查看更多>>摘要：We present PINNs-MPF framework, an application of Physics-Informed Neural Networks (PINNs) to handle Multi-Phase-Field (MPF) simulations of microstructure evolution. A combination of optimization techniques within PINNs and in direct relation to MPF method are extended and adapted. The numerical resolution is realized through a multi-variable time-series problem by using fully discrete resolution. Within each interval, space, time, and phases/grains are treated separately, constituting discrete subdomains. PINNs-MPF is equipped with an extended multi-networking (parallelization) concept to subdivide the simulation domain into multiple batches, with each batch associated with an independent NN trained to predict the solution. To ensure continuity across the spatio-temporal-phasic subdomains, a Master NN efficiently is to handle interactions among the multiple networks and facilitates the transfer of learning. A pyramidal training approach is proposed to the PINN community as a dual-impact method: to facilitate the initialization of training when dealing with multiple networks, and to unify the solution through an extended transfer of learning. Furthermore, a comprehensive approach is adopted to specifically focus the attention on the interfacial regions through a dynamic meshing process, significantly simplifying the tuning of hyper-parameters, serving as a key concept for addressing MPF problems using machine learning. We perform a set of systematic simulations that benchmark foundational aspects of MPF simulations, i.e., the curvature-driven dynamics of a diffuse interface, in the presence and absence of an external driving force, and the evolution and equilibrium of a triple junction. The proposed PINNs-MPF framework successfully reproduces benchmark tests with high fidelity and Mean Squared Error (MSE) loss values ranging from 10~(-6) to 10~(-4) compared to ground truth solutions.
查看更多>>摘要：The problem of radiation diffusion is extremely challenging due to the complex physical processes and nonlinear characteristics of the equation involved. In this paper, we propose a class of high-precision meshless methods for 3D nonlinear radiation diffusion equations applicable to spherical and cylindrical walls. Firstly, when the energy density is linearly related to temperature, we use a full-implicit difference scheme to discretize the time term, and then approximate the spatial term using radial basis functions to construct a new solution scheme for solving the 3D linear radiation diffusion equation. Secondly, when dealing with the nonlinear relationship between energy density and temperature, we successfully reduced the complexity of problem to be by linearizing T~4. Then, we use radial basis functions to approximate unknown functions and established a large class of solving schemes, which solved by the Kansa's method. Finally, we validate the efficiency and high accuracy of the proposed methods through a series of numerical examples on spherical and cylindrical walls. In summary, the meshless numerical solution methods proposed in this paper not only avoids the complexity of meshing in irregular areas, but also provides a new and high-precision numerical solution method for the 3D radiation diffusion equation.
查看更多>>摘要：The traditional boundary element method (BEM) often faces challenges in efficiently solving inhomogeneous problems, particularly in thin-walled geometries, due to the need for domain discretization and the handling of nearly singular integrals. In this study, we propose an efficient hybrid algorithm that combines the BEM with physics-informed neural networks (PINNs) to solve inhomogeneous potential problems in thin-walled structures. The approach transforms inhomogeneous equations into equivalent homogeneous ones by subtracting a closed-form particular solution, which is derived using the learning capabilities of PINNs. This methodology not only simplifies the problem formulation but also enhances computational efficiency by eliminating the need for domain discretization, making it particularly well-suited for thin-walled geometries. Additionally, the scaled coordinate transformation BEM, a recently developed technique for solving domain integrals, is also employed for comparative analysis. Finally, a nonlinear coordinate transformation is employed to effectively regularize nearly singular integrals, which are critical in BEM for thin structures. The proposed method achieves accurate and reliable results with a small number of boundary elements, even for structures with extremely small thickness-to-length ratios, as low as 10~(-9). This makes the method highly suitable for modeling thin films and thin-walled structures, particularly in the context of advanced smart materials. The unique contribution of this work lies in the integration of PINNs with BEM to tackle challenges specific to thin-walled inhomogeneous problems, offering a more efficient and accurate solution compared to traditional BEM-based method.
查看更多>>摘要：Fracture initiation in solids fundamentally arises from pre-existing discontinuities, such as crack networks and void distributions, which are ubiquitously observed in engineering structures. This paper presents an innovative unsupervised learning framework, termed Kolmogorov-Arnold representation theorem enhanced peridynamic-informed neural network (PD-KINN), designed to address challenges in elastic deformation characterization and brittle damage prediction. The framework integrates the novel Kolmogorov-Arnold networks (KANs) with traditional physics-informed neural networks (PINNs), this hybrid architecture demonstrates parameter-efficient learning while maintaining similar or better predictive performance. Notably, the network leverages the nonlocal integral operator of peridynamics to naturally describe discontinuous variables, making it effective in modeling material deformation and fracture. Moreover, the transfer learning technique is implemented to account for the incremental loading histories and crack path evolution. Finally, comparative validation against analytical and numerical solutions confirms PD-KINN's superiority in handling fracture analysis of various solid structures under quasi-static loadings.
