Abstract: In this article, a novel meshless method using space–time radial polynomial basis function (SRPBF) for solving backward heat conduction problems is proposed. The SRPBF is constructed by incorporating time-dependent exponential function into the radial polynomial basis function. Dierent from the conventional radial basis function (RBF) collocation method that applies the RBF at each center point coinciding with the inner point, an innovative source collocation scheme using the sources instead of the centers is first developed for the proposed method. The randomly unstructured source, boundary, and inner points are collocated in the space–time domain, where both boundary as well as initial data may be regarded as space–time boundary conditions. The backward heat conduction problem is converted into an inverse boundary value problem such that the conventional time–marching scheme is not required. Because the SRPBF is infinitely dierentiable and the corresponding derivative is a nonsingular and smooth function, solutions can be approximated by applying the SRPBF without the shape parameter. Numerical examples including the direct and backward heat conduction problems are conducted. Results show that more accurate numerical solutions than those of the conventional methods are obtained. Additionally, it is found that the error does not propagate with time such that absent temperature on the inaccessible boundaries can be recovered with high accuracy.

Figure 1. Configuration of the collocation points for the direct heat conduction problem.

Figure 2. Illustration of the space–time collocation scheme: (a) The DHCP; (b) The BHCP.

Keywords: collocation method; space–time; radial polynomial; basis function; heat conduction

doi:10.3390/app10093215

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Abstract: https://www.mdpi.com/2076-3417/10/9/3215

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In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

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On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Abstract: In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.
Keywords: free surface; nonlinear; heterogeneity; the collocation Trefftz method; nonlinear boundary condition

doi:10.3390/w11040835

The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme

Abstract: In this article, we present a meshless method based on the method of fundamental solutions (MFS) capable of solving free surface flow in three dimensions. Since the basis function of the MFS satisfies the governing equation, the advantage of the MFS is that only the problem boundary
needs to be placed in the collocation points. For solving the three-dimensional free surface with nonlinear boundary conditions, the relaxation method in conjunction with the MFS is used, in which
the three-dimensional free surface is iterated as a movable boundary until the nonlinear boundary conditions are satisfied. The proposed method is verified and application examples are conducted. Comparing results with those from other methods shows that the method is robust and provides high accuracy and reliability. The effectiveness and ease of use for solving nonlinear free surface flows in three dimensions are also revealed.

Keywords: three–dimensional; tree surface; nonlinear; the method of fundamental solutions (MFS); meshless method

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doi:10.3390/app9081715

Abstract

In this article, we propose a novel meshless method for solving two-dimensional stationary heat conduction problems in layered materials. The proposed method is a recently developed boundary-type meshless method which combines the collocation scheme from the method of fundamental solutions (MFS) with the collocation Trefftz method (CTM) to improve the applicability of the method for solving boundary value problems. Particular non-singular basis functions from cylindrical harmonics are adopted in which the numerical approximation is based on the superposition principle using the non-singular basis functions expressed in terms of many source
points. For the modeling of multi-layer composite materials, we adopted the domain decomposition method (DDM), which splits the domain into smaller subdomains. The continuity of the flux and the temperature has to be satisfied at the interface of subdomains for the problem. The validity of
the proposed method is investigated for several test problems. Numerical applications were also carried out. Comparison of the proposed method with other meshless methods showed that it is highly accurate and computationally efficient for modeling heat conduction problems, especially in heterogeneous multi-layer composite materials.
Keywords: heat conduction problems; the collocation scheme; the meshless method; the domain decomposition method; layered materials.

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A Novel Boundary-Type Meshless Method for Modeling Geofluid
Flow in Heterogeneous Geological Media

Jing-En Xiao, Cheng-Yu Ku , Chih-Yu Liu, and Wei-Chung Yeih

Abstract

A novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media was developed. The numerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which are expressed in terms of sources located outside the domain of the problem. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method. To deal with the subsurface flow problems of heterogeneous geological media, the domain decomposition method was adopted so that flux conservation and the continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model. The validity of the model is established for a number of test problems. Application examples of subsurface flow problems with free surface in homogenous and layered heterogeneous geological media were also carried out. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for solving subsurface flow with nonlinear free surface in layered heterogeneous geological media even with large contrasts in the hydraulic conductivity.

Full paper link:

https://doi.org/10.1155/2018/9804291

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Modeling of Transient Flow in Unsaturated Geomaterials for Rainfall-Induced Landslides Using a Novel Spacetime Collocation Method