Numerical Method Research Articles

Credit default swap pricing with counterparty risk in a reduced form model with Hawkes process

Credit default swap pricing with counterparty risk in a reduced form model with Hawkes process

De-projecting the EFThedron

The space of Wilson coefficients of EFT that can be UV completed into consistent theories was recently shown to be described analytically by a positive geometry, termed the EFThedron. However, this geometry, as well as complementary numerical methods of semi-definite programming, have so far focused on the positivity of the partial wave expansion, which allows bounding only ratios of couplings. In this paper we describe how the unitarity upper bound of the partial waves can be incorporated. This new problem can be formulated in terms of the well known L-moment problem, which we generalize and solve from a geometrical perspective. We find the non-projective generalization of the EFThedron has an infinite number of non-linear facets, which in some cases have remarkably simple descriptions. We use these results to derive bounds on single couplings, finding that the leading derivative operators are bounded by unity, when normalized by the cut-off scale and loop factors. For general operators of mass dimension 2k we find the upper bound is heavily suppressed at large k, with an 1/k fall-off.

Semi-inclusive single-jet production in DIS at next-to-leading order in the Color Glass Condensate

Within the Color Glass Condensate (CGC) effective field theory, we derive the next-to-leading order (NLO) cross-section for the single-jet semi-inclusive cross-section in deep inelastic scattering (DIS) at small x, for both longitudinally and transversely polarized virtual photons. We provide analytic expressions, valid at finite Nc and suitable for numerical evaluation, for both the cross-section differential in rapidity and transverse momentum and the cross-section differential in rapidity only. Our NLO formulae demonstrate that the very forward rapidity regime is plagued by large double logarithmic corrections coming from phase space constraints on soft gluons close to the kinematic threshold for jet production. A joint resummation of small-x and threshold logarithms at single logarithmic accuracy is proposed to remedy the instability of the cross-section in this regime. By integrating over the single-jet phase space, we recover known results for the NLO DIS structure functions at small x, previously obtained using the optical theorem.

Numerical evaluation of low-head inverted cross-flow turbine design method: A comprehensive study

This study aimed to address the challenges of micro-hydropower generation in low-head environments to expand the proportion of hydropower in renewable energy and achieve a carbon neutrality policy. To achieve these goals, various factors affecting the micro-hydro systems, including geographical constraints, construction costs, and grid connectivity, were reviewed and assessed. The results concluded that a novel type of cross-flow turbine with inverted structures, improved the economic feasibility, productivity, and installability of micro-hydropower plants by reducing head losses in open channels or small streams. The parametric studies consisted of four steps: first, designing the initial inverted cross-flow turbine based on the traditional design rules; for the second and third steps, a parametric study for comparing the runner diameter ratio and the number of blades is conducted; and in the last step optimized turbine was designed. The best efficiency point of the optimized inverted cross-flow turbine was 72.92 % at a runner rotational speed of 170 RPM with a 0.75 runner diameter ratio and 35 blades. The optimized turbine was improved by 7.8 % in power and 7.35 % in efficiency, higher than the initial turbine. From the hill chart comparison between the initial and optimized turbine, operation range expansion was confirmed. Overall, this research demonstrates the potential of the inverted cross-flow turbine under low-head conditions.

Robustness of embedded fibre optic sensor mesh configurations for monitoring composite structures

Fibre optic sensors are attractive for monitoring of composite structures but the significant risk of damage to fibres where they enter the structure and within the structure itself can make their use unreliable. Mesh configurations of embedded fibre Bragg grating sensors with improved robustness are considered for monitoring composite structures. The configurations of single ended line, bidirectional line, loop, and a novel 2×2 mesh are analysed using a probabilistic model of the sensor system robustness. Numerical evaluation of these configurations has shown that the 2×2 mesh configuration provides the greatest robustness of all of the configurations across all inlet port failure probabilities. The probability of a sensor becoming isolated in a 2×2 mesh is approximately 3 times lower than for the bidirectional line configuration. Using a new matrix approach, the 2×2 mesh configuration is analysed in terms of disaggregation of individual sensor spectra for both fault free and faulty conditions. Simulation and experimental evaluation have demonstrated that successful recovery of individual grating spectra is possible with small spurious signals.