查看更多>>摘要：Numerical solutions of initial value problems (IVPs) for stiff differential equations via explicit methods such as Euler's method, trapezoidal method and Runge-Kutta methods suffer from stability issues and demand unacceptably small time steps. Backward differentiation formulas (BDF), a class of implicit methods, have been successfully used for resolving stiff IVPs. Classical BDF methods are derived using polynomial basis functions. In this paper, we develop radial basis function based finite difference (RBF-FD) type BDF methods for solving stiff problems. Therefore, we obtain analytical expressions for Gaussian and Multiquadric based RBF-BDF schemes along with their local truncation errors. Then we discuss the stability, order, consistency and convergence of RBF-BDF methods, which also depend on the free shape parameter. Finally, we validate the proposed methods by solving some benchmark problems. In order to gain enhanced accuracy, we adaptively choose the shape parameter such that local truncation error is minimized at each time-step. RBF-BDF methods of order two to six achieve at least one order greater accuracy and order of convergence than corresponding classical BDF schemes.
Sergey I. FomenkoMikhail V. GolubYanzheng WangAli Chen...
106217.1-106217.11页
查看更多>>摘要：Detailed studies of peculiar wave phenomena in piezoelectric metamaterials require advanced and accurate numerical methods. An extended boundary integral equation method based on the employment of the Fourier transform of Green's matrices and the Bubnov-Galerkin method is presented for the wave motion simulation of a multi-layered piezoelectric laminate with electrode arrays connected pairwise via electrical circuits. The semi-analytical nature of the method allows its application to find optimal configuration and tuning parameters of the considered piezoelectric laminated structures with arrays of electrodes. The proposed semi-analytical method is verified by the comparison with other numerical methods, and the convergence and the accuracy of the developed method are demonstrated using several representative examples of metamaterial configuration for piezoelectric bimorphs. Examples of tuning of stop bands and regimes of mode conversion are shown.
Tulio R.E. MarquesGabriela M. FonsecaRafael M. LinsFelicio B. Barros...
106219.1-106219.14页
查看更多>>摘要：In this work, the ZZ-BD recovered stress field is first used to enhance the data transferred from the global to the local scale models in the Generalized Finite Element Method with Global-Local enrichments (GFEM~(gl))-The recovered stress field is constructed by solving a block-diagonal system of equations resulting from an L_2 approximate function projection associated with the singular stress field in the crack tip neighboring. In GFEM~(gl) analysis, the global solution is imposed as Dirichlet or Cauchy-type boundary conditions in the local domain. In the former case, only displacements are considered. The main contribution of this work lies in the definition of the Cauchy boundary conditions, where the stress field is combined with the displacements. A two-dimensional plate problem with an edge crack under mixed opening mode is solved using GFEM~(gl). Stress intensity factors are extracted from global and local problems using the Interaction Integral strategy. Numerical results indicate that the Cauchy boundary conditions with the ZZ-BD recovered stress field provide a more accurate solution than raw or average stress fields, as well as regular Dirichlet boundary conditions. The effects of using a buffer zone in the local problem are also examined. Finally, the Interaction Integral performance strategy is investigated, with the key parameter being the circumference radius that intersects the elements where the stress intensity factors are extracted. An investigation is performed into the local and global problems, and a range of these parameters is identified to minimize errors in the stress intensity factors.
查看更多>>摘要：In this paper, we consider the numerical solutions of three-dimensional axisymmetric nonlinear boundary integral equations with logarithmic kernel. A numerical algorithm with using extrapolation twice is developed to solve the equations, which possesses the low computing complexities and high accuracy. The asymptotic compact operator theory is used to prove the convergence of the algorithm. The efficiency of the algorithm is illustrated by numerical examples.
查看更多>>摘要：The generalized nth-order perturbation method for the quantitative uncertainty analysis in half-space acoustic problems proposed in this study is based on the isogeometric boundary element method, where the acoustic wave frequency is defined as a stochastic variable. We derive the Taylor series expansion and the kernel function formulation of the acoustic boundary integral equation for the half-space acoustic problem, and obtain the sound pressure's nth-order derivative with respect to the acoustic wave frequency. In addition, we employ Burton-Miller method to deal with the fictitious frequency problem of external sound field and apply fast multipole method to accelerate the matrix-vector product computation. The statistical characterization of the acoustic state function is obtained based on the nth-order perturbation theory. Finally, the accuracy and efficacy of the uncertainty quantization algorithm is confirmed by three numerical examples.
查看更多>>摘要：This study introduces a novel spring element model for efficient simulation of nonlinear seepage in porous media. The model discretizes the simulation domain into tetrahedral elements and constructs orthogonal Three-dimensional permeability networks within each element, establishing a quantitative relationship between pipe flow and nodal pressure differences. By developing a mathematical model linking network flow to nodal pressure differences, the method enables precise allocation of pipe flow in the local coordinate system and accurate transformation to the global coordinate system, thereby determining nodal flow and velocity. The Three-Dimensional Seepage Spring Element Method (3D-SSEM) simplifies the element flow matrix in finite element analysis to three essential pipe permeability stiffness values, thereby reducing computational complexity. Coupled with parallel computing strategies, the algorithm achieves significant improvements in computational efficiency and memory usage. The method is validated through four numerical examples, demonstrating high efficiency and accuracy in solving saturated-unsaturated seepage problems. Compared with analytical solutions and other numerical methods, it exhibits superior convergence and reduced solution time while maintaining precision. Additionally, the method effectively simulates complex coupled processes in large-scale real-world environments, offering robust support for practical engineering design optimization.
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