Numerical Method Research Articles

Benchmarking numerical simulation of induced fault slip with semi-analytical solutions

AbstractPore pressure fluctuation in subsurface reservoirs and its resulting mechanical response can cause fault reactivation. Numerical simulation of such induced seismicity is important to develop reliable seismic hazard and risk assessments. However, modeling of fault reactivation is quite challenging, especially in the case of displaced faults, i.e., faults with non-zero offset. In this paper, we perform a systematic benchmarking study to validate two recently developed numerical methods for fault slip simulation. Reference solutions are based on a semi-analytical approach that makes use of inclusion theory and Cauchy-type singular integral equations. The two numerical methods both use finite volume discretizations, but they employ different approaches to represent faults. One of them uses a conformal discrete fault model (DFM) while the other employs an embedded (non-conformal) fault model. The semi-analytical test cases cover a vertical frictionless fault, and inclined displaced faults with constant friction and slip-weakening friction. It was found that both numerical methods accurately represent pre-slip stress fields caused by pore pressure changes. Moreover, they also successfully cope with a vertical frictionless fault. However, for the case with an inclined displaced fault with a constant friction coefficient, the embedded method can not converge for the post-slip phase, whereas the DFM successfully coped with both constant and slip-weakening friction coefficients. In its current implementation, the DFM is therefore the model of choice when accurate simulation of local faulted systems is required.

Development of a simple test to quantify the consolidation properties of liquefied granular materials

AbstractLiquefaction can have devastating consequences by causing increased mobility of debris flows, tailings dam breaches, and settlement following seismic shaking. Observations on the consolidation behaviour of liquefied soils in 1-g or centrifuge shake table tests have permitted significant advancements in analytical and numerical methods to predict the rate and magnitude of consolidation settlement. However, advanced consolidation models introduce material parameters which are currently difficult to define quickly and at low cost. The objective of this paper is to demonstrate the development of a simple, low-cost test that can be used to quickly estimate the spatially and temporally varying coefficient of consolidation, which controls post-liquefaction settlements. Using the proposed simple setup, experiments were conducted on four uniformly graded sands of varying grain size and on one well-graded mixture. A range of analytical and numerical approaches from literature were assessed for their ability to back-analyse the observed pore pressure dissipation and settlement. Estimated values for the coefficient of consolidation were comparable between models and estimates of surface settlement matched well the experimental results. The simplicity of the proposed test combined with its low-cost and mobile nature, raise significant possibilities for quick estimations of the evolution of the coefficient of consolidation post-liquefaction.

A Dataset of 3M Single-DOF Planar 4-, 6-, and 8-Bar Linkage Mechanisms With Open and Closed Coupler Curves for Machine Learning-Driven Path Synthesis

Abstract In recent years, there has been a strong interest in applying machine learning techniques to path synthesis of linkage mechanisms. However, progress has been stymied due to a scarcity of high-quality datasets. In this article, we present a comprehensive dataset comprising nearly three million samples of 4-, 6-, and 8-bar linkage mechanisms with open and closed coupler curves. Current machine learning approaches to path synthesis also lack standardized metrics for evaluating outcomes. To address this gap, we propose six key metrics to quantify results, providing a foundational framework for researchers to compare new models with existing ones. We also present a variational autoencoder-based model in conjunction with a k-nearest neighbor search approach to demonstrate the utility of our dataset. In the end, we provide example mechanisms that generate various curves along with a numerical evaluation of the proposed metrics.

Two-Dimensional Prediction of Transient Cavitating Flow Around Hydrofoils Using a DeepCFD Model

Cavitation is a common phenomenon in naval and ocean engineering, typically occurring in the wakes of high-speed rotating propellers and on the surfaces of fast-moving underwater vehicles. To investigate cavitation phenomena, computational fluid dynamics (CFD) simulations are indispensable. Nevertheless, the inherently complex nature of cavitation, which involves phase transitions, heat transfer, and significant pressure fluctuations, often results in high computational costs for these simulations. To address the computational challenges associated with cavitation simulations, a DeepCFD model, which leverages convolutional neural networks (CNNs), was employed to accurately predict cavitation around hydrofoils. Through specific modifications, the DeepCFD model was trained on 400 hydrofoil configurations, learned from CFD simulations. The numerical methods were validated against a modified NACA66 hydrofoil. It was found that the model could accurately predict cavitation shapes under various flow conditions, although it showed some discrepancies in velocity predictions, especially for detached cavitating flows. The significance of this study lies in its potential to simply predict cavitating flows and expedite marine vehicle design through the application of CNNs in cavitation prediction, offering a novel and impactful approach to computational fluid dynamics in the field.

