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.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
This paper presents a combined seismic metamaterial model based on the principle of local resonance to effectively shield Rayleigh waves at wide low-frequency bandgap in two-layered soil. In the ...proposed combined seismic metamaterial model, the unit-cell structure comprises two parts: a partially embedded underground concrete column and a multi-layer composite material structure enclosing the underground portion of the embedded concrete columns. By using the finite element method, the band structure and propagation characteristics of the proposed combined seismic metamaterial model are numerically investigated. Later the concrete column height effect is investigated, such as the combined seismic metamaterial model with positive gradient arrangement, negative gradient arrangement, concave arrangement, and convex arrangement. Then, the El-Centro seismic wave is considered and found that it can be considerably attenuated by the designed combined seismic metamaterial model. Additionally, this work does parameter analysis on items such as concrete density and column height.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
3.
A mesh-free algorithm for ROF model Khan, Mushtaq Ahmad; Chen, Wen; Ullah, Asmat ...
EURASIP Journal on Advances in Signal Processing,
07/2017, Volume:
2017, Issue:
1
Journal Article
Peer reviewed
Open access
The total variation (TV) denoising method is a PDE-based technique that preserves the edges well but has undesirable staircase effect in some cases, namely, the translation of smooth regions (ramps) ...into piecewise constant regions (stairs). This paper introduces a novel mesh-free approach using TV (ROF model) regularization and radial basis function (RBF) for the numerical approximation of TV-based model to remove the additive noise from the measurements. This approach is structured on local collocation and multiquadric radial basis function. These features enable this strategy not only to eliminate noise from images and preserve the edges but also has the advantage to minimize the staircase effect substantially from real and artificial images which cause the image to look blocky. Experimental results demonstrate that the proposed mesh-free approach is robust and performs well in visual improvement as well as peak signal-to-noise ratio compared with the recent partial differential equation (PDE)-based traditional methods.
A machine learning strategy based on the semi-analytical singular boundary method (SBM) is presented for the thermal conductivity identification of functionally graded materials (FGMs). In this ...study, only the temperature or heat flux on the surface or interior of FGMs can be measured by the thermal sensors, and the SBM is used to construct the database of the relationship between the thermal conductivity and the temperature distribution of the functionally graded structure. Based on the aforementioned constructed database, the artificial neural network-based machine learning strategy was implemented to identify the thermal conductivity of FGMs. Finally, several benchmark examples are presented to verify the feasibility, robustness, and applicability of the proposed machine learning strategy.
In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. ...PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the extrinsic approach with PINNs and then express that loss function in extrinsic form. Subsequently, the loss function could be minimized extrinsically with respect to the NN parameters. Numerical results demonstrate that the extrinsic approach based on PINNs for surface problems has good accuracy and higher efficiency compared with the embedding approach based on PINNs. In addition, the strong nonlinear mapping ability of NNs makes this approach robust in solving time-dependent nonlinear problems on more complex surfaces.
This paper firstly employs the fast multipole method (FMM) to accelerate the singular boundary method (SBM) solution of the Stokes equation. We present a fast multipole singular boundary method ...(FMSBM) based on the combination of the SBM and the FMM. The proposed FMSBM scheme reduces CPU operations and memory requirements by one order of magnitude, namely O(N) (where N is the number of boundary nodes). Thus, the strategy overcomes costly expenses of the SBM due to its dense interpolation matrix while keeping its major merits being free of mesh, boundary-only discretization, and high accuracy in the solution of the Stokes equation. The performance of this scheme is tested to a few benchmark problems. Numerical results demonstrate its efficiency, accuracy and applicability.
•A fast multipole singular boundary method is presented.•The fast multipole formulations of the singular boundary method are derived for both the velocity and traction.•The CPU time and the memory requirement of the singular boundary method are both reduced to O(N).•The new strategy overcomes costly expenses of the singular boundary method due to its dense interpolation matrix.
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A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small ...subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo‐spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo‐spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large‐scale simulations. The present method can be viewed as a competitive alternative for solving numerically large‐scale boundary value problems with complex‐shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified‐Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.
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In this paper, a novel semi-analytical collocation solver, the spatial–temporal radial Trefftz collocation method (STRTCM) is proposed to solve 3D transient wave equations with specified sound source ...excitations. Unlike the traditional time discretization strategies, the proposed numerical scheme introduces the spatial–temporal radial Trefftz functions (STRTFs) as the basis functions for the spatial and temporal discretization of the transient wave equations. The STRTFs are constructed in the spatial–temporal domain, which is a combination of 3D Euclidean space and time into a 4D manifold. Moreover, since the initial and boundary conditions are imposed on the spatial–temporal domain boundaries, the original transient wave propagation problem can be converted to an inverse boundary value problem. To deal with the specified time-dependent sound source excitations, the composite multiple reciprocity technique is extended from the spatial domain to the spatial–temporal domain, which transforms the original problem with a source term into a high-order problem without a source term. By deriving the related STRTFs for the considered high-order problem, the proposed scheme only requires the node discretization on the spatial–temporal domain boundaries. The efficiency of the proposed method is numerically verified by four benchmark examples under 3D transient wave equations with specified time-dependent sound source excitation.
A strong-form boundary collocation method, the singular boundary method (SBM), is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures. The ...SBM is of several advantages including mathematically simple, easy-to-program, meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method. Due to the periodic behaviors of the structures, the SBM coefficient matrix can be represented as a block Toeplitz matrix. By employing three different fast Toeplitz-matrix solvers, the computational time and storage requirements are significantly reduced in the proposed SBM analysis. To demonstrate the effectiveness of the proposed SBM formulation for wave propagation analysis in periodic structures, several benchmark examples are presented and discussed The proposed SBM results are compared with the analytical solutions, the reference results and the COMSOL software.
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The paper makes the first attempt at applying the Kansa method, a radial basis function meshless collocation method, to the space-fractional advection-dispersion equations, which have recently been ...observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer-order derivative. However, because of non-local property of Integra-differential operator of space-fractional derivative, numerical solution of these novel models is very challenging and little has been reported in literature. It is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non-local property. Thus, the global methods appear to have certain advantages in numerical simulation of these non-local models because of their high accuracy and smaller size resultant matrix equation. Compared with the finite difference method, popular in the solution of fractional equations, the Kansa method is a recent meshless global technique and is promising for high-dimensional irregular domain problems. In this study, the resultant matrix of the Kansa method is accurately calculated by the Gauss-Jacobi quadrature rule. Numerical results show that the Kansa method is highly accurate and computationally efficient for space-fractional advection-dispersion problems.
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