Discretization Size Research Articles

Discrete modeling of elastic heterogeneous media

Discrete models provide advantages in simulating fracture in quasi-brittle materials due, in part, to their simplicity in representing cracking and other forms of displacement discontinuity. However, the stress analyses that form the basis for fracture simulation are complicated by difficulties in modeling the Poisson effect and other aspects of elastic behavior. The capabilities of Voronoi-cell lattice models, which are a form of particle-based lattice model, for elastic stress analysis are evaluated. It is found that the conventional means for representing the Poisson effect in particle-based lattice models result in spatially correlated stress oscillations that, at first glance, mimic the effects of material heterogeneity. The correlation length is dependent on discretization size. Alternatively, material heterogeneity can be introduced into elastically uniform lattice models via random assignments of material properties, independent of mesh size and geometry.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

Influence of Image Discretization and Patch Size on Microbubble Localization Precision.

For ultrasound localization microscopy, the localization of microbubbles (MBs) is an essential part to obtain super-resolved maps of the vasculature. This paper analyzes the impact of image discretization and patch size on the precision of different MB localization methods to reconcile different observations from previous studies, provide an estimate of feasible localization precision, and derive guidelines for an optimal parameter selection. For this purpose, images of MBs were simulated with Gaussian point-spread functions (PSF) of varying width parameter σ at randomly generated subpixel positions, and Rician distributed noise was added. Four localization methods were tested on patches of different sizes (number of pixels N × N): Gaussian fit, radial symmetry method, calculation of center of mass, and peak detection. Additionally, the Cramér-Rao lower bound (CRLB) for the given estimation problem was calculated. Our results show that the localization precision is strongly influenced by the ratio of the PSF width parameter σ to the pixel size Δ, as well as the patch size N. The best parameter combination depends on the localization method. Generally, very small σ/Δ ratios as well as large σ/Δ ratios in combination with small N lead to performance degradation. The Gaussian fit as representative of a model-based fit comes close to the CRLB and always performs best for the σ/Δ ratios given by image discretization if N is adapted to the PSF. To achieve good results with the Gaussian fit and the radial symmetry method, a good rule of thumb is to set the pixel sizes Δ ≤ σ/0.6 and the patch sizes N ≥ 2σ/Δ + 3.

A new diagonalization based method for parallel-in-time solution of linear-quadratic optimal control problems

A new diagonalization technique for the parallel-in-time solution of linear-quadratic optimal control problems with time-invariant system matrices is introduced. The target problems are often derived from a semi-discretization of a Partial Differential Equation (PDE)-constrained optimization problem. The solution of large-scale time dependent optimal control problems is computationally challenging as the states, controls, and adjoints are coupled to each other throughout the whole time domain. This computational difficulty motivates the use of parallel-in-time methods. For time-periodic problems our diagonalization efficiently transforms the discretized optimality system into nt (=number of time steps) decoupled complex valued 2ny × 2ny systems, where ny is the dimension of the state space. These systems resemble optimality systems corresponding to a steady-state version of the optimal control problem and they can be solved in parallel across the time steps, but are complex valued. For optimal control problems with initial value state equations a direct solution via diagonalization is not possible, but an efficient preconditioner can be constructed from the corresponding time periodic optimal control problem. The preconditioner can be efficiently applied parallel-in-time using the diagonalization technique. The observed number of preconditioned GMRES iterations is small and insensitive to the size of the problem discretization.

A Novel, Efficient Approach to Simulate the 3D Resolved Spatial and Temporal Development of Space-Charge Layers in Solid Electrolytes