A proof of convergence and equivalence for 1D finite element methods and ReLU neural networks

This paper investigates the convergence and equivalence properties of the Finite Element Method (FEM) and Rectified Linear Unit Neural Networks (ReLU NNs) in solving differential equations. We provide a detailed comparison of the two approaches, highlighting their mutual capabilities in function space approximation. Our analysis proves the subset and superset inclusions that establish the equivalence between FEM and ReLU NNs for approximating piecewise linear functions. Furthermore, a comprehensive numerical evaluation is presented, demonstrating the error convergence behavior of ReLU NNs as the number of neurons per basis function varies. Our results show that while increasing the number of neurons improves approximation accuracy, this benefit diminishes beyond a certain threshold. The maximum observed error between FEM and ReLU NNs is 10-4, reflecting excellent accuracy in solving partial differential equations (PDEs). These findings lay the groundwork for integrating FEM and ReLU NNs, with important implications for computational mathematics and engineering applications.

On Some Works of Boris Teodorovich Polyak on the Convergence of Gradient Methods and Their Development

The paper presents a review of the current state of subgradient and accelerated convex optimization methods, including the cases with the presence of noise and access to various information about the objective function (function value, gradient, stochastic gradient, higher derivatives). For nonconvex problems, the Polyak–Lojasiewicz condition is considered and a review of the main results is given. The behavior of numerical methods in the presence of a sharp minimum is considered. The aim of this review is to show the influence of the works of B.T. Polyak (1935–2023) on gradient optimization methods and their surroundings on the modern development of numerical optimization methods.

Comparative evaluation of machine learning models for assessment of seabed liquefaction using finite element data

Predicting wave-induced liquefaction around submarine pipelines is crucial for marine engineering safety. However, the complex of interactions between ocean dynamics and seabed sediments makes rapid and accurate assessments challenging with traditional numerical methods. Although machine learning approaches are increasingly applied to wave-induced liquefaction problems, the comparative accuracy of different models remains under-explored. We evaluate the predictive accuracy of four classical machine learning models: Gradient Boosting (GB), Support Vector Machine (SVM), Multi-Layer Perceptron (MLP), and Random Forest (RF). The results indicate that the GB model exhibits high stability and accuracy in predicting wave-induced liquefaction, due to its strong ability to handle complex nonlinear geological data. Prediction accuracy varies across output parameters, with higher accuracy for seabed predictions than for pipeline surroundings. The combination of different input parameters significantly influences model predictive accuracy. Compared to traditional finite element numerical methods, employing machine learning models significantly reduces computation time, offering an effective tool for rapid disaster assessment and early warning in marine engineering. This research contributes to the safety of marine pipeline protections and provides new insights into the intersection of marine geological engineering and artificial intelligence.

Extension of a 3D Geological Entity Modeling Method for Discontinuous Deformation Analysis

AbstractWith the development of 3D discontinuous deformation analysis (DDA) in precise stress fields and crack propagation problems, it has also demonstrated outstanding capabilities in solving continuous–discontinuous problems. However, currently, 3D DDA modeling primarily focuses on generating rock joint networks and developing 3D cutting algorithms. Correspondingly, 3D geological modeling methods are not yet mature, and establishing 3D models often demands substantial time. The lack of supporting preprocessing modeling methods and corresponding visual operation interfaces significantly hampers the development of 3D DDA. This method builds upon advanced research achievements in unmanned aerial vehicle oblique photography, 3D reconstruction, 3D cutting, computer graphics, and visualization program design. This research establishes a 3D geological entity modeling method for 3D DDA and constructs a comprehensive program using relevant C++ libraries and C language interfaces. In this method, a 3D geological model that incorporates geological elements such as strata and faults is initially established using non‐uniform rational B‐splines (NURBSs) surfaces as the boundary of the solid model. Subsequently, finite element meshing is applied, followed by corresponding topology transformation, resulting in a 3D block system model suitable for 3D DDA calculation. To cater for diverse application scenarios, continuous–discontinuous models integrated with subblocks and models of arbitrary polyhedra can be established. The proposed method has been validated through several typical modeling examples, showing its ability to rapidly and generate 3D high‐precision geological reality models suitable for 3D DDA calculations. Additionally, some techniques used in this method can be extended for modeling other numerical simulation methods, warranting further research.

