Computers & Mathematics with Applications最新文献

In this paper, we demonstrate the applicability of an exact truncation method for the solution of waves scattering problems in unbounded media, known as the Jami-Lenoir method, to linear elasticity. Our approach avoids the usual splitting of waves as the sum of longitudinal and transversal waves in the analysis and in the numerical modeling of elastodynamic waves scattering problems. The exact absorbing condition imposed on the computational boundary gathers the outgoing behavior of scattered waves given by their Green's integral representation formula with a modified Kupradze radiation condition that ensures uniqueness results and improve the system's conditioning. The truncation boundary can even be closely located from the obstacle with a distance of a few element lengths. Numerical experiments show the accuracy of the Jami-Lenoir approach and the efficiency of the Schwarz preconditioner for the solution of the exterior Neumann problem with a Krylov iterative solver.

We propose conservative type numerical method to simulate thermoelectrical semiconductor device problem, in which mixed finite element method used for electric potential equation, conservative characteristic finite element method for electron and hole concentration equations, and standard finite element method for heat conduction equation. By temporal-spatial error splitting argument, the optimal error estimates without certain time step restriction are derived, and low order convergence rate of electrostatic potential and electric field intensity will not affect the accuracy of the electron, hole density and temperature. Numerical tests are performed to validate the theoretical results and application performance of the given method.

The alternating direction method of multipliers within a shape optimization framework is developed for solving geometric inverse problems, focusing on a cavity identification problem from the perspective of non-destructive testing and evaluation techniques. The rationale behind this method is to achieve more accurate detection of unknown inclusions with pronounced concavities, emphasizing the aspect of shape optimization. Several numerical results to illustrate the applicability and efficiency of the method are presented for various shape detection problems. These numerical experiments are conducted in both two- and three-dimensional settings, with a focus on cases involving noise-contaminated data. The main finding of the study is that the proposed method significantly outperforms conventional shape optimization methods based on first-order optimality conditions in reconstructing unknown cavity shapes. This superior performance is demonstrated through more numerically accurate constructions compared to classical methods.

Simulations of large-scale dynamical systems require expensive computations and large amounts of storage. Low-dimensional representations of high-dimensional states such as in reduced order models deriving from, say, Proper Orthogonal Decomposition (POD) trade in a reduced model complexity against accuracy and can be a solution to lessen the computational burdens. However, for really low-dimensional parametrizations of the states as they may be needed for example for controller design, linear methods like the POD come to their natural limits so that nonlinear approaches will be the methods of choice. In this work, we propose a convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider a deep clustering model where a CAE is integrated with k-means clustering for improved encoding performance. The proposed set of methods is compared to the standard POD approach in three scenarios: single- and double-cylinder wakes modeled by incompressible Navier-Stokes equations and flow setup described by viscous Burgers' equations.

In this paper, we derive the improved uniform error bounds for the long-time dynamics of the d-dimensional $(d=2,3)$ nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by ${\epsilon }^2$ where $0<\epsilon \le 1$ is a dimensionless parameter. The second-order time-splitting method is applied to the temporal discretization and the Fourier pseudo-spectral method is used for the spatial discretization. To obtain the explicit relation between the numerical errors and the parameter ε, we introduce the regularity compensation oscillation technique to the convergence analysis of fractional models. Then we establish the improved uniform error bounds $O\left({\epsilon }^2{\tau }^2\right)$ for the semi-discretization scheme and $O(h^m+{\epsilon }^2{\tau }^2)$ for the full-discretization scheme up to the long time at $O(1/{\epsilon }^2)$. Further, we extend the time-splitting Fourier pseudo-spectral method to the complex NSFSGE as well as the oscillatory complex NSFSGE, and the improved uniform error bounds for them are also given. Finally, extensive numerical examples in two-dimension or three-dimension are provided to support the theoretical analysis. The differences in dynamic behaviors between the fractional sine-Gordon equation and classical sine-Gordon equation are also discussed.

