Nonuniform Discretization Research Articles

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

An adaptive non-uniform L2 discretization for the one-dimensional space-fractional Gray–Scott system

This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order α∈(1,2) in one dimension. The fractional Laplacian in PDE is computed by using Riemann–Liouville (R–L) derivatives, incorporating a boundary condition of the form u=0 in R∖Ω. The proposed approach extends the L2 method to non-uniform meshes for calculating the R–L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray–Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order α: from self-replication to standing waves and from travelling waves to self-replication.

Variable horizon ordinary state-based peridynamic analysis in ANSYS framework

This study presents the implementation of ordinary state-based peridynamic (OSB PD) analysis in a finite element framework using ANSYS, a commercial software. The novelty of the present PD formulation is its ability to accommodate nonuniform discretization of the PD solution domain, which may be crucial for the realistic modeling of complex structures. It enables the use of variable horizon while eliminating the need for surface and volumetric correction procedures. The PD governing equations are constructed by using MATRIX27, an element native to ANSYS. The PD bond breakage during the implicit solution of these equations is achieved through the EKILL option in ANSYS. The efficacy and robustness of the present PD approach are demonstrated by considering four distinct geometric configurations, each subjected to specific boundary conditions. The results from the present PD approach exhibit a good agreement with those obtained from traditional finite element method (FEM) and experimental data.

Viscoelastic Fractional Model with a Non-Uniform Time Discretization for Laminated Glass: Experimental Validation

We discuss a novel approach, based on fractional calculus with a non-uniform time discretization, to numerically simulate interlayer viscoelastic behaviour and associated time-dependent deformation of laminated glass. Reference is made to the classic example of a simply supported laminated glass beam under long-duration loads. The fractional model is compared with some results obtained using the widely used finite element software ABAQUS 2021, which for the viscoelastic properties of the polymeric interlayer, utilizes the more traditional approach based on the Wiechert model and approximation via Prony series of the relaxation function and a uniform discretization of time for the numerical solution. The model is also validated through the comparison with experimental test. The novel approach based on fractional calculus presents two main advantages: 1) the definition of the model parameters from experimental data is simplified; and 2) the numerical implementation is easier and computationally more efficient. When a long observation time is considered, the use of a non-uniform time discretization presents the great advantage of not neglecting any part of the relaxation function. Use of traditional uniform time discretization requires the use of large time steps making it impossible to describe all the changes of the relaxation curve within the large time interval. Practical examples will be presented using viscoelastic models for Trosifol® Extra Stiff (PVB) and SentryGlas® interlayers. This methodology also shows potential to advance next generation standards for the design of structural laminated glass.

A peridynamic differential operator modeling approach for ceramic matrix composites microstructure with uniform or non-uniform discretization

Based on the theory of peridynamic differential operator (PDDO), a novel three-dimensional (3D) peridynamic (PD) model for isotropic and orthotropic material is proposed to investigate the elastic mechanical behaviors and stress distribution of Ceramic Matrix Composites (CMC) microstructure. The non-local expressions of strain and stress tensors are derived by employing PDDO and the classical constitutive equations. The bond forces in interface region crossing two different constituent materials are obtained by converting the partial differential terms into the non-local forms. The weak form of PD equations of motion is derived by using the principle of virtual work and PDDO. The validity of this approach is verified by predicting the elastic response and stress distributions of isotropic and orthotropic materials under uniaxial tension and pure shear deformation. Finally, simulations are conducted on a CMC microstructure with fiber and matrix under periodic boundary conditions. It is demonstrated that the current approach can effectively build 3D CMC microstructure with uniform or non-uniform discretization, and accurately predict the stress fields and effective elastic properties.

Optimal discretization of geothermal boreholes for the calculation of g-functions

The effect of the discretization of geothermal boreholes on the accuracy of g-function evaluations is studied. A data set of 557,056 bore field configurations covering a large range of geometrical parameters is generated using pygfunction. A nonuniform discretization of borehole segments geometrically expanding in length toward the middle of the borehole is proposed. A nonuniform discretization is shown to achieve better accuracy than a uniform discretization. The nonuniform discretization is optimized to minimize the maximum absolute percentage error over the entire data set. The discretization is optimized for each bore field configuration, and an artificial neural network (ANN) is trained to predict the optimal discretization given only geometrical and thermal parameters of the boreholes, excluding the borehole positions. Thermal parameters that quantify the bore field temperature distribution are introduced as inputs to the ANN. The maximum absolute percentage error using a uniform discretization is 99.0% in the worst studied case of a dense rectangular field of Nb = 1116 boreholes with lengths of 418.8 m and spacings of 3.14 m and 3.18 m along rows and columns, while only 1% of the cases feature an error above 26.7%. The error is reduced to 3.6% using the global optimal discretization and 3.3% using the ANN.