Influence of Specimen Width on Crack Propagation Process in Lightly Reinforced Concrete Beams

Models used to study the fracture process of concrete are often considered 2D, ignoring the influence of specimen width. However, during the fracture process in pre-cracked concrete beams, the crack length varies along the thickness direction, especially in reinforced concrete. To study the influence of specimen width on reinforced concrete fracture behavior, a 3D numerical method was used to simulate the crack propagation processes of lightly reinforced concrete beams based on Fracture Mechanics. Nonlinear spring elements with different stress-displacement constitutive laws were employed to characterize the softening behavior of concrete and the bond-slip behavior between the steel bars and concrete, respectively. It is assumed that the crack begins to propagate when the maximum stress intensity factor at the crack tip along the beam width reaches the initial fracture toughness of concrete. To verify the validity of the proposed method, the completed crack propagation processes of lightly reinforced concrete three-point bending notched beams were simulated, and the calculated load-crack mouth opening displacement curves showed a reasonable agreement with the experimental data. Moreover, the impact of the 2D reinforced concrete beam model on the crack propagation process was analyzed. The results indicate that at the initial loading stage, the external load P obtained from the 2D model is significantly larger than the result from the presented 3D model.

Parameter estimation for allometric trophic network models: A variational Bayesian inverse problem approach

Abstract Differential equation models are powerful tools for predicting biological systems, capable of projecting far into the future and incorporating data recorded at arbitrary times. However, estimating these models' parameters from observations can be challenging because numerical methods are often required to approximate their solution. An example of such a model is the allometric trophic network model, for which studies considering its inverse problem are limited, particularly in the Bayesian framework. Here we develop a variational Bayesian method for parameter inference of the allometric trophic network model and explore how accurately we can recover its parameter values. We represent the model as a Bayesian neural network, which combines an artificial neural network with Bayesian inference, using a surrogate for the posterior distribution of model parameters, and train this model by evolutionary optimization to avoid potentially costly computation of the gradient with respect to the model parameters. Using synthetic data, we compare the accuracy of this variational inference to ordinary least squares estimation. To reduce the number of estimated parameters, we focus on the inference of functional response parameters. Our variational Bayesian method yields parameter estimates that are comparable to the ordinary least squares results in terms of accuracy. The method provides a promising approach for including uncertainty quantification in parameter estimation, which the simple ordinary least squares approach as it is does not address. Regardless of the method, potential multimodality of the inference problem is nonetheless important to keep in mind. The present study suggests a technique for parameter inference of ordinary differential equation models in the Bayesian context. We propose the method especially for validation of the allometric trophic network model against empirical data.

Theoretical and numerical study of a Burgers viscous equation type with moving boundary

In this article, we investigate the existence, uniqueness, and numerical aspects of a one‐ and two‐dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one‐ and two‐dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

Numerical Simulation of the Horizontal Water-Entry Process of High-Speed Vehicles

Based on the RNG k-ε turbulence model and VOF multiphase flow model, a numerical model of horizontal water-entry of the vehicle was established, and the numerical method was verified by experimental results. The cavitation characteristics, fluid resistance, and motion of the vehicle under different conditions were studied during the vehicle’s water-entry process. The results show that the cavitation process can be divided into the cavity development stage, saturation stage, and collapse stage. With the increase in initial velocity and mass of the vehicle, more water vapor will be generated during the water-entry process. The initial velocity of the vehicle had a limited effect on the resistance coefficient. The resistance coefficient in the stable stage remained almost unchanged for vehicles with different masses. Nevertheless, the time interval of the stable stage was shortened, and the resistance coefficient was greater in the gradually increasing stage for the vehicle with a smaller mass. For vehicles with higher initial velocity or smaller mass, the instantaneous velocity decreased faster after it entered the water. The vehicle with a streamlined design was able to reduce the generation of water vapor and decrease fluid resistance and its coefficient, and the vehicle can run farther during the water-entry process.

Nuclear burning in an accretion flow around a stellar-mass black hole embedded within an AGN disk

Abstract A stellar-mass black hole, embedded within the accretion disk of an active galactic nuclei (AGN), has the potential to accrete gas at a rate that can reach approximately ∼109 times the Eddington limit. This study explores the potential for nuclear burning in the rapidly accreting flow towards this black hole and studies how nucleosynthesis affects metal production. Using numerical methods, we have obtained the disk structure while considering nuclear burning and assessed the stability of the disk. In contrast to gas accretion onto the surface of a neutron star or white dwarf, the disk remains stable against the thermal and secular instabilities because advection cooling offsets the nuclear heating effects. The absence of a solid surface for a black hole prevents excessive mass accumulation in the inner disk region. Notably, nuclear fusion predominantly takes place in the inner disk region, resulting in substantial burning of $\rm ^{12}C$ and $\rm ^{3}He$, particularly for black holes around M = 10 M⊙ with accretion rates exceeding approximately ∼107 times the Eddington rate. The ejection of carbon-depleted gas through outflows can lead to an increase in the mass ratio of oxygen or nitrogen to carbon, which may be reflected in observed line ratios such as $\rm N\, V/C\, IV$ and $\rm O\, IV/C\, IV$. Consequently, these elevated spectral line ratios could be interpreted as indications of super-solar metallicity in the broad line region.

Interface phonon transport in nanomaterials: numerical methods and modulation strategies.

The thermal properties of interfaces in nanomaterials are critical for various technological applications, including thermal management in electronic and photonic devices, thermoelectric conversion and thermal insulation. Recent advancements in numerical simulation tools (the non-equilibrium Green's approach, the Boltzmann transport equation and the Monte Carlo method, molecular dynamics simulations) have significantly enhanced our understanding of phonon transport and scattering processes in nanomaterials. These advances have led to the discovery of new thermal interfacial materials and enabled precise modulation of phonon thermal conductance to achieve desired thermal performance. This review summarizes recent research progress in interface thermal transport, focusing on intriguing heat phenomena such as finite size effect and phonon coherent property. Additionally, it discusses strategies for modulating thermal conductance through disorder and roughness. Finally, the review proposes the opportunities and challenges associated with modulating interface thermal transport.

A Prediction Method for Calculating Fracturing Initiation Pressure Considering the Modification of Rock Mechanical Parameters After CO2 Treatment

The establishment of a more realistic CO2 fracturing model serves to elucidate the intricate mechanisms underlying CO2 fracturing transformation. Additionally, it furnishes a foundational framework for devising comprehensive fracturing construction plans. However, current research has neglected to consider the influence of CO2 on rock properties during CO2 fracturing, resulting in an inability to precisely replicate the alterations in the reservoir post-CO2 injection into the formation. This disparity from the actual conditions poses a substantial limitation to the application and advancement of CO2 fracturing technology. This work integrates variations in the physical parameters of rocks after complete contact and reaction with CO2 into the numerical model of crack propagation. This comprehensive approach fully acknowledges the impact of pre-CO2 exposure on the mechanical parameters of reservoir rocks. Consequently, it authentically restores the reservoir state following CO2 injection, ensuring a more accurate representation of the post-fracturing conditions. In comparison with conventional numerical simulation methods, the approach outlined in this paper yields a reduction in the error associated with predicting fracturing pressure by 9.8%.

Analysis of flow and energy dissipation in pump turbine with small guide vane opening

In this study, a detailed flow state analysis of a reversible pump turbine (RPT) under small guide vane opening conditions was conducted using both numerical simulation and experimental methods on three analysis planes at 0.5 span. The entropy production rate (EPR) in entropy production theory was utilized to visualize the location and magnitude of energy losses in the flow process. The focus was placed on the coherent vortex structures in the volute, stay vane, guide vane, and runner flow path. The results demonstrate that, throughout the runner’s cycle, the leading edge of the guide vane experiences low-pressure regions with uneven pressure distribution within the runner channel. Velocity streamlines reveal flow separation and vortex generation at the typical guide vane trailing edge, with a maximum velocity reaching 49.7. Analysis of radial force and torque on typical guide vanes and the runner indicates that radial force and torque on the guide vanes do not significantly change, with the radial force coefficient CF r−x varying between 0.254 and 1.3. Major energy losses are concentrated behind the trailing edge of the guide vane, primarily in the form of jet wake. Coherent vortex structures are observed in the volute, stay vane, guide vane, and runner flow path, with slight turbulent wake at the trailing edge of the stay vane, clear shedding vortex streets, and vortex build-up in the bladeless area between the guide vane and the runner.