The role of space-charge layers (SCLs) on the overall battery performance of solid-state batteries is heavily discussed in the scientific community ranging from dominant to negligible [1]. Reasons for this are that direct measurements of SCLs in experiments are difficult and so far, simulation models that can efficiently resolve the transient development of the shape of SCLs while being thermodynamically consistent are missing. The different length scales of SCLs on the one hand and of the microstructure of a battery cell on the other hand make a spatial discretization of realistic microstructures with resolved SCLs infeasible. To date, this was circumvented by using phenomenological models that do not require a spatial discretization of the SCL, by applying thermodynamically consistent models to simplified one-dimensional geometries, or by performing simulations on the sub-continuum scale, e.g. DFT simulations or kinetic Monte Carlo simulations. As all models, the mentioned models have limitations. In this case, they are for example either thermodynamically not consistent or limited to small, not representative geometries.In this work [2], we propose a model that enables the prediction of the spatial and temporal development of SCLs within geometrically resolved microstructures while being thermodynamically consistent [3]. We exploit that effects in SCLs are predominantly one-dimensional. Furthermore, it can be shown, that SCLs develop orthogonal to the interface between the solid electrolyte and the electrode in the steady state. This allows to project the equations in the regions of the solid electrolyte where SCLs are expected in one dimension. In the remaining region of the solid electrolyte, the equations remain three-dimensional. A consistent coupling strategy between the three-dimensional and the one-dimensional region is derived and is irrevocably connected to the fulfilment of conservation properties. This modelling approach allows to drastically reduce the size of the numerical discretization required to resolve SCLs in space, as a fine discretization is only required in one direction embedded into a coarse three-dimensional discretization which represents the geometric microstructure. Thus, the required computational effort significantly reduces.The continuous equations are discretized in space using the finite element method. The obtained nonlinear system of equations is solved with the Newton-Raphson scheme using one monolithic system, i.e. the concurrent solution of the differently discretized regions to ensure robustness and efficiency of the solver. Tailoring an iterative solver by utilizing physical information of the system (similar to [4]), enables to apply the model to realistic microstructures with more than 10 million unknowns.With the proposed approach it is possible for the first time to quantify the influence of realistic microstructures on the transient and spatial formation process of SCLs. This work is the first to show that the formation process of SCLs at the interface between solid electrolyte and electrode is strongly influenced by the geometry of the microstructure (see Fig. 1): The dominating trend is a propagation of the SCL through the composite electrode, i.e. from the separator side to the current collector side of each electrode. Simultaneously, an inhomogeneity in the plane being orthogonal to the dominant direction is observed. This can be attributed to different conduction paths within this two-dimensional manifold due to the irregularity of the investigated microstructures and their geometric complexity. Furthermore, characteristic points in the microstructure can be identified where more charge is stored in the SCL during the transient development compared to the steady state. Again, this is attributed to the complex microstructures in interplay with the impedance of the bulk electrolyte and of the SCL. For these observations a modelling approach that resolves SCLs in realistic microstructures is mandatory.

Application of extreme value statistics to internal pore distribution in ceramics and prediction of size dependency of strength scatter

To implement ceramics in high-reliability components, it is necessary to understand the strength scatter caused by the stochastic distribution of defects in ceramics. In this study, we propose a numerical simulation method to predict the strength scatter of ceramics and its size dependency using microstructural data obtained using X-ray computed tomography and scanning electron microscopy. The generalized Pareto (GP) model, a type of extreme value statistic, was applied to treat the three-dimensional distribution of the pore size. Further, we predicted the bending strengths of the four types of bending tests with different effective volumes, and the results corresponded with the experimental results. Additionally, the method was applied to specimen models with different discretization sizes to verify the advantages of the GP model. The results show the effectiveness of the proposed method for the specimen size dependence of the bending strength and specimen discretization size independence.

Effect of meshing technique and time discretization size on thickness strain localization during hole-flanging simulation of DP980 sheet at high strain level

