Conventional Numerical Methods Research Articles

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%.

A novel temporal finite element method to solve static viscoelastic problems

A novel temporal finite element method is presented to solve static viscoelastic problems, which is more flexible and appropriate to describe temporal variation of displacement than conventional finite difference or numerical integral methods. By using virtual work principle, a spatial finite element based governing equation is derived in terms of displacement and its derivatives. Then two temporal finite element models are developed using Gurtin variational principle and weighted residual technique. A kind of hybrid shape functions with polynomial and trigonometric basis is stressed to give more flexible descriptions of time varying variables. A criterion of stability analysis is derived, which can numerically be conducted when the constitution of shape functions and step size are prescribed. A recursive algorithm toward end is developed by which the temporal FE analysis can be conducted via a matrix power product with the initial condition, instead of step marching. The proposed approach is available for the viscoelastic model described by linear differential equations with order h ≤ 2, and can conveniently be combined with well-developed numerical algorithms to tackle with boundary value problems, such as FEM, SBFEM etc. Various numerical examples, including those with static/harmonic loads, creep, stress singularity and heterogeneous structures, etc. are provided to illustrate the efficiency of the proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with analytical or ABAQUS-based solutions.

A Novel Quantitative Water Channeling Identification Method of Offshore Oil Reservoirs

Offshore oilfields are characterized by loose sandstone reservoirs, strong heterogeneity and high injection and production intensity. Water channeling gradually develops after entering the high water cut stage, which weakens production performance. Current identification methods usually have high computational costs and low efficiency. A quantitative identification model of water channeling based on inter-well connection units has been established by simplifying the complex reservoir system into a connection network between injectors and producers, which can quickly and accurately obtain strength characteristic parameters for waterflow channels. In addition, a comprehensive evaluation factor M and classification standard for water channeling suitable for offshore heterogeneous reservoirs have been proposed. It indicates a thief zone when M is larger than 0.65, a predominant waterflow channel when M is between 0.55 and 0.65, and no water channeling when M is smaller than 0.55. The application of (an) offshore S oilfield demonstrates that the new method successfully identifies 18 segments of the thief zone and 19 segments of the predominant waterflow channel and improves computational speed by 100 times compared with the conventional numerical modeling method. This novel method allows for rapid and accurate identification and prediction of water channeling, including location, directions, and strengths, thereby providing timely and practical guidance for inefficient water channel treatment.

Modeling the Time Evolution of Compact Binary Systems with Machine Learning

This work introduces advanced computational techniques for modeling the time evolution of compact binary systems using machine learning. The dynamics of compact binary systems, such as black holes and neutron stars, present significant nonlinear challenges due to the strong gravitational interactions and the requirement for precise numerical simulations. Traditional methods, like the post-Newtonian approximation, often require significant computational resources and face challenges in accuracy and efficiency. Here, we employed machine learning algorithms, including deep learning models like long short-term memory (LSTM) and temporal convolutional network (TCN), to predict the future evolution of these systems based on extensive simulation data. Our results demonstrate that employing both LSTM and TCN even as black-box predictors for sequence prediction can also significantly improve the prediction accuracy without physics-informed neural networks (as partial differential equation solvers with prior knowledge or inductive bias. By employing LSTM and TCN, we obtained R 2 values of 99.74% and 99.19% for the evolutionary orbits of the compact binaries data set, respectively. Our models demonstrate the ability to effectively capture the dynamics of the binaries, achieving high prediction performance with significantly reduced computational overhead by a factor of 40, compared to conventional numerical methods. This study paves the way for more effective and computationally scalable approaches to the understanding of gravitational phenomena and predictive modeling in gravitational-wave astronomy.