An efficient and accurate DSRFG method via nonuniform energy spectra discretization

An appropriate set-up of inflow boundary conditions is crucial for accurate prediction of wind effects on the target structures using Large Eddy Simulation (LES). The discretizing and synthesizing random flow generation (DSRFG) method is a commonly used method which however requires considerable computational cost. This paper attempts to improve the efficiency and accuracy of the DSRFG method by modifying the way of discretizing the energy spectra. Since the DSRFG method is originated from the Kraichnan’s method, the grid effects on both methods are analyzed theoretically and examined numerically. Research results show that the Kraichnan’s method is sensitive to the grid effects while the improved DSRFG method is slightly affected. Additionally, the improved DSRFG method is applied to simulate the wind effects on a low-rise building by LES and the results agree well with the NIST aerodynamic database.

Real-time adaptive entry trajectory generation with modular policy and deep reinforcement learning

Recently, data-driven entry trajectory generation algorithms of hypersonic vehicles were highlighted for their high accuracy with low computational costs. Learning a unique black box model mapping the state to control variables is expected. However, insufficient adaptability remains a challenging problem when applying the algorithms in practice, especially for multi-missions. In this article, a real-time entry trajectory generation algorithm with modular policy is proposed to achieve adaptability to various missions, such as changing targets and emergency entry. Based on the modular idea, the entry trajectory problem is decomposed into two decision problems, i.e., adjusting the profile of the angle of attack (AOA) to determine the flight capability and planning the bank angle to ensure the satisfaction of task constraints, which correspond to AOA module and bank module respectively. Then, algorithms are developed by utilizing deep reinforcement learning (DRL) to train the two modules with which an intelligent entry trajectory generation algorithm is proposed to achieve real-time trajectory design in a wide range of reachable areas. Moreover, a non-uniform discretization approach with a state-related independent variable is proposed to deal with state misalignment and contradiction in stepsize settings. The simulation results demonstrate that the proposed algorithm can provide a reliable trajectory in a very short time and adapt to various tasks.

An improved peridynamic model for quasi-static and dynamic fracture and failure of reinforced concrete

In this paper, an improved peridynamic model is established to characterize the bond-slip behavior between ribbed steel rebar and concrete. An interfacial force term is introduced into the description of interaction between rebar and concrete considering the chemical bonding force, the friction, as well as the mechanical bite force, and a size-dependent non-uniform discretization algorithm for rebar and concrete is developped to reduce the calculation cost in numerical simulations. An equivalent modeling approach for ribs is presented as well, which can accurately reflect the interaction between ribs and concrete. Validation of the proposed model and approach is established through modeling pull-out problem of ribbed steel rebar, in which the crack initiation and propagation during the pull-out process can be accurately captured. Moreover, the dynamic fracture and impact failure of reinforced concrete are numerically modeled and analyzed. The numerical predictions are consistent with available experimental results, in terms of fracture characteristics of concrete, deformation of rebars, as well as the force–displacement curves.

A meshfree model of hard-magnetic soft materials

Hard-magnetic soft materials comprising a soft matrix embedded with hard-magnetic particles can exhibit large deformations under external magnetic fields, which have attracted much interest due to their non-contact activation, flexible programmability, and rapid response in various applications such as biomedical devices, soft robotics and flexible electronics. Precise predictions of large deformations of hard-magnetic soft materials would be a key for relevant applications. Given that the rational designs in the programmable shape morphing of ferromagnetic structures requires high precision, here we develop a meshfree model based on the radial point interpolation method to quantitatively predict large deformations of hard-magnetic soft materials. We apply the stabilized conforming nodal integration instead of direct nodal integration to eliminate the influence of zero-energy mode, and the central difference scheme is implemented for the resolution procedure. We explore the bending of a hard-magnetic beam and snap buckling of a bistable hard-magnetic beam under external magnetic stimuli. Uniform and non-uniform discretizations are compared to show the insensitivity of the radial point interpolation method to nodal mesh. To demonstrate the versatile programmability of our model in the design of magnetic robotics, we further simulate complex locomotion (crawling, walking and rolling) of hard-magnetic soft robots under external magnetic excitation. The results could be used to guide rational designs of ferromagnetic structures and soft continuum robots.

Real-Time Trajectory Planning for Hypersonic Entry Using Adaptive Non-Uniform Discretization and Convex Optimization

This paper introduces an improved sequential convex programming algorithm using adaptive non-uniform discretization for the hypersonic entry problem. In order to ensure real-time performance, an inverse-free precise discretization based on first-order hold discretization is adopted to obtain a high-accuracy solution with fewer temporal nodes, which would lead to constraint violation between the temporal nodes due to the sparse time grid. To deal with this limitation, an adaptive non-uniform discretization is developed, which provides a search direction for purposeful clustering of discrete points by adding penalty terms in the problem construction process. Numerical results show that the proposed method has fast convergence with high accuracy while all the path constraints are satisfied over the time horizon, thus giving potential to real-time trajectory planning.

Multi-adaptive spatial discretization of bond-based peridynamics

Peridynamic (PD) models are commonly implemented by exploiting a particle-based method referred to as standard scheme. Compared to numerical methods based on classical theories (e.g., the finite element method), PD models using the meshfree standard scheme are typically computationally more expensive mainly for two reasons. First, the nonlocal nature of PD requires advanced quadrature schemes. Second, non-uniform discretizations of the standard scheme are inaccurate and thus typically avoided. Hence, very fine uniform discretizations are applied in the whole domain even in cases where a fine resolution is per se required only in a small part of it (e.g., close to discontinuities and interfaces). In the present study, a new framework is devised to enhance the computational performance of PD models substantially. It applies the standard scheme only to localized regions where discontinuities and interfaces emerge, and a less demanding quadrature scheme to the rest of the domain. Moreover, it uses a multi-grid approach with a fine grid spacing only in critical regions. Because these regions are identified dynamically over time, our framework is referred to as multi-adaptive. The performance of the proposed approach is examined by means of two real-world problems, the Kalthoff–Winkler experiment and the bio-degradation of a magnesium-based bone implant screw. It is demonstrated that our novel framework can vastly reduce the computational cost (for given accuracy requirements) compared to a simple application of the standard scheme.

Hydrogen-like Plasmas under Endohedral Cavity

Over the past few decades, confined quantum systems have emerged to be a subject of considerable importance in physical, chemical and biological sciences. Under such stressed conditions, they display many fascinating and notable physical and chemical properties. Here we address this situation by using two plasma models, namely a weakly coupled plasma environment mimicked by a Debye-Hückel potential (DHP) and an exponential cosine screened Coulomb potential (ECSCP). On the other hand, the endohedral confinement is achieved via a Woods-Saxon (WS) potential. The critical screening constant, dipole oscillator strength (OS) and polarizability are investigated for an arbitrary state. A Shannon entropy-based strategy has been invoked to study the phase transition here. An increase in Z leads to larger critical screening. Moreover, a detailed investigation reveals that there exists at least one bound state in such plasmas. Pilot calculations are conducted for some low-lying states (ℓ=1−5) using a generalized pseudo spectral scheme, providing optimal, non-uniform radial discretization.

Rapid surrogate modeling of magnetotelluric in the frequency domain using physics-driven deep neural networks

The magnetotelluric (MT) forward modeling problem primarily relies on spatially discretization of Maxwell's equations using polynomials into an algebraic system with finite dimensions. It is computationally prohibitive to solve the algebraic system, resulting in a slow computational speed. The inversion scheme requires a significant number of forward computations, and the efficiency of the inversion is determined by the forward modeling speed. Therefore, constructing an economical surrogate model as a fast solver for the forward problem can considerably improve the efficiency of inversion. Because of their capacity to approximate, deep neural networks (DNNs) have showed significant potential for surrogating. We present a physics-driven model (PDM) to solve the MT governing equation without using any labeled data. Specifically, the product of conductivity and frequency is used as the input to the DNNs, and the loss function is given by the governing equation to ”drive” the training. The trained model is capable in predicting electromagnetic fields at any frequency within the range of trained datasets, even ones that are not presented in the training. Numerical experiments are conducted on 2-D conductivity structures with uniform and non-uniform discretization. The results show excellent agreement on the MT responses between the PDM predictions and the finite-difference method (FDM). In addition, the computing speed of PDM exceeds by multiple times that of FDM.

A coupling approach of the isogeometric–meshfree method and peridynamics for static and dynamic crack propagation

A coupling approach of the isogeometric–meshfree method and the peridynamic method is developed for static and dynamic crack propagation. The coupling approach exhibits advantages in the flexibility of modeling cracks and the exactness of geometry representation. The isogeometric–meshfree method, which adopts the moving least-squares approximations to establish the equivalence between meshfree shape functions and isogeometric basis functions, is capable of obtaining the exact geometry. The isogeometric–meshfree nodes located on the geometry can be directly coupled with peridynamic points by the balanced force principle, which is straightforward and effective. With this approach, the boundary conditions can be enforced directly using the isogeometric–meshfree method, and the surface effects of peridynamics are effectively eliminated. Moreover, the present approach is flexible in switching the isogeometric–meshfree nodes to peridynamic points and achieving adaptive coupling with the same convergence rate as the one for the isogeometric–meshfree method, which is higher than the one for the original peridynamic method. The dual-horizon peridynamic method is adopted for non-uniform discretization. Based on the exact geometry representation, the coupling approach is further extended to crack problems with contact loading. Several representative examples are presented to validate the effectiveness of the present approach in solving static/dynamic crack problems and studying fractures caused by contact.