In the simulation of the hole-flanging process, many had reported that the effect of mesh sensitivity on strain localizations is insignificant. In this study, mesh layouts including the number of elements along the hole circumference (CD), element sizes in radial directions (R1) and the time discretization (or increment) sizes (T) are varied in the 3D FE simulation of 10-mm hole flanging process of DP980 thin sheets using a conical punch. Its effect on wall thickness distributions and forming load profiles are investigated. Also, the effect of punch surface meshing techniques i.e., analytical and discrete rigid on the distributions is investigated. The simulation results showed that the largest % wall thinning range around the expanded hole edges for punch displacement, S = 15 mm and 20 mm are 25.8–29.2% and 33.3–64.2%, respectively. The load-displacement curves during the continuous hole expansion are not linear for R1 ≥ 0.5 mm. For CD70 & R1 = 0.1 mm, the increase in T values from 0.1 s to 1.0 s has significantly increased the total simulation time by 25% due to the convergence problems. The automatic time increment mode in each step at T = 1.0 s has failed to reduce the total simulation time. The discrete rigid punch surface produces more concentrated wall thickness distributions than the analytical one, leading to higher amount of wall thinning, particularly for high S values. Since the model stiffness reduces with decreasing mesh size. It is recommended to match the simulated peak load with the experimental peak under the instability condition with a smooth hole e.g., a laser cut hole to determine the appropriate meshing method for the mesh sensitivity study. With that, the stiffness of the simulation model is calibrated with the experiment.

On geometric convergence for the Metropolis-adjusted Langevin algorithm under simple conditions

Summary While the Metropolis-adjusted Langevin algorithm is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors’ knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish V-uniformly geometric convergence for the Metropolis-adjusted Langevin algorithm under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quite common in the Markov chain Monte Carlo literature. Finally, we pay special attention to the dependence of the bounds we derive on the step size of the Euler–Maruyama discretization, which corresponds to the proposed Markov kernel of the Metropolis-adjusted Langevin algorithm.

An efficient solid shell material point method for large deformation of thin structures

AbstractThe standard material point method (MPM) encounters severe numerical difficulties in simulating shell structures. In order to overcome the shortcomings of locking effects, the discretization size of background grid should be small enough, usually smaller than 1/5 of the shell thickness, which however will lead to prohibitive computational cost. A novel solid shell material point method (SSMPM) is proposed to efficiently model the large deformation of thin structures. The SSMPM describes the material domain of shell structures by shell particles with hexahedral particle domains. The locking treatments of solid shell element are then introduced in SSMPM, which results in the correction of strain field throughout the shell thickness. Namely, the assumed natural strain (ANS) method is adopted to eliminate the shear locking and trapezoidal locking, while the enhanced assumed strain (EAS) method is employed to eliminate the thickness locking. With the precise description of bending modes, a single layer of particles and a coarse background grid could be used in shell structure simulations with the SSMPM, which dramatically increases the computational efficiency. A local multi‐mesh contact method is presented to naturally couple SSMPM and MPM for the contact situations of shells with other objects. Several numerical examples, including beam vibration, pinched cylinder with free edges and full hemispherical shell, are performed to verify and validate the SSMPM, which shows that the SSMPM considerably outperforms the standard MPM in these situations. A fluid–structure interaction problem and the penetration of a thin plate are investigated based on the contact method and the results are in good agreement with those in the literature.

Multi-frequency wavefield modeling of acoustic VTI wave equation using physics informed neural networks

Incorporating anisotropy is crucial for accurately modeling seismic wave propagation. However, numerical solutions are susceptible to dispersion artifacts, and they often require considerable computational resources. Moreover, their accuracy is dependent on the size of discretization, which is a function of the operating frequency. Physics informed neural networks (PINNs) have demonstrated the potential to tackle long-standing challenges in seismic modeling and inversion, addressing the associated computational bottleneck and numerical dispersion artifacts. Despite progress, PINNs exhibit spectral bias, resulting in a stronger capability to learn low-frequency features over high-frequency ones. This paper proposes the use of a simple fully-connected PINN model, and evaluates its potential to interpolate and extrapolate scattered wavefields that correspond to the acoustic VTI wave equation across multiple frequencies. The issue of spectral bias is tackled by incorporating the Kronecker neural network architecture with composite activation function formed using the inverse tangent (atan), exponential linear unit (elu), locally adaptive sine (l-sin), and locally adaptive cosine (l-cos) activation functions. This allows the construction of an effectively wider neural network with a minimal increase in the number of trainable parameters. The proposed scheme keeps the network size fixed for multiple frequencies and does not require repeated training at each frequency. Numerical results demonstrate the efficacy of the proposed approach in fast and accurate, anisotropic multi-frequency wavefield modeling.