Prediction of directional solidification in freeze casting of biomaterial scaffolds using physics-informed neural networks

Freeze casting, a manufacturing technique widely applied in biomedical fields for fabricating biomaterial scaffolds, poses challenges for predicting directional solidification due to its highly nonlinear behavior and complex interplay of process parameters. Conventional numerical methods, such as computational fluid dynamics (CFD), require adequate and accurate boundary condition knowledge, limiting their utility in real-world transient solidification applications due to technical limitations. In this study, we address this challenge by developing a physics-informed neural networks (PINNs) model to predict directional solidification in freeze-casting processes. The PINNs model integrates physical constraints with neural network predictions, requiring significantly fewer predetermined boundary conditions compared to CFD. Through a comparison with CFD simulations, the PINNs model demonstrates comparable accuracy in predicting temperature distribution and solidification patterns. This promising model achieves such a performance with only 5000 data points in space and time, equivalent to 250,000 timesteps, showcasing its ability to predict solidification dynamics with high accuracy. The study’s major contributions lie in providing insights into solidification patterns during freeze-casting scaffold fabrication, facilitating the design of biomaterial scaffolds with finely tuned microstructures essential for various tissue engineering applications. Furthermore, the reduced computational demands of the PINNs model offer potential cost and time savings in scaffold fabrication, promising advancements in biomedical engineering research and development.

Derivation of analytic formulae for several resonance frequencies of the SparkJet actuator

Building on previous research that emphasized the importance of focusing on resonance frequencies for a fundamental understanding of the thrust and complex internal flow phenomena of the SparkJet actuators, this study theoretically derives analytic formulae for several important resonance frequencies. The research addresses the typical configuration of the SparkJet actuators, which consists of a cylindrical cavity and orifice connected by a conical converging nozzle. While the resonance frequencies of the SparkJet actuators can be obtained by solving the eigenvalue problem presented in previous studies, this eigenvalue problem, despite being a typical boundary value problem in the form of an elliptic partial differential equation, is challenging to solve using conventional numerical methods such as iterative methods, because the eigenvalue is included as an unknown in the operator. Consequently, Boundary Element Method (BEM) or methods using the Green functions have been proposed to obtain numerical solutions, but these still require handling large matrix data, resulting in significant computational costs and memory consumption. To overcome this, the current study first simplifies the eigenvalue problem using the conformal mapping presented in previous research. Then, referencing prior studies that claim that three types of resonance frequencies (Helmholtz resonance frequency and resonance frequencies representing streamwise or radial directional oscillation) significantly affect the thrust of the SparkJet actuators, characteristics of these frequencies are mathematically defined. Using these mathematical characteristics, the study derives the asymptotic and approximate solutions of the simplified eigenvalue problem, from which the resonance frequencies are obtained. The analytic formulae proposed in this study theoretically explain the already known geometric tendencies of the resonance frequencies and reveal new geometric factors influencing resonance frequencies, which were previously unknown. This approach is expected to facilitate obtaining several important resonance frequencies of the SparkJet actuator more promptly and accurately and to provide a deeper understanding of the nature of the complex oscillation phenomena inside the actuator.

Accurate numerical calculation method for liquid crystal phase shifter loaded with metallic NaM

ABSTRACT Metallic Nanowire Filled Membrane (NaM) exhibit immense potential in enhancing liquid crystal phase shifters (LCPSs) performance. However, conventional numerical calculation methods are inadequate for accurately simulating LCPSs loaded with NaM due to the significant dimensional disparities between the nanoscale NaM and the millimetre or even centimetre-scale LCPS. In this paper, a novel calculation technology is proposed to accurately simulate the electromagnetic (EM) response of LCPSs integrated with NaM, including phase shift and figure of merit (FoM). The effective constitutive tensors of the NaM are extracted and utilised to construct the EM model for characterising the NaM properties. Three NaM-LCPSs are simulated using the proposed EM model and compared with measurement results. The findings reveal that the proposed calculation method exhibits an error of less than 20%, while previously reported numerical calculation method demonstrated error as high as 60%. Furthermore, leveraging the proposed calculation method, the performance of a NaM-LCPS is optimised by systematically varying its dimensional parameters. The results demonstrate a substantial improvement in the FoM of the NaM-LCPS, increasing from 67°/dB to 120°/dB, significantly expanding the potential applications of NaM-LCPSs.

A novel least squares approach generating approximations orthogonal to the null space of the operator

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.