In this work, a least-square finite difference-based physics-informed neural network (LSFD-PINN) is proposed to simulate steady incompressible flows. The original PINN employs the automatic differentiation (AD) method to compute differential operators. However, the AD method, which is essentially based on the chain rule, requires a series of matrix operations to obtain derivatives during the training process. This may reduce computational efficiency, especially for large-scale networks. Additionally, the AD method still needs to compute lower-order derivative terms even if the partial differential equation (PDE) involves only higher-order derivatives, leading to unnecessary calculations. Although conventional finite difference (FD) methods can effectively mitigate these limitations, they only consider information in a single direction. Moreover, they require introducing extra virtual collocation points for each collocation point to assist in computing differential operators when using randomly distributed collocation points. This increases the computational effort and storage requirements, especially in scenarios involving high-order discretization schemes or a large number of collocation points. To address these issues, we introduced the least squares finite difference (LSFD) method to calculate the differential operators required in PINN. Compared to the AD method, the LSFD method relies only on the network's output for calculating derivatives, thus avoiding a series of matrix operations. In comparison to the FD method, the LSFD not only considers multi-directional information but also can be applied to random point distributions without the need for introducing virtual points. To demonstrate its effectiveness, LSFD-PINN is tested on representative problems such as lid-driven cavity flow, flow around a backward-facing step, and flow around a circular cylinder in a pipe. Numerical results indicate that LSFD-PINN achieves satisfactory accuracy without any labeled data, significantly outperforming AD-PINN and FD-PINN, especially in high Reynolds number flows. Additionally, the computational efficiency of LSFD-PINN is superior to that of AD-PINN.

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

We present a family of generic a posteriori error estimators for the two-field Biot contact problem. While every family member of these estimators is reliable only certain members are also efficient. A crucial property of our error estimator is that it can measure the error of any approximation, not only of approximations with Galerkin orthogonality. Hence, it can be easily coupled with primal-dual active set algorithms. Additionally, we present explicitly an hp-finite element discretization and its residual based a posteriori error estimator based on the generic setup. Several numerical experiments underline the theoretical results.

In this paper, a high-order generalised differential quadrature element method (GDQE) is proposed to simulate two-dimensional (2D) and three-dimensional (3D) incompressible flows on unstructured meshes. In this method, the computational domain is decomposed into unstructured elements. In each element, the high-order generalised differential quadrature (GDQ) discretisation is applied. Specifically, the GDQ method is utilised to approximate the partial derivatives of flow variables and fluxes with high-order accuracy inside each element. At the shared interfaces between different GDQ elements, the common flux is computed to account for the information exchange, which is achieved by the lattice Boltzmann flux solver (LBFS) in the present work. Since the solution in each GDQ element solely relies on information from itself and its direct neighbouring element, the developed method is authentically compact, and it is naturally suitable for parallel computing. Furthermore, by selecting the order of elemental GDQ discretisation, arbitrary accuracy orders can be achieved with ease. Representative incompressible flow problems, including 2D laminar flows as well as 3D turbulent simulations, are considered to evaluate the accuracy, efficiency, and robustness of the present method. Successful numerical simulations, especially for scale-resolving 3D turbulent flow problems, confirm that the present method is efficient and high-order accurate.

In this paper we present a novel approach for the design of high order general boundary conditions when approximating solutions of the Euler equations on domains with curved boundaries, using meshes which may not be boundary conformal. When dealing with curved boundaries and/or unfitted discretizations, the consistency of boundary conditions is a well-known challenge, especially in the context of high order schemes. In order to tackle such consistency problems, the so-called Reconstruction for Off-site Data (ROD) method has been recently introduced in the finite volume framework: it is based on performing a boundary polynomial reconstruction that embeds the considered boundary treatment thanks to the implementation of a constrained minimization problem. This work is devoted to the development of the ROD approach in the context of discontinuous finite elements. We use the genuine space-time nature of the local ADER predictors to reformulate the ROD as a single space-time reconstruction procedure. This allows us to avoid a new reconstruction (linear system inversion) at each sub-time node and retrieve a single space-time polynomial that embeds the considered boundary conditions for the entire space-time element. Several numerical experiments are presented proving the consistency of the new approach for all kinds of boundary conditions. Computations involving the interaction of shocks with embedded curved boundaries are made possible through an a posteriori limiting technique.