In this review paper we try to describe some recent results on modeling and analysis of economic risks by using the techniques of fractional calculus. The use of fractional order operators in the risk theory is due to the presence of long and short memory in most of economic models and their nonlocality over time. We emphasize on the use and interpretation of the Dzherbashian-Caputo fractional derivative

This paper presents a numerical study of the effects of the particle's shape and its interaction with surrounding fluid on the mechanism of sandpiles formation in air and water, respectively. This study is motivated by the fact that seabed sediments are predominantly deposited in water and consist of non-spherical particles. In our study, a non-linear contact model is employed in the Discrete Element

The elastic viscoplastic properties are non-negligible factors of clays that cause the deviation between the theoretical and the measured values, which were rarely incorporated in layered consolidation problems. Thus, this paper extended a generalized bi-layered one-dimensional (1D) consolidation model for soft ground under ramp load, simultaneously incorporating elastic viscoplastic deformation, non-Darcy

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2182-2208, October 2023. Abstract. This paper describes a locally corrected trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter

We consider a stationary convection–diffusion–reaction model problem in a two- or three-dimensional bounded domain. We approximate this model by a non-stationary problem and propose a numerical method that combines the method of characteristics with an augmented mixed finite element procedure. We show that this scheme has a unique solution. We also derive a residual-based a posteriori error indicator

Dynamic wave reflections would interfere with the laboratory observation of the soil arching under cyclic loading when a traditional trapdoor model with rigid boundaries was used. This problem would be prominent when investigating the soil arching in pile-supported embankments where the stationary support (pile) would be small, making the rigid boundary close to the domain of interest affected by dynamic

Frost heave can lead to both the ground uplifting and frost heave pressure under different circumstances, and cause many engineering problems. To describe the characteristics of frost heave under various thermo-hydro-mechanical (THM) coupling conditions and calculate both the frost heave amount and the frost heave pressure, the coupled THM process, as well as the phase change of the pore water, should

In this paper a new approach to handle stress-based topology optimization problems by using the Topology Optimization of Binary Structures method is presented. The design update is carried out with binary values (0 or 1) and a boundary identification scheme is employed to smooth the structural contours to avoid artificial stress concentrations that can occur because of the jagged nature of the topology

This paper is interested in developing reduced order models (ROMs) for repeated simulation of fractional elliptic partial differential equations (PDEs) for multiple values of the parameters (e.g., diffusion coefficients or fractional exponent) governing these models. These problems arise in many applications including simulating Gaussian processes, geophysical electromagnetics. The approach uses the

In this paper, we study the cyclic anti-periodic boundary value problem of a nonlinear tripled system of fractional Langevin equations. By means of Krasnoselskii fixed point theorem and Banach contraction mapping theorem, we obtain sufficient conditions for the existence and uniqueness of results. Moreover, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the proposed problem. Finally

Bituminized waste products (BWPs) were produced by conditioning in bitumen the co-precipitation sludge resulting from industrial reprocessing of spent nuclear fuel. Underground geological disposal is a solution for the long-term disposal of some intermediate level long-lived (ILW-LL) categorized BWPs in France. After one or several hundred thousand years, the water from the host rock will fully saturate

Strainburst often occurs in the loading process of tangential stress concentration from surrounding rocks after excavation and unloading of deep rock mass. The concentrated tangential stress is relatively larger on the tunnel walls, which decreases towards the interior of surrounding rocks with a certain gradient. To explore the fracture evolution characteristics of surrounding rocks under different

Landslide motion is often simulated with interface-like laws able to capture changes in frictional strength caused by the growth of the pore water pressure and the consequent reduction of the effective stress normal to the plane of sliding. Here it is argued that, although often neglected, the evolution of all the 3D stress components within the basal shear zone of landslides also contributes to changes

The paper presented a novel analytical solution to evaluate the seismic-induced stresses and displacements of tunnels covered by an isolation coating layer subjected to SH waves. The solution is derived based on the assumptions that the ground, isolation coating layer, and liner are linear elastic and in steady-state motion. The analytical solution is obtained using the Bessel-Fourier series. Comparisons

Rotating electrical machines are becoming ubiquitous as we move towards a more electrified and sustainable world. Torque calculation is an essential task in the design process of rotating machines. In the frame of finite element analysis, the Maxwell stress tensor method is a common technique to calculate the torque exerted on a rigid body. However, the computed torque is strongly mesh-dependent and