Multi-Fidelity Learned Emulator for Waves and Porous Coastal Structures Interaction Modelling

A thorough understanding and treatment of wave-structure interaction (WSI) mechanics is essential for the rigorous engineering design of coastal protections. Conventional numerical analysis methods are accurate and generalize well, but are heavily dependent on the adopted mesh resolution and frequently incur substantial computational costs. To bypass these limitations, a meshless multi-fidelity residual neural network (MRNN) emulator is introduced in this study to infer the spatio-temporal responses arising from WSI. MRNN first employs a ‘low-fidelity’ simulator to learn basic WSI relationships by training on simulations obtained using a coarse numerical mesh. Subsequently, a ‘high-fidelity’ (HF) simulator is then employed to learn the mapping between numerical simulations performed using the coarse mesh and additional detailed fine meshes. The results indicate that MRNN is a highly robust emulator which requires significantly less HF data through its hierarchical framework compared to conventional single-fidelity data-driven strategies. By way of example, the MRNN emulator is applied to the cases of a porous dam break and breakwater. A broad spectrum of WSI responses, such as water through the porous dam medium, can be accurately captured using the MRNN emulator which is benchmarked against a conventional numerical modelling with a fine mesh. The computational efficiency of the MRNN is shown to be independent of the mesh resolution and complexity of the studied partial differential equations. It provides a generic and utilitarian emulator for any engineering problem of interest.

A surrogate-assisted extended generative adversarial network for parameter optimization in free-form metasurface design

Metasurfaces have widespread applications in fifth-generation (5G) microwave communication. Among the metasurface family, free-form metasurfaces excel in achieving intricate spectral responses compared to regular-shape counterparts. However, conventional numerical methods for free-form metasurfaces are time-consuming and demand specialized expertise. Alternatively, recent studies demonstrate that deep learning has great potential to accelerate and refine metasurface designs. Here, we present XGAN, an extended generative adversarial network (GAN) with a surrogate for high-quality free-form metasurface designs. The proposed surrogate provides a physical constraint to XGAN so that XGAN can accurately generate metasurfaces monolithically from input spectral responses. In comparative experiments involving 20000 free-form metasurface designs, XGAN achieves 0.9734 average accuracy and is 500 times faster than the conventional methodology. This method facilitates the metasurface library building for specific spectral responses and can be extended to various inverse design problems, including optical metamaterials, nanophotonic devices, and drug discovery.

A weighted stabilized lagrange interpolation collocation method for boundary condition identification in 3D electromagnetic inverse scattering

The identification of boundary conditions in electromagnetic inverse scattering is of importance in various engineering applications, ranging from geophysical exploration to wireless communication. Conventional numerical methods solving this problem often suffer from the iterative process, leading to inefficiencies and non-convergence. This paper introduces a weighted scheme of the stabilized Lagrange interpolation collocation method (weighted SLICM) to resolve this problem. Weighted SLICM efficiently integrates governing equations, boundary conditions, and measurement conditions using a weighted least squares approach, offering a straightforward single-step solution and obviating the need for iterative processes in traditional methods like the finite element method. By incorporating regularization techniques, weighted SLICM decreases measurement errors which are unavoidable in engineering problems, thereby ensuring high efficiency and accuracy. In addition, characterized as a strong-form collocation method that relies solely on point information but not on grid connectivity, the weighted SLICM is readily extendible to complex three-dimensional applications in electromagnetic inverse scattering. Extensive simulations of benchmark problems show its ability to achieve accurate and stable results in boundary condition identification in electromagnetic inverse scattering problems including 1D, 2D, and 3D environments, highlighting the effectiveness of the weighted SLICM in navigating complex engineering challenges and substantially enriching research methodologies in this area.

Unveiling advection–dominated interactions: Efficacy of neural networks in natural systems modelling

Studying the interactions between advection and dispersion in natural systems, especially in cases where advection predominates, are important because it is necessary to accurately model physical phenomena in domains like hydrology, atmospheric science, environmental engineering, etc. Conventional analytical and numerical methods often have drawbacks, such as high computational costs and challenges in accurately modeling complex behaviors. Improved simulation and a deeper understanding of these interactions are made possible by neural networks’ capacity to learn from big datasets and simulate complex relationships. To demonstrate that deep neural networks effectively capture scenarios within physical processes where advection plays a dominant role, the effectiveness of applying deep neural networks to a third-order dispersive partial differential equation incorporating advection and dispersion terms has been investigated in this study.