An extended peridynamic model for analyzing interfacial failure of composite materials with non-uniform discretization

This work presents an extended ordinary state-based peridynamic model and approaches to analyze the fracture behavior of composite materials considering interfacial failure. An interface force term is introduced to characterize the interaction among different compositions in composite materials, which enables naturally the non-uniform discretization to be conveniently adopted, especially where sharp interfaces exist. The calculation of force states in different compositions is independent of each other, and the use of non-uniform discretization can improve computational efficiency significantly. Further, inspired by the cohesive zone model (CZM), the deformation coefficient and degradation behavior of interface bonds are derived by using the traction-separation law, and a mixed-mode fracture criterion for interface bonds based on the bond energy density is established as well. Validation of the proposed model and approach is established through a classical rectangular plate example, and then the proposed model is employed to analyze the failure of typical composites including laminated glass and concrete. The results show good agreement with corresponding experimental observations and it indicates that the proposed extended peridynamic model can accurately and efficiently capture and describe the fracture of composite materials with sharp interfaces.

The non-uniform L1-type scheme coupling the finite volume method for the time–space fractional diffusion equation with variable coefficients

In this paper, we consider the time–space fractional diffusion equation on a bounded interval with the Caputo time fractional derivative and the two-sided Riemann–Liouville spatial fractional derivative. |The diffusion coefficients are variable with respect to x and t. The temporal non-uniform L1-type discretization and the finite volume method based on the nodal basis functions are adopted to discrete this fractional partial derivative equation. We strictly prove that our scheme is conditionally stable with convergence rate Oh2+N−min{2−α,rα}, where h is the spatial step and N,α,r are the temporal nodal numbers, the fractional parameter and the graded parameter respectively. Finally, some numerical results are presented to validate the theoretical analysis.

Weak form of bond-associated peridynamic differential operator for thermo-mechanical analysis of orthotropic structures

In this study, a weak form of bond-associated peridynamic differential operator (WBA-PDDO) is presented for thermo-mechanical analysis of orthotropic structures. The proposed method inherits the advantage of the original peridynamic differential operator and eliminates the zero-energy mode by introducing the concept of bond-associated family. The WBA-PDDO for thermo-mechanical equations of orthotropic structures is obtained by using the variational principle. The developed model is applied to heat transfer analysis of orthotropic square plate, thermo-mechanical analysis of orthotropic square plate with a hole and orthotropic connecting rod. By comparing with the exact solutions and finite element results, the convergence and accuracy of the proposed method are demonstrated in uniform and non-uniform discretization. The presented model provides an alternative approach for thermo-mechanical analysis of orthotropic structures.

Error and stability estimates of a time-fractional option pricing model under fully spatial–temporal graded meshes

To price vanilla European and American options via the fractional Black–Scholes model, first a (2−α)-order discretization scheme for the Caputo fractional derivative based upon graded meshes along time is presented. This is fruitful for problems having nonsmooth data at the initial time. Second, a nonuniform discretization of space is also considered and the coefficient weights based upon meshless radial basis function generated finite difference (RBF-FD) methodology are contributed under the inverse quadratic function. The fully space–time nonuniform solution method is then constructed via the presented discretization formulas. Numerical tests are given to show effectiveness and accuracy of our numerical scheme.

A Semianalytical Mathematical Model and Seepage Characteristics of Off‐Center Fractured Vertical Wells with Multiwing Asymmetrical Fractures

The complexity of formation conditions leads to multiwing asymmetric fractures after large‐scale fracturing. According to a well pattern model and reservoir characteristics, a testing well is away from the center of the reservoir, and the existing well‐test mathematical model cannot meet the field data analysis demand. Therefore, the mathematical model for the fractured wells with multiwing asymmetrical fractures is established. The model solution of the Laplace domain is obtained by the Laplace transform, nonuniform fracture discretization, and pressure drop superposition principle. The model is compared with the numerical model in this paper, and the result shows that the presented semianalytical model and the calculation method are correct. The eight main flow stages are divided according to pressure derivative characteristics and the seepage process of off‐center fractured vertical wells with multiwing asymmetrical fractures. The larger off‐center distance will lead to higher pressures and derivative curves. Larger fracture asymmetry factors are higher pressures and derivative curves during bilinear and linear flow regimes. The field data show that the model is applicable and that the model can give the guidance of fractured well formation evaluation.