Improving the stability of high-speed trains using an innovative passive yaw damper

Ensuring lateral stability is essential for achieving excellent operation safety and ride comfort of high-speed trains. However, complicated operational environments, particularly various wheel-rail contact geometries, would result in the occurrence of hunting instability. This study presents an innovative passive yaw damper to improve the stability of a high-speed train undergoing extreme wheel-rail contact conicities. A sensitivity analysis of the vehicle stability is first performed to demonstrate the importance of the dynamic stiffness of yaw dampers, promoting the application of the innovative damper. Experimental characterisation procedures enable the detailed investigation into the static and dynamic characteristics of the innovative damper, which are compared with those of conventional passive dampers to highlight the difference. To simulate the dynamic behaviour of the yaw damper, a numerical modelling method based on the Maxwell model is proposed. Using this method, the effectiveness of the innovative damper is assessed in terms of improving the vehicle stability at various wheel-rail conicities. The results show that the innovative yaw damper can ensure carbody stability at low conicity and bogie stability at high conicity, superior to conventional dampers, which fail to balance the vehicle stability at distinct conicities and usually result in hunting instability in specific wheel-rail contact states.

Simulation of time-invariant three-dimensional Freak waves and their scattering identification based on the JONSWAP spectrum and Donelan direction function

Abstract The results of recent maritime accident investigations show that many maritime accidents are related to the attack of freak waves, which can sink ships instantaneously with colossal energy. To identify and forecast freak waves accurately and avoid the danger of freak waves to marine structures and people, we have conducted further research on freak waves. In this paper, we propose an efficient time-invariant three-dimensional (3-D) freak wave computational model to study the scattering characteristics of time-invarian 3-D freak waves and obtain a more explicit identification of their scattering differences. First, this paper adopts linear superposition method to simulate a 3-D random rough sea surface based on the JONSWAP wave spectrum and Donelan directional distribution function. Moreover, we analyze the effect of different angular frequencies on the simulated sea surface. At the same time, the Monte Carlo method is used to further simulate the capillary waves based on the random sea surface, and this method vastly improves the computational accuracy of the random sea surface. Then, based on the two-wave train superposition wave energy focusing mode, a numerical calculation method of time-invariant 3-D freak waves is proposed, and the time-series evolution process of 3-D freak waves is simulated. Finally, the backscattering coefficients of the 3-D freak wave surface are calculated using the two-scale method, and the electromagnetic scattering characteristics of the freak wave surface under different evolution stages, wind speeds, and radar incidence angles are analyzed. By this method, we conclude that the recognition of freak waves is more accurate under medium-high wind speed and significant incidence angle conditions. The experiments show that the research in this paper can identify and predict freak waves more effectively, further improve the forecast accuracy of future freak waves, and also have some application value for freak wave risk warnings.

Explicit Runge–Kutta Numerical Manifold Method for Solving the Burgers’ Equation via the Hopf–Cole Transformation

This paper presents an efficient numerical manifold method for solving the Burgers’ equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers’ equation.

Studying a cylindrical moving boundary problem in ductal carcinoma in situ

In this paper, a mathematical model of ductal carcinoma in situ (DCIS) is considered as a moving boundary problem in cylindrical coordinates. Using the front-fixing technique, first, we transform the problem into a nonlinear parabolic problem on a fixed domain, and then some essential properties of the solution, such as the uniqueness of the solution, are shown. A numerical approach based on the finite difference method is established for solving the fixed domain problem. The solvability of the proposed finite difference method is proved, and the stability and convergence of the numerical solution are shown. For a test case problem, analytical and numerical results are compared to demonstrate the ability and effectiveness of the numerical method.