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In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

In this paper, we study the orthogonal Gauss collocation method (OGCM) with an arbitrary polynomial degree for the numerical solution of a two-dimensional (2D) fourth-order subdiffusion model. This numerical method involves solving a coupled system of partial differential equations by using OGCM in space together with the L1 scheme in time on a graded mesh. The approximations w h n and v h n of w ( · , t n ) and Δ w ( · , t n ) are constructed. The stability of w h n and v h n are proved, and the a priori bounds of ‖ w h n ‖ and ‖ v h n ‖ are established, remaining α -robust as α → 1 - . Then, the error ‖ w ( · , t n ) - w h n ‖ and ‖ Δ w ( · , t n ) - v h n ‖ are estimated with α -robust at each time level. In addition, superconvergence results of the first-order and second-order derivative approximations are proved. These new error bounds are desirable and natural, as that they are optimal in both temporal and spatial mesh parameters for each fixed α . Finally some numerical results are provided to support our theoretical findings.

This paper presents a summary of various localized collocation schemes and their engineering applications. The basic concepts of localized collocation methods (LCMs) are first introduced, such as approximation theory, semianalytical collocation methods and localization strategies. Based on these basic concepts, five different formulations of localized collocation methods are introduced, including the localized radial basis function collocation method (LRBFCM) and the generalized finite difference method (GFDM), the localized method of fundamental solutions (LMFS), the localized radial Trefftz collocation method (LRTCM), and the localized collocation Trefftz method (LCTM). Then, several additional schemes, such as the generalized reciprocity method, Laplace and Fourier transformations, and Krylov deferred correction, are introduced to enable the application of the LCM to large-scale engineering and scientific computing for solving inhomogeneous, nonisotropic and time-dependent partial differential equations. Several typical benchmark examples are presented to show the recent developments and applications on the LCM solution of some selected boundary value problems, such as numerical wave flume, potential-based inverse electrocardiography, wave propagation analysis and 2D phononic crystals, elasticity and in-plane crack problems, heat conduction problems in heterogeneous material and nonlinear time-dependent Burgers’ equations. Finally, some conclusions and outlooks of the LCMs are summarized.

In many dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also possess a continuous nature in a sense that their order is distributed over a given range. This paper is concerned with the optimization of systems whose governing equations contain a fractional derivative of distributed order, in the Caputo sense. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration-by-parts formula, the necessary optimality conditions are derived in terms of a nonlinear two-point distributed-order fractional boundary value problem. A Legendre spectral collocation method is developed for solving such problem. The solution method involves the use of three-term recurrence relations for both the left- and right-sided fractional integrals of the shifted Legendre polynomials. The convergence of the proposed method is discussed. The optimal profiles show the performance of the numerical solution and the effect of the fractional derivatives in the optimal results.

In this work, we present a deep collocation method (DCM) for three-dimensional potential problems in non-homogeneous media. This approach utilizes a physics-informed neural network with material transfer learning reducing the solution of the non-homogeneous partial differential equations to an optimization problem. We tested different configurations of the physics-informed neural network including smooth activation functions, sampling methods for collocation points generation and combined optimizers. A material transfer learning technique is utilized for non-homogeneous media with different material gradations and parameters, which enhance the generality and robustness of the proposed method. In order to identify the most influential parameters of the network configuration, we carried out a global sensitivity analysis. Finally, we provide a convergence proof of our DCM. The approach is validated through several benchmark problems, also testing different material variations.

The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.

The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations. This paper provides a comprehensive review of collocation methods and their applications, focused on elasticity, heat conduction, electromagnetic field analysis, and fluid dynamics. The merits of the collocation method can be attributed to the need for element mesh, simple implementation, high computational efficiency, and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method. Beginning with the fundamental principles of the collocation method, the discretization process in the continuous domain is elucidated, and how the collocation method approximation solutions for solving differential equations are explained. Delving into the historical development of the collocation methods, their earliest applications and key milestones are traced, thereby demonstrating their evolution within the realm of numerical computation. The mathematical foundations of collocation methods, encompassing the selection of interpolation functions, definition of weighting functions, and derivation of integration rules, are examined in detail, emphasizing their significance in comprehending the method’s effectiveness and stability. At last, the practical application of the collocation methods in engineering contexts is emphasized, including heat conduction simulations, electromagnetic coupled field analysis, and fluid dynamics simulations. These specific case studies can underscore collocation method’s broad applicability and effectiveness in addressing complex engineering challenges. In conclusion, this paper puts forward the future development trend of the collocation method through rigorous analysis and discussion, thereby facilitating further advancements in research and practical applications within these fields.

In the present paper, we construct the numerical solution for time fractional (1 + 1)- and (1 + 2)-dimensional Schrödinger equations (TFSEs) subject to initial boundary. The solution is expanded in a series of shifted Jacobi polynomials in time and space. A collocation method in two steps is developed and applied. First step depends mainly on application of shifted Jacobi Gauss-Lobatto-collocation method for spatial discretization on the approximate solution and its spatial derivatives occurring in the TFSE and substitution in the boundary conditions or treatment of the non-local conservation conditions by the Jacobi Gauss-Lobatto quadrature rule. As a result, a system of fractional differential equation for the expansion coefficients is obtained. The second step is to use a shifted Jacobi Gauss-Radau- collocation scheme, for temporal discretization, to reduce such system into a system of nonlinear Newton iterative method. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithms demonstrating superiority over other methods.

The reproducing kernel (RK) approximation based direct collocation method (DCM) requires the complex and time-consuming derivatives calculation of the approximation function, and the DCM has the poor accuracy and stability, which hinder the extensive application of this method. Therefore, in this work, we propose a gradient reproducing kernel (GRK) based stabilized collocation method (SCM) which can manage the computational complexity of the RK derivatives calculation and significantly improve the efficiency by directly constructing the GRK approximations, and handle the accuracy and stability problems of the DCM by employing the SCM. Moreover, the computation cost of the SCM is about the same level of the DCM, which is much more efficient than the Galerkin meshfree methods. The proposed method is particularly suitable for solving the thin beam and plate problems which requests the fourth-order differentiation in the strong form. The implementations of this method for the static and dynamical problems are detailedly exhibited. Numerical examples confirm that the presented algorithm provides high efficiency and good performance for the beam and plate simulations.

A general arbitrary order recursive gradient formulation is presented for meshfree approximation. According to this method, an n th order recursive meshfree gradient is formulated as an interpolation of the ( n  − 1)th order gradients by standard first order meshfree gradients, which finally can be expressed as a successive multiplication of standard first order meshfree gradients. This formulation avoids the complex and costly computation of conventional high order derivatives of meshfree shape functions. One crucial ingredient of the proposed methodology is that the resulting recursive meshfree gradients with a p th degree basis function not only meet the conventional p th order consistency conditions for standard gradients, but also satisfy ( p  + 1)th to ( p  +  n  − 1)th extra high order consistency conditions. This important property leads to superconvergent meshfree collocation algorithms and here we focus on the classical fourth order Kirchhoff plate problems. An accuracy analysis of the proposed recursive gradient meshfree collocation formulation for Kirchhoff plates reveals that superconvergence is simultaneously achieved for both even and odd degrees of basis functions. More specifically, two and four additional orders of accuracy are respectively gained by the proposed method for even and odd degree basis functions, compared with the standard meshfree collocation scheme. Furthermore, the extra high order consistency conditions of recursive meshfree gradients enable superconvergent meshfree collocation analysis of Kirchhoff plates using low order basis functions of less than 4th degree, while the standard meshfree collocation approach requires at least a 4th degree basis function to maintain convergence. The accuracy and efficiency of the proposed methodology are holistically demonstrated by numerical results.