviour. It was found that severe corrosion degradation could cause an excessive reduction of structural capacity even at 50%. Finally, the results of the present study are compared with the already studied corrosion models showing a highly non-conservative solution of the latter, where the plate thickness is reduced only due to the corrosion degradation progress.

Numerical Simulation of the Elastic–Ideal Plastic Material Behavior of Short Fiber-Reinforced Composites Including Its Spatial Distribution with an Experimental Validation

For the numerical simulation of components made of short fiber-reinforced composites, the correct prediction of the deformation including the elastic and plastic behavior and its spatial distribution is essential. When using purely deterministic modeling approaches, the information of the probabilistic microstructure is not included in the simulation process. One possible approach for the integration of stochastic information is the use of random fields. In this study, numerical simulations of tensile test specimens were conducted utilizing a finite deformation elastic–ideal plastic material model. A selection of the material parameters covering the elastic and plastic domain are represented by cross-correlated second-order Gaussian random fields to incorporate the probabilistic nature of the material parameters. To validate the modeling approach, tensile tests until failure were carried out experimentally, which confirmed the assumption of the spatially distributed material behavior in both the elastic and plastic domain. Since the correlation lengths of the random fields cannot be determined by pure analytic treatments, additionally numerical simulations were performed for different values of the correlation length. The numerical simulations endorsed the influence of the correlation length on the overall behavior. For a correlation length of 5 mm, a good conformity with the experimental results was obtained. Therefore, it was concluded that the presented modeling approach was suitable to predict the elastic and plastic deformation of a set of tensile test specimens made of short fiber-reinforced composite sufficiently.

Correlation analysis of the elastic‐ideal plastic material behavior of short fiber‐reinforced composites

AbstractFor the numerical simulation of short fiber‐reinforced composites and the correct analysis of the deformation, information about the plastic behavior and its spatial distribution is essential. When using purely deterministic modeling approaches information of the probabilistic microstructure is not included in the simulation process. One possible approach for the integration of stochastic information is the use of random fields, which requires information about the correlation structure of all material input parameters. In this study the correlation structure for finite strain elasto‐plastic material behavior of short fiber‐reinforced composites is analyzed. This approach combines the use of already established procedures for linear‐elastic material behavior with a homogenization method for plasticity. The obtained results reveal a complex correlation structure, which is approximated with triangle and exponential correlation functions influenced by the window size. Due to the dependence of the hyperelastic and plastic material parameters on the fiber mass fraction, the strain‐energy density function coefficients are cross‐correlated with the yield strength of the composite. With this knowledge at hand, in a subsequent work numerical simulations of tensile tests are conducted that cover the elastic and plastic domain and include spatially distributed material properties.

Probabilistic analysis of concrete cracking using stochastic finite element methods: application to nuclear containment buildings at early age

In the case of quasi-homogeneously applied tensile loads, the intrinsic scattering of concrete properties leads to spatially random strain localization, crack initiation and propagation. The modelling of such spatial randomness, in the case of Equivalent-Homogeneous-Material Finite Elements based approaches, can be achieved thanks to the use of Random Fields. However, when aiming at probabilistic analyses, numerous realizations are required which induces a hefty computational time and restricts their applicability to the modelling of large concrete structures. In this contribution, an original probabilistic coupling strategy is provided based on non-intrusive Stochastic Finite Elements Methods. It consists of defining an explicit Surface Response of the cracking patterns expressed in terms of the most influential inputs using an Adaptive Surface Response Method combined to a Polynomial Chaos Expansion Method. Direct Monte Carlo Method is then applied—to the explicit Surface Response of the cracking patterns—to achieve Global Sensitivity Analysis, Uncertainties Quantification and probabilistic modelling at a reasonable cost. The defined strategy is validated based on a Representative Structural Volume of a 1:3 scaled experimental Containment Building at early age using a weakly coupled thermo-mechanical model. As a result, the study quantifies the effect of the most influential parameters (the Young’s modulus—the tensile strength—the coefficients of thermal expansion and autogenous shrinkages) on concrete cracking at early age and provides accurate numerical prediction of the cracking patterns (cracks’ number, opening and spacing values) observed on site and their frequencies.

Risk-averse structural topology optimization under random fields using stochastic expansion methods

This work proposes a level-set based approach for solving risk-averse structural topology optimization problems considering random field loading and material uncertainty. The use of random fields increases the dimensionality of the stochastic domain, which poses several computational challenges related to the minimization of the Excess Probability as a measure of risk awareness. This problem is addressed both from the theoretical and numerical viewpoints. First, an existence result under a typical geometrical constraint on the set of admissible shapes is proved. Second, a level-set continuous approach to find the numerical solution of the problem is proposed. Since the considered cost functional has a discontinuous integrand, the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic Polynomial Chaos (PC) approach with a Monte-Carlo (MC) sampling method for uncertainty propagation. Furthermore, to address the increment of dimensionality induced by the random field, an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the PC coefficients. A key point is that the non-intrusive nature of such an approach facilitates the use of High Performance Computing (HPC) to alleviate the computational burden of the problem. Several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal.

Analysis, Simulation and Prediction of Multivariate Random Fields with PackageRandomFields

Modeling of and inference on multivariate data that have been measured in space, such as temperature and pressure, are challenging tasks in environmental sciences, physics and materials science. We give an overview over and some background on modeling with crosscovariance models. The R package RandomFields supports the simulation, the parameter estimation and the prediction in particular for the linear model of coregionalization, the multivariate Matern models, the delay model, and a spectrum of physically motivated vector valued models. An example on weather data is considered, illustrating the use of RandomFields for parameter estimation and prediction.

Homogenization of geomaterials containing voids by random fields and finite elements

The paper describes the use of random fields and finite elements to assess the influence of porosity and void size on the effective elastic stiffness of geomaterials. A finite element model is developed involving “tied freedoms” that allows analysis of an ideal block of materials leading to direct evaluation of the effective Young’s modulus and Poisson’s ratio. The influence of block size and representative volume elements (RVE) are discussed. The use of random fields and Monte-Carlo simulations deliver a mean and standard deviation of the elastic parameters that lead naturally to a probabilistic interpretation. The methodology is extended to a foundation problem involving a footing on an elastic foundation containing voids. The approach enables estimates to be made of the probability of excessive settlement.

Estimation of forming limit diagrams by the use of the finite element method and Monte Carlo simulation

A stochastic finite element-based approach for forming limit calculations of sheets is proposed and evaluated. Material inhomogeneities are represented by spatial thickness variations of the sheet which are modelled by the use of random fields. The effects of changing the smoothness, wavelengths, amplitude and anisotropy of the field realisations on the forming limit diagram are investigated. Further, the effects of the patch size and plastic anisotropy on the forming limit diagram are studied. The assumed thickness variations result in a quite wide scatter band, and changes of the characteristics of the thickness field result in changes of the shape and variance of the forming limit diagram.

Nonlinear plate aeroelastic response with uncertain stiffness and boundary conditions

The probabilistic response of a nonlinear panel in supersonic flow is investigated using a new computational methodology. The aeroelastic system is modelled by coupling the von Karman plate equations with piston theory aerodynamics. Baseline deterministic results are compared to published data to establish validity of the aeroelastic model. Uncertainties in modulus of elasticity and boundary conditions (BCs) are considered, with their impact on panel response being quantified in terms of limit-cycle oscillation (LCO) onset pressure and panel amplitude. The panel's uncertain BCs are modelled using rotational springs in place of ideally fixed boundary conditions. The panel's uncertain elastic modulus is modelled as a second-order, spatially homogeneous, isotropic, random field. The boundary spring stiffness values are spatially constant for each random realization assuming either a uniform or Weibull distribution. Probabilistic response distributions are obtained through Monte Carlo simulation (MCS) for selected values of flowfield dynamic pressure. MCS is run for the case of random modulus only, as well as random modulus combined with either random BC distribution. It is observed that uncertainties have the greatest nonlinear influence on LCO amplitude near the deterministic point of LCO onset. Furthermore, panel response is much more sensitive to location of the modulus fields' extreme values than the correlation length used to generate the field. Novel aspects of the computational methodology include (1) the use of random fields (as opposed to a random variable) to model uncertainty in the elastic modulus, and (2) the use of proper orthogonal decomposition (POD) to yield insight into the local factors that govern the global response of the panel, permitting enhanced response prediction based on the nature of the random field realizations.

A lattice model for self-avoiding polymers with controlled length distributions. II. Corrections to Flory–Huggins mean field

We continue the analysis of the lattice spin-field theory which is introduced in paper I and which formally gives the exact entropy for a set of completely flexible self-avoiding polymers on a lattice. The model allows for arbitrary chain length distributions and arbitrary polymer volume fractions. Use of random fields produces a field theory from which Flory–Huggins results are recovered in the mean field limit. Here we recast the mean field formulation to allow for the rigorous and systematic evaluation of corrections by means of a cluster expansion. We then calculate corrections to the mean field directly up to fourth order and provide a systematic diagrammatic method for evaluating general order corrections. Our results illustrate the polymer concentration dependence of the corrections to the Flory–Huggins mean field results as well as provide the origins for the entropic contribution to the Flory χ parameter and its concentration dependence. Generalizations to treat rods and semiflexible polymers will be given elsewhere.