Comparison of machine learning and electrical resistivity arrays to inverse modeling for locating and characterizing subsurface targets

This study evaluates the performance of multiple machine learning (ML) algorithms and electrical resistivity (ER) arrays for inversion with comparison to a conventional Gauss-Newton numerical inversion method. Four different ML models and four arrays were used for the estimation of only six variables for locating and characterizing hypothetical subsurface targets. The combination of dipole-dipole with Multilayer Perceptron Neural Network (MLP-NN) had the highest accuracy. Evaluation showed that both MLP-NN and Gauss-Newton methods performed well for estimating the matrix resistivity while target resistivity accuracy was lower, and MLP-NN produced sharper contrast at target boundaries for the field and hypothetical data. Both methods exhibited comparable target characterization performance, whereas MLP-NN had increased accuracy compared to Gauss-Newton in prediction of target width and height, which was attributed to numerical smoothing present in the Gauss-Newton approach. MLP-NN was also applied to a field dataset acquired at U.S. DOE Hanford site.

Advanced neural network approaches for coupled equations with fractional derivatives

We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods.Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems.

Pressure swing adsorption process modeling using physics-informed machine learning with transfer learning and labeled data

Pressure swing adsorption (PSA) modeling remains a challenging task since it exhibits strong dynamic and cyclic behavior. This study presents a systematic physics-informed machine learning method that integrates transfer learning and labeled data to construct a spatiotemporal model of the PSA process. To approximate the latent solutions of partial differential equations (PDEs) in the specific steps of pressurization, adsorption, heavy reflux, counter-current depressurization, and light reflux, the system's network representation is decomposed into five lightweight sub-networks. On this basis, we propose a parameter-based transfer learning (TL) combined with domain decomposition to address the long-term integration of periodic PDEs and expedite the network training process. Moreover, to tackle challenges related to sharp adsorption fronts, our method allows for the inclusion of a specified amount of labeled data at the boundaries and/or within the system in the loss function. The results show that the proposed method closely matches the outcomes achieved through the conventional numerical method, effectively simulating all steps and cyclic behavior within the PSA processes.

Walkin’ Robin: Walk on Stars with Robin Boundary Conditions

Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes the walk on stars (WoSt) algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt produces bounded-variance estimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as the walk on boundary algorithm. We also develop bidirectional and boundary value caching strategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.

Inf-sup neural networks for high-dimensional elliptic PDE problems

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

Cubic Hermite interpolators on the space of probability measures

In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures . First, we develop new approaches to capture the Riemannian geometric structure of when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on , namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail.

A parameter estimation method for chromatographic separation process based on physics-informed neural network

Chromatographic separation processes are most often modeled in the form of partial differential equations (PDEs) to describe the complex adsorption equilibria and kinetics. However, identifying parameters in such a model requires substantial computational effort. In this work, a novel parameter estimation approach using a Physics-informed Neural Network (PINN) model is developed and tested for a binary component system. Numerical accuracy of our PINN model is confirmed by validating its simulations against those of the finite element method (FEM). Furthermore, model parameters in the kinetic model are estimated by the PINN model with sufficient accuracy from the observed data at the column outlet, where parameter fitting error can be reduced by up to 35.0 % from the conventional method. In a comparison with the conventional numerical method, our approach can reduce the computational time by up to 95 %. The robustness of the PINN model has also been demonstrated by estimating model parameters from noisy artificial experimental data.

A deep learning operator-based numerical scheme method for solving 1D wave equations

Abstract In this paper, we introduce the deep numerical technique DeepNM, which is designed for solving one-dimensional (1D) hyperbolic conservation laws, particularly wave equations. By creatively integrating traditional numerical schemes with deep learning techniques, the method yields improvements over conventional approaches. Specifically, we compare this approach against two established classical numerical methods: the discontinuous Galerkin method (DG) and the Lax–Wendroff correction method (LWC). While maintaining a comparable level of accuracy, DeepNM significantly improves computational speed, surpassing conventional numerical methods in this aspect by more than tenfold, and reducing storage requirements by over 1000 times. Furthermore, DeepNM facilitates the utilization of higher-order numerical schemes and allows for an increased number of grid points, thereby enhancing precision. In contrast to the more prevalent PINN method, DeepNM optimally combines the strengths of conventional mathematical techniques with deep learning, resulting in heightened accuracy and expedited computations for solving partial differential equations. Notably, DeepNM introduces a novel research paradigm for numerical equation-solving that can be seamlessly integrated with various traditional numerical methods.