In order to ensure the reliability of a numerical simulation software, verification and validation are unavoidable tasks. In this paper, we present a new rigorous code verification strategy based on manufactured solutions for the static analysis of geometrically non-linear Kirchhoff–Love shells and apply it to Isogeometric Analysis (IGA). While IGA is based on a parametric surface description, we advocate

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2133-2156, October 2023. Abstract. We develop a unifying framework for interpolatory [math]-optimal reduced-order modeling for a wide class of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for [math]-optimal model

The development and calibration of soil models under the framework of plasticity is notoriously challenging given the prismatic features in soil's shear behaviors. Data-driven deep neural networks (DNNs) offer an alternative approach to this formidable task. However, classical DNN models struggle to accurately capture soil mechanical responses using limited training data. To address this issue, a unified

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2106-2132, October 2023. Abstract. This work establishes [math]-norm stability and convergence for an L2 method on general nonuniform meshes when applied to the subdiffusion equation. Under mild constraints on the time step ratio [math], such as [math] for [math], the positive semidefiniteness of a crucial bilinear form associated with the

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2084-2105, October 2023. Abstract. We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2062-2083, October 2023. Abstract. Solution methods for the nonlinear PDE of the Rudin–Osher–Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First-order methods are common. They are popular due to their simplicity and easy implementation. Some second-order

The smoothed particle finite element method (SPFEM) is an effective framework for large deformation analysis. The original SPFEM possesses the rank deficiency issue due to the direct nodal integration technique, which can be overcome by incorporating the strain gradient stabilization method. However, an extra volumetric locking due to the strain gradient stabilization term has been found. In this study

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2035-2061, October 2023. Abstract. In this paper, we develop a conditional Monte Carlo method for solving PDEs involving an integral fractional Laplacian on any bounded domain in arbitrary dimensions. We first construct the Feynman–Kac representation based on the Green function and Poisson kernel for the fractional Laplacian operator on the

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2011-2034, October 2023. Abstract. The solution of the nonlinear initial-value problem [math] for [math] with [math], where [math] is the Caputo derivative of order [math] and [math] are positive parameters, is known to exhibit [math] decay as [math]. No corresponding result for any discretization of this problem has previously been proved

This note presents an exact analytical solution of Fredlund–Hasan consolidation for unsaturated soils under an arbitrary loading using the mode superposition method. Air pressure and water pressure are expressed in a series of products of Eigen functions with respect to depth and normal coordinates with respect to time, respectively. A pressure function substitutes for the related homogeneous problem

This paper provides the solution of a homogenization model that simultaneously considers the plastic and elastic strain fields in a single analytical formulation. In addition, the authors aim to decouple plastic flows triggered in the inclusions, in the matrix at the interfaces with the inclusions, and in the matrix at a large distance from the inclusions. The authors propose an admissible displacement

Excavation damaged zone (EDZ) widely exists in the surrounding rock of underground roadway due to blasting excavation, which can further change the surrounding rock property and deteriorate the stability of the roadway. In the presented research, a damaged roadway model was developed to investigate the dynamic stress concentration factor (DSCF) distribution around the damaged roadway under a transient

In very thin ply laminates, delamination failure initiation occurs at much higher stress levels as compared to conventional ply laminates. This results in significant plastic deformation in the matrix accompanied by large fiber rotations. A closer look reveals that microstructure of fiber reinforced composites at large strains do not rotate with the plastic spin induced by the total deformation gradient

In the context of MOR techniques for parametrized PDEs, a novel computational method relying on NURBS-based geometric mappings and PGD-based space separated representations has recently been developed. Such approach has opened new perspectives to classical PGD formulations. In particular, it has extended the use of the PGD to complex non-separable an non-simply-connected domains. Moreover, the domain

In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as Lq(Rn) norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of C2 diffeomorphisms in specific cases. Lastly, we extend

Given only a finite collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to numerically solve elliptic and parabolic partial differential equations (PDEs) supplemented with boundary conditions. Since the construction of triangulations on unknown manifolds can be both difficult and expensive, both in terms of computational and data

We introduce new frames, called metaplectic Gabor frames, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions, cf. [7], [8], [5], [17], [27], [28]. Namely, we develop the theory of metaplectic atoms in a full-general setting and prove an inversion formula for metaplectic Wigner distributions on Rd. Its discretization provides metaplectic Gabor frames. Next