An information theoretic metric for measurement vulnerability to data integrity attacks on smart grids

AbstractA novel metric that describes the vulnerability of the measurements in power systems to data integrity attacks is proposed. The new metric, coined vulnerability index (VuIx), leverages information theoretic measures to assess the attack effect in terms of the fundamental limits of the disruption and detection tradeoff. The result of computing the VuIx of the measurements in the system yields an ordering of their vulnerability based on the degree of exposure to data integrity attacks. This new framework is used to assess the measurement vulnerability of IEEE 9‐bus and 30‐bus test systems and it is observed that power injection measurements are significantly more vulnerable to data integrity attacks than power flow measurements. A detailed numerical evaluation of the VuIx values for IEEE test systems is provided.

Modal analysis and output characteristics of an ultrasonic motor with a square prismatic shell stator

Abstract A rotary traveling wave ultrasonic motor having a square prismatic shell stator is proposed. The stator contacts with the rotor through interference fit, and elastically deforms slightly, and then presses against the rotor. Then the pre-pressure is created by the initial deformation of the stator with simple structure. The in-plane modal problem of the stator is analyzed by numerical method as a folded beam and the results are compared with those from finite element method (FEM) and laser doppler measurement. The 3rd and 5th degenerate modes with repeat natural frequencies are selected as working modes, and the influence of actuating frequency and voltage on the no-load speed and stall torque is studied by multi-physics simulation for the operation of these two modes. A prototype is fabricated and the output characteristics were tested and compared with the simulated results. The 3rd working mode is found to have better performance and can reach a no-load speed of 205 rpm and maximum torque of 3.9 mN·m when the input is 28.0 kHz and 100V. The effect of the pre-pressure for the 3rd working mode is discussed further.

Research on the analysis and management measures of land subsidence caused by groundwater exploitation in Tianjin, China

Since the 20th century, there has been an increasing reliance on groundwater due to the rapid development of agriculture and industry. This has resulted in the exploitation of groundwater in large quantities, leading to global land subsidence problems. Tianjin is one of the cities where this problem first emerged, but few studies and surveys have been conducted. As a result, the lack of management of groundwater extraction has caused the issue of land subsidence to become more prominent in recent years. By investigating and collating global research on land subsidence problems caused by groundwater extraction, it is concluded that mathematical and numerical methods, global information systems, remote sensing, and other technologies can be utilized to monitor and predict groundwater levels. Regional management measures can then be developed based on simulation results. However, the application of such technology in Tianjin is insufficient, and there is a need to strengthen the connection between management measures and research data analysis results.

Normal form and Hopf bifurcation for the memory‐based reaction‐diffusion equation with nonlocal effect

In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods.

An efficient high‐order two‐level explicit/implicit numerical scheme for two‐dimensional time fractional mobile/immobile advection‐dispersion model

AbstractThis article constructs a new two‐level explicit/implicit numerical scheme in an approximate solution for the two‐dimensional time fractional mobile/immobile advection‐dispersion problem. The stability and error estimates of the proposed technique are deeply analyzed in the ‐norm. The developed approach is less time consuming, fourth‐order in space and temporal accurate of order , where is the time step and denotes a positive parameter less than 1. This result shows that the two‐level explicit/implicit formulation is faster and more efficient than a large class of numerical schemes widely discussed in the literature for the considered problem. Numerical experiments are performed to verify the theoretical studies and to demonstrate the efficiency of the new numerical method.

A synergistic approach to optimizing the performance of a concentrating solar segmented variable area leg thermoelectric generator using numerical methods and neural networks

AbstractThis study presents an optimized design for segmented variable area leg thermoelectric modules using finite element methods and Bayesian regularized neural networks. We explored the impact of geometry and thermal parameters on module performance using ANSYS software, identifying optimal parameters for power output and efficiency. Key findings revealed the higher influence of geometric parameters and confirmed the advantages of segmented thermoelectric generators for high-temperature applications like concentrated solar systems. With this optimization, power output and efficiency of the module increased by 875% and 165%, respectively, under 25 Suns. To refine the optimization process, a Bayesian regularized neural network was utilized, proving effective in predicting module performance with a low mean squared error and high coefficient of determination. This research provides important insights into high-performance thermoelectric modules for sustainable energy applications, demonstrating the significant role of advanced computational methods in energy solutions.

Assessment of structural mechanical effects related to torsional deformations of propellers

AbstractLifting propellers are of increasing interest for Advanced Air Mobility. All propellers and rotors are initially twisted beams, showing significant extension–twist coupling and centrifugal twisting. Torsional deformations severely impact aerodynamic performance. This paper presents a novel approach to assess different reasons for torsional deformations. A reduced-order model runs large parameter sweeps with algebraic formulations and numerical solution procedures. Generic beams represent three different propeller types for General Aviation, Commercial Aviation, and Advanced Air Mobility. Simulations include solid and hollow cross-sections made of aluminum, steel, and carbon fiber-reinforced polymer. The investigation shows that centrifugal twisting moments depend on both the elastic and initial twist. The determination of the centrifugal twisting moment solely based on the initial twist suffers from errors exceeding 5% in some cases. The nonlinear parts of the torsional rigidity do not significantly impact the overall torsional rigidity for the investigated propeller types. The extension–twist coupling related to the initial and elastic twist in combination with tension forces significantly impacts the net cross-sectional torsional loads. While the increase in torsional stiffness due to initial twist contributes to the overall stiffness for General and Commercial Aviation propellers, its contribution to the lift propeller’s stiffness is limited. The paper closes with the presentation of approximations for each effect identified as significant. Numerical evaluations are necessary to determine each effect for inhomogeneous cross-sections made of anisotropic material.

Semi-empirical cross-section formulas for electron-impact ionization and rate coefficient calculations

We propose a semi-empirical formula for the cross section of ionization by electron impact. The formula involves adjustable parameters which are determined by comparison with measured or numerically calculated cross sections. In the latter case, the ions are perturbed by their environment which is a high-density plasma. As a consequence, the cross section is significantly modified. We investigate Be-like carbon, nitrogen and oxygen as well as aluminum ions. We also show that the formula is well-suited for interpolation and extrapolation. Knowing the cross section, we calculate the rate coefficient within the Boltzmann and Fermi–Dirac statistics. In the first case, the rate can be calculated analytically. In the second one, it can be expressed in terms of special functions, but the numerical evaluation is more convenient while providing accurate results. Our results are compared to experiment and to other calculations.

Numerical Simulation of the Density Effect on the Macroscopic Transport Process of Tracer in the Ruhrstahl–Heraeus (RH) Vacuum Degasser

Silicon steel (electrical steel) has been used in electric motors that are important components in sustainable new energy Electrical Vehicles (EVs). The Ruhrstahl–Heraeus process is commonly used in the refining process of silicon steel. The refining effect inside the RH degasser is closely related to the flow and mixing of molten steel. In this study, a 260 t RH was used as the prototype, and the transport process of the passive scalar tracer (virtual tracer) and salt tracer (considering density effect) was studied using numerical simulation and water model research methods. The results indicate that the tracer transports from the up snorkel of the down snorkel to the bottom of the ladle, and then upwards from the bottom of the ladle to the top of the ladle. Density and gravity, respectively, play a promoting and hindering role in these two stages. In different areas of the ladle, density and gravity play a different degree of promotion and obstruction. Moreover, in different regions of the ladle, the different circulation strength leads to the different promotion degrees and obstruction degrees of the density. This results in the difference between the concentration growth rate of the salt tracer and the passive scalar in different regions of the ladle top. From the perspective of mixing time, density and gravity have no effect on the mixing time at the bottom of the ladle, and the difference between the passive scalar and NaCl solution tracer is within the range of 1–5%. For a larger dosage of tracer case, the difference range is reduced. However, at the top of the ladle, the average mixing time for the NaCl solution case is significantly longer than that of the passive scalar case, within the range of 3–14.7%. For a larger dosage of tracer case, the difference range is increased to 17.4–41.1%. It indicates that density and gravity delay the mixing of substances at the top area of the ladle, and this should be paid more attention when adding denser alloys in RH degasser.

Weak-formulated physics-informed modeling and optimization for heterogeneous digital materials

Abstract Numerical solutions to partial differential equations (PDEs) are instrumental for material structural design where extensive data screening is needed. However, traditional numerical methods demand significant computational resources, highlighting the need for innovative optimization algorithms to streamline design exploration. Direct gradient-based optimization algorithms, while effective, rely on design initialization and require complex, problem-specific sensitivity derivations. The advent of machine learning offers a promising alternative to handling large parameter spaces. To further mitigate data dependency, researchers have developed physics-informed neural networks (PINNs) to learn directly from PDEs. However, the intrinsic continuity requirement of PINNs restricts their application in structural mechanics problems, especially for composite materials. Our work addresses this discontinuity issue by substituting the PDE residual with a weak formulation in the physics-informed training process. The proposed approach is exemplified in modeling digital materials, which are mathematical representations of complex composites that possess extreme structural discontinuity. This paper also introduces an interactive process that integrates physics-informed loss with design objectives, eliminating the need for pre-trained surrogate models or analytical sensitivity derivations. The results demonstrate that our approach can preserve the physical accuracy in data-free material surrogate modeling but also accelerates the direct optimization process without model pre-training.

Adjusting exceptional points using saturable nonlinearities

We study the impact of saturable nonlinearity on the presence and location of exceptional points in a non-Hermitian dimer system. The inclusion of the saturable nonlinearity leads to the emergence of multiple eigenvalues, exceeding the typical two found in the linear counterpart. To identify the exceptional points, we calculate the nonlinear eigenvalues both from a polynomial equation for the defined population imbalance and through a fully numerical method. Our results reveal that exceptional points can be precisely located by adjusting the non-equal saturable nonlinearities in the detuning space.

Swirling flow field reconstruction and cooling performance analysis based on experimental observations using physics-informed neural networks

The design of thermal protection modules (such as film cooling) for combustion chambers requires a high-fidelity swirling flow field. Although numerical methods provide insights into three-dimensional mechanisms of swirling flow, their predictions of key features such as recirculation zones and swirling jets are often unsatisfactory due to inherent anisotropy and the isotropic nature of the Boussinesq hypothesis. Experimental methods, such as hot wire, laser Doppler velocimetry, and planar particle image velocimetry (PIV), offer more accurate reference data but are limited by sparse or planar observations. In this study, considering the outperformed capability of solving inverse problems, physics-informed neural network (PINN) was adopted to reconstruct the mean swirling flow field based on limited experimental observations from two-dimensional and two-component (2D2C) results. It was found that adding partial information characterizing the swirling flow, such as the swirling jet, could significantly improve the reconstruction of flow field. In addition, film cooling effectiveness was the key variable to evaluate the film cooling performance, which was relatively measurable in the scalar field. To further improve the accuracy of the reconstruction, the multi-source strategy was adopted into the neural network, where the film cooling effectiveness (FCE) of the effusion plate was imported as the scalar source. It was found that the prediction of the flow field near the target plate was improved, where the highest error reduction could reach 76.5%. Finally, through the reconstructed three-dimensional vortex distribution, it was found that swirling flow vortex structures near the swirler exit had a significant impact on cooling effectiveness, causing a non-uniform cooling distribution. This study aims to diagnose the three-dimensional swirling flow field with deep learning by leveraging limited experimental data and deepen the understanding of effusion cooling under swirling flow condition so that obtains a more accurate reference in the design of thermal protection modules.

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

Bearing Characteristics with Effect of Bond–Slip Behavior in Massive Ring-Type Reinforced Concrete Structures

The bond–slip behavior of the steel–concrete interface is critical in reinforced concrete (RC) structures since the bond action is the mechanism that ensures the two materials work in co-operation. However, there is little research considering the bond–slip behavior in massive ring-type reinforced concrete (MRRC) structure bearing analyses due to the complexity of modeling the interfacial behavior. Hence, the influence of the bond–slip behavior on the bearing characteristics of MRRC structures remains unclear. Steel-lined reinforced concrete penstock is such an MRRC structure, composed of steel liner and reinforced concrete and commonly used in diversion pipelines. This paper aims to explore the bearing characteristics considering the bond–slip behavior in the composite penstock by using a promising numerical method, the cohesive zone model. Three interface models were proposed to represent the different interaction conditions at the steel–concrete interface. Moreover, a sensitivity analysis was performed to study the impact of the bond strength on the bond performance and structural behavior. The simulation results showed that the prediction results (steel stress and crack process) considering the bond–slip behavior were in good agreement with the experimental results. The steel stresses near the cracks were smaller and more uniform after considering the bond–slip behavior, since the stresses were no longer concentrated on the crack but distributed in an area near the crack. However, the steel stress differences in these models were within 10%, which means that the bond performance had a limited effect on the structural safety design. The crack widths were greatly influenced by the bond conditions, and the maximum crack width (0.461 mm) in poor conditions was beyond the limiting value (0.3 mm). Consequently, bond–slip behavior must be paid more attention in durability design.

The fundamental solution of the master equation for a jump‐diffusion Ornstein–Uhlenbeck process

AbstractAn integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods.