On damping created by heterogeneous yielding in the numerical analysis of nonlinear reinforced concrete frame elements

In the dynamic analysis of structural engineering systems, it is common practice to introduce damping models to reproduce experimentally observed features. These models, for instance Rayleigh damping, account for the damping sources in the system altogether and often lack physical basis. We report on an alternative path for reproducing damping coming from material nonlinear response through the consideration of the heterogeneous character of material mechanical properties. The parameterization of that heterogeneity is performed through a stochastic model. It is shown that such a variability creates the patterns in the concrete cyclic response that are classically regarded as source of damping

Fast r-adaptivity for multiple queries of heterogeneous stochastic material fields

We present an r-adaptivity approach for boundary value problems with randomly fluctuating material parameters solved through the Monte Carlo or stochastic collocation methods. This approach tailors a specific mesh for each sample of the problem. It only requires the computation of the solution of a single deterministic problem with the same geometry and the average parameter, whose numerical cost becomes marginal for large number of samples. Starting from the mesh used to solve that deterministic problem, the nodes are moved depending on the particular sample of mechanical parameter field. The reduction in the error is small for each sample but sums up to reduce the overall bias on the statistics estimated through the Monte Carlo scheme. Several numerical examples in 2D are presented.Peer ReviewedPostprint (author's final draft

Fast r-adaptivity for multiple queries of heterogeneous stochastic material fields

We present an r-adaptivity approach for boundary value problems with randomly fluctuating material parameters solved through the Monte Carlo or stochastic collocation methods. This approach tailors a specific mesh for each sample of the problem. It only requires the computation of the solution of a single deterministic problem with the same geometry and the average parameter, whose numerical cost becomes marginal for large number of samples. Starting from the mesh used to solve that deterministic problem, the nodes are moved depending on the particular sample of mechanical parameter field. The reduction in the error is small for each sample but sums up to reduce the overall bias on the statistics estimated through the Monte Carlo scheme. Several numerical examples in 2D are presented

Influence of the spatial correlation structure of an elastic random medium on its scattering properties

International audienceIn the weakly heterogeneous regime of elastic wave propagation through a random medium, transport and diffusion models for the energy densities can be set up. In the isotropic case, the scattering cross sections are explicitly known as a function of the wave number and the correlations of the Lamé parameters and density. In this paper, we discuss the precise influence of the correlation structure on the scattering cross sections, mean free paths and diffusion parameter, and separate that influence from that of the correlation length and variance. We also analyze the convergence rates towards the low-and high-frequency ranges. For all analyses, we consider five different correlation structures, that allow us to explore a wide range of behaviors. We identify that the controlling factors for the low-frequency behavior are the value of the Power Spectral Density Function (PSDF) and its first non-vanishing derivative at the origin. In the high frequency range, the controlling factor is the third moment of the PSDF (which may be unbounded)

Weak localization in radiative transfer of acoustic waves in a randomly-fluctuating slab

This paper concerns the derivation of radiative transfer equations for acoustic waves propagating in a randomly fluctuating slab (between two parallel planes) in the weak-scattering regime, and the study of boundary effects through an asymptotic analysis of the Wigner transform of the wave solution. These radiative transfer equations allow to model the transport of wave energy density, taking into account the scattering by random heterogeneities. The approach builds on the method of images, where the slab is extended to a full-space, with a periodic map of mechanical properties and a series of sources located along a periodic pattern. Two types of boundary effects, both on the (small) scale of the wavelength, are observed: one at the boundaries of the slab, and one inside the domain. The former impact the entire energy density (coherent as well as incoherent) and is also observed in half-spaces. The latter, more specific to slabs, corresponds to the constructive interference of waves that have reflected at least twice on the boundaries of the slab and only impacts the coherent part of the energy density.Comment: 7 figure

Influence of periodically fluctuating material parameters on the stability of explicit high-order spectral element methods

This paper presents a mathematical analysis of the stability of high-order spectral elemetns with explicit time marching for the solution of acoustic wave propagation in heterogeneous media. Using a Von Neumann stability analysis, the origin for the instability that is often observed when the stability limit derived for homegenous media is adapted for heterogeneous media. In large scale projects such as the simulation of seismic wave propagation, this known issue results either in simulations that are too expensive due to a conservative choice of the time step or to simulations that lead to unstable results due to the inability to predict the suitability of the time step. The proposed work not only explain the mathematical origin of this problem but also derives exact stability bounds are given for cases when the heterogeneity is periodic

Multi-level Neural Networks for Accurate Solutions of Boundary-Value Problems

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy, classical discretization methods such as the finite element methods, remains a significant challenge. Deep learning methods usually struggle to reliably decrease the error in their approximate solution. A new methodology to better control the error for deep learning methods is presented here. The main idea consists in computing an initial approximation to the problem using a simple neural network and in estimating, in an iterative manner, a correction by solving the problem for the residual error with a new network of increasing complexity. This sequential reduction of the residual of the partial differential equation allows one to decrease the solution error, which, in some cases, can be reduced to machine precision. The underlying explanation is that the method is able to capture at each level smaller scales of the solution using a new network. Numerical examples in 1D and 2D are presented to demonstrate the effectiveness of the proposed approach. This approach applies not only to physics informed neural networks but to other neural network solvers based on weak or strong formulations of the residual.Comment: 34 pages, 20 figure

A stable XFEM formulation for multi-phase problems enforcing the accuracy of the fluxes through Lagrange multipliers

When applied to diffusion problems in a multi-phase setup, the popular extended finite element method (XFEM) strategy suffers from an inaccurate representation of the local fluxes in the vicinity of the interface. The XFEM enrichment improves the global quality of the solution but it is not enforcing any local feature to the fluxes. Thus, the resulting numerical fluxes in the vicinity of the interface are not realistic, in particular when conductivity ratios between the different phases are very high. This paper introduces an additional restriction to the XFEM formulation aiming at properly reproducing the features of the local fluxes in the transition zone. This restriction is implemented through Lagrange multipliers, and the stability of the resulting mixed formulation is tested satisfactorily through the Chapelle–Bathe numerical procedure. Several examples are presented, and the solutions obtained show a spectacular improvement with respect to the standard XFEM.Peer ReviewedPostprint (author’s final draft

Assessing the influence of around-source deep crustal heterogeneities on the seismic wave propagation by 3-D broad-band numerical modelling

Tectonic and seismic activities induce crustal rock fracturing in the immediate surroundings of buried fault discontinuities. The heterogeneous nature of this 3-D scatter distribution leaves a high-frequency footprint on the recorded wave-field at surface. This study represents an attempt to clarify the complex relationship between the propagated wave-field and the statistical properties of the regional medium. This objective is pursued by inspecting the broadband (0-25Hz) synthetic wave-forms obtained by sourceto-site 3-D numerical simulations in regional-size scenarios. The presented numerical study compares the major differences obtained when including heterogeneous properties in the model. Coda-waves are analysed and the high-frequency attenuation is assessed by computing the k coefficient for different source-station configurations.This work was performed using HPC resources from the Mésocentre computing center of CentraleSupélec and École Normale Supérieure Paris-Saclay (http://mesocentre.centralesupelec.fr/)