Clay rocks are multiphase porous media having a complex structure and behaviour characterised by heterogeneity, damage and viscosity, existing on a wide range of scales. The mesoscopic scale of mineral inclusions embedded in a clay matrix has an important role in the mechanisms of deformation under mechanical loading by cracking and creeping. This study introduces a micromechanical approach to model

We propose a new Ritz vector-based dynamic substructuring method which substitutes the unit pseudo-forces applied at the adjacent degrees of freedom (DOFs) using distributed forces. One of the main problems of the Ritz vector and unit pseudo-force-based dynamic substructuring method is the strong dependence of the number of reduction bases on the interface DOF, which is not reduced by substructuring

This study proposes a new model in explicit form to predict the diameter of the jet grouting column of three popular jet grouting systems (i.e., single, double, and triple). The proposed model can quantify the uncertainty associated with the prediction. Bayesian model selection was used to determine the optimal models and, the bootstrap sampling method was adopted to avoid bias in the collected database

In deep learning, it is common to use more network parameters than training points. In such scenario of over-parameterization, there are usually multiple networks that achieve zero training error so that the training algorithm induces an implicit bias on the computed solution. In practice, (stochastic) gradient descent tends to prefer solutions which generalize well, which provides a possible explanation

In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean–Vlasov stochastic differential equations, possibly with a drift term exhibiting super-linear growth in the state component. We prove strong convergence of order one and moment stability, making use of techniques from variational calculus on the space of probability measures with finite

Reliability analysis of earth slopes considering soil spatial variability has garnered significant attention from researchers. However, previous studies have predominantly focused on long slopes with infinite or ample width, such as dams and subgrades, but rare research was dedicated to slopes with width restricted by boundary constraints. In this regard, this paper proposes an efficient reliability

In this paper, a novel strategy based on nonlinear thermal analysis has been developed using finite element simulations and artificial neural networks in order to predict the time-temperature distributions in arc welding process. The highly nonlinear and transient thermal finite element methodology pertaining to simulations of arc welding process is investigated through various combinations of numerical

We consider the homogenization for time-fractional diffusion equations in a periodic structure. First, we derive the homogenized time-fractional diffusion equations. Next, we prove the stability in determining a constant diffusion coefficient by minimum data. Moreover, we investigate the inverse problems of estimating the homogenized diffusion coefficient by the data for non-homogenized structure.

A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its

In this paper, we propose a novel ROM stabilization strategy for under-resolved convection-dominated flows, the approximate deconvolution Leray ROM (ADL-ROM). The new ADL-ROM introduces AD as a new means to increase the accuracy of the classical Leray ROM (L-ROM) without degrading its numerical stability. We also introduce two new AD ROM strategies: the Tikhonov and van Cittert methods. Our numerical

Hydrothermal ore-forming systems usually involve coupled processes among pore-fluid flow, heat transfer, mass transport and chemical reactions in fluid-saturated porous media. The finite element method (FEM) has provided a powerful tool to solve hydrothermal ore-forming problems associated with pore-fluid convective flow. Depending on different interests of research, a hydrothermal ore-forming system

In this work, following the discrete de Rham approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete

This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added

Many geo-engineering applications, for example, enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase-field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However

We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) that include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation

We present new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math., 148, 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1989-2010, August 2023. Abstract. We propose and analyze a semidiscrete parametric finite element scheme for solving the area-preserving curve shortening flow. The scheme is based on Dziuk’s approach [SIAM J. Numer. Anal., 36 (1999), pp. 1808–1830] for the anisotropic curve shortening flow. We prove that the scheme preserves two fundamental

We present a hypoplastic model to capture the mechanical responses of hydrate-bearing sands by incorporating hydrate saturation and grain breakage. In our model, a new degradation scalar is introduced to account for the effects of hydrate bonding evolution on strength. We also adopt the variable critical state line to assess the effects of hydrate saturation and relative breakage ratio at high confining

Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators

This study introduces a novel topology optimization method for transient two-phase fluid problem with continuous behavior, which remains a challenging task despite advances in computing capabilities. The application of gradient-based optimizer to continuous two-phase fluid systems is complicated because it requires some modifications of the governing equations to reflect changes in the interface between

The susceptibility of a granular soil to suffusion is strongly dependent on its grain size distribution (GSD) and the mechanical and hydraulic conditions it is subjected to. This study investigates the onset of suffusion considering the effect of confining pressure and stress anisotropy using a fully resolved computational fluid dynamics and discrete element method (CFD–DEM). Three benchmarks, including