Bilateral XVA pricing under stochastic default intensity: PDE modelling and computation

Adjusting derivative prices to take into account default risk has attracted the attention of several researchers and practitioners, especially after the 2007-2008 financial crisis. We derive a novel partial differential equation (PDE) model for derivative pricing including the adjustment for default risk, assuming that the default risk of one of the counterparties (the buyer) follows a Cox-Ingersoll-Ross (CIR) process, while the other party has constant default risk. The time-dependent PDE derived is of Black-Scholes type and involves two “space” variables, namely the asset price and the buyer default intensity, as well as a nonlinear source term. We formulate boundary conditions appropriate for the default intensity variable. The numerical solution of the PDE is based on standard finite differences, and a penalty-like iteration for handling the nonlinearity. We also develop and analyze a novel asymptotic approximation formula for the adjusted price of derivatives, resulting in a very efficient, accurate, and convenient for practitioners formula. We present numerical results that indicate stable second order convergence for the 2D PDE solution in terms of the discretization size. We compare the effectiveness of the 2D PDE and asymptotic approximations. We study the effect of various numerical and market parameters to the values of the adjusted prices and to the accuracy of the computed solutions.

Ready-to-use deep-learning surrogate models for problems with spatially variable inputs and outputs

Data-driven intelligent surrogate models gain popularity recently. Particularly in Monte-Carlo-style stochastic analysis, the influencing factors are considered as inputs, the quantities of interest are considered as outputs, and cheaper-to-evaluate surrogates models are built from a small amount of sample data and are used for the full Monte-Carlo analysis. This paper presents a framework with three innovations: (1) we build surrogate models for a particular problem that covers any possible material properties or boundary conditions commonly encountered in practice, so the models are ready to use, and do not require new data or training anymore. (2) The inputs and outputs to the problem are both spatially variable. Even after discretization, the input and output sizes are in the order of tens of thousands, which is challenging for traditional machine-learning algorithms. We take the footing failure mechanism as an example. Two types of neural networks are examined, fully connected networks and deep neural networks with complicated non-sequential structures (a modified U-Net). (3) This study is also the first attempt to use U-Nets as surrogate models for geotechnical problems. Results show that fully connected networks can fit well simple problems with a small input and output size, but fail for complex problems. Deep neural networks that account for the data structure give better results.

Multi-index ensemble Kalman filtering

In this work we combine ideas from multi-index Monte Carlo and ensemble Kalman filtering (EnKF) to produce a highly efficient filtering method called multi-index EnKF (MIEnKF). MIEnKF is based on independent samples of four-coupled EnKF estimators on a multi-index hierarchy of resolution levels, and it may be viewed as an extension of the multilevel EnKF (MLEnKF) method developed by the same authors in 2020. Multi-index here refers to a two-index method, consisting of a hierarchy of EnKF estimators that are coupled in two degrees of freedom: time discretization and ensemble size. Under certain assumptions, when strong coupling between solutions on neighboring numerical resolutions is attainable, the MIEnKF method is proven to be more tractable than EnKF and MLEnKF. Said efficiency gains are also verified numerically in a series of test problems.

Identifying parameters of a discrete numerical model of soil from a geotechnical field test

A discrete numerical model of a pressuremeter test was developed in this paper. The model is constituted of spherical particles with a contact law involving an inter-particle rolling resistance in addition to the classical Coulomb friction model. The model is calibrated from a pressuremeter test performed in the Hostun sand. An adaptive size of discretization of the soil domain is considered to limit the computational cost. Different parameter sets are defined to identify the initial density of the numerical granular assembly. Then, each parameter set leads to a different calibration of the mechanical parameters of the model in order to reproduce the pressuremeter test results. Simulations show that experimental data from a pressuremeter test alone are not enough to fully calibrate the discrete model. Other laboratory test results need to be considered in the calibration methodology. Eventually, the prediction capability of the model is assessed by comparing the simulated constitutive responses to those measured experimentally on different triaxial loading paths.

Manifold alignment-based multi-fidelity reduced-order modeling applied to structural analysis

This work presents the application of a recently developed parametric, non-intrusive, and multi-fidelity reduced-order modeling method on high-dimensional displacement and stress fields arising from the structural analysis of geometries that differ in the size of discretization and structural topology. The proposed approach leverages manifold alignment to fuse inconsistent field outputs from high- and low-fidelity simulations by individually projecting their solution onto a common subspace. The effectiveness of the method is demonstrated on two multi-fidelity scenarios involving the structural analysis of a benchmark wing geometry. Results show that outputs from structural simulations using incompatible grids, or related yet different topologies, are easily combined into a single predictive model, thus eliminating the need for additional pre-processing of the data. The new multi-fidelity reduced-order model achieves a relatively higher predictive accuracy at a lower computational cost when compared to a single-fidelity model.

Spatio‐temporal discretization uncertainty of distributed hydrological models

AbstractQuantifying the uncertainty linked to the degree to which the spatio‐temporal variability of the catchment descriptors (CDs), and consequently calibration parameters (CPs), represented in the distributed hydrology models and its impacts on the simulation of flooding events is the main objective of this paper. Here, we introduce a methodology based on ensemble approach principles to characterize the uncertainties of spatio‐temporal variations. We use two distributed hydrological models (water balance simulation model and Hydrotel) and six catchments with different sizes and characteristics, located in southern Quebec, to address this objective. We calibrate the models across four spatial (100, 250, 500 and 1000 m2) and two temporal (3 and 24 h) resolutions. Afterwards, all combinations of CDs‐CPs pairs are fed to the hydrological models to create an ensemble of simulations for characterizing the uncertainty related to the spatial resolution of the modelling, for each catchment. The catchments are further grouped into large (>1000 km2), medium (between 500 and 1000 km2) and small (<500 km2) to examine multiple hypotheses. The ensemble approach shows a significant degree of uncertainty (over 100% error for estimation of extreme streamflow) linked to the spatial discretization of the modelling. Regarding the role of CDs, results show that first, there is no meaningful link between the uncertainty of the spatial discretization and catchment size, as spatio‐temporal discretization uncertainty can be seen across different catchment sizes. Second, the temporal scale plays only a minor role in determining the uncertainty related to spatial discretization. Third, the more physically representative a model is, the more sensitive it is to changes in spatial resolution. Finally, the uncertainty related to model parameters is larger than that of CDs for most of the catchments. Yet, there are exceptions for which a change in spatio‐temporal resolution can alter the distribution of state and flux variables, change the hydrologic response of the catchments and cause large uncertainties.

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier–Stokes equations

The primary emphasis of this work is the development of a finite element based space–time discretization for solving the stochastic Lagrangian averaged Navier–Stokes (LANS-α) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of Rd, d∈{2,3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale α, namely we first assume α to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when α is fixed. A preparatory analysis of uniform estimates in both α and discretization parameters is carried out. Starting out from the stochastic LANS-α model, we achieve convergence toward the continuous strong solutions of the stochastic Navier–Stokes equations in 2D when α vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-α model is accomplished if α is fixed. Neither of the mentioned convergences involves the Skorokhod theorem.

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

Fast analysis of narrow-band transient electromagnetic scattering using the ACA enhanced envelope-tracking marching-on-in-time solver

In this paper, an efficient adaptive cross approximation (ACA) algorithm enhanced envelope-tracking marching-on-in-time (ET-MOT) solver is proposed to analyze the narrow-band transient electromagnetic scattering from three-dimensional perfect electrically conducting (PEC) objects. In the ET-MOT method, the time step size is inversely proportional to the bandwidth of the incident signal. However, the size of spatial discretization is still inversely proportional to the maximum frequency component of the incident wave. Therefore, the transmission distance of electromagnetic wave in one-time step is usually much larger than the size of spatial discretization. This results in the immediate interaction matrix Z0 being a dense matrix or even a full rank matrix for narrow-band problems. The ACA algorithm is used to accelerate the matrix-vector multiplication during the iteration solution at each time marching step. A series of numerical examples show the accuracy and efficiency of the proposed method.

A non-standard discretized SIS model of epidemics.

In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation.