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Mr. Jing-En Xiao
Ph. D. candidate

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0 Geotechnical Engineering
0 Groundwater
0 Hydrogeology
0 Numerical Modeling
0 Soil Mechanics

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Journal article
Published: 11 April 2021 in Applied Sciences
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In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

ACS Style

Cheng-Yu Ku; Li-Dan Hong; Chih-Yu Liu; Jing-En Xiao; Wei-Po Huang. Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method. Applied Sciences 2021, 11, 3421 .

AMA Style

Cheng-Yu Ku, Li-Dan Hong, Chih-Yu Liu, Jing-En Xiao, Wei-Po Huang. Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method. Applied Sciences. 2021; 11 (8):3421.

Chicago/Turabian Style

Cheng-Yu Ku; Li-Dan Hong; Chih-Yu Liu; Jing-En Xiao; Wei-Po Huang. 2021. "Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method." Applied Sciences 11, no. 8: 3421.

Journal article
Published: 28 December 2020 in Mathematics and Computers in Simulation
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This paper presents the meshless method using radial polynomials with the combination of the multiple source collocation scheme for solving elliptic boundary value problems. In the proposed method, the basis function is based on the radial polynomials, which is different from the conventional radial basis functions that approximate the solution using the specific function such as the multiquadric function with the shape parameter for infinitely differentiable. The radial polynomial basis function is a non-singular series function in nature which is infinitely smooth and differentiable in nature without using the shape parameter. With the combination of the multiple source collocation scheme, the center point is regarded as the source point for the interpolation of the radial polynomials. Numerical solutions in multiple dimensions are approximated by applying the radial polynomials with given terms of the radial polynomials. The comparison of the proposed method with the radial basis function collocation method (RBFCM) using the multiquadric and polyharmonic spline functions is conducted. Results demonstrate that the accuracy obtained from the proposed method is better than that of the conventional RBFCM with the same number of collocation points. In addition, highly accurate solutions with the increase of radial polynomial terms may be obtained.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu; Der-Guey Lin. On solving elliptic boundary value problems using a meshless method with radial polynomials. Mathematics and Computers in Simulation 2020, 185, 153 -173.

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Chih-Yu Liu, Der-Guey Lin. On solving elliptic boundary value problems using a meshless method with radial polynomials. Mathematics and Computers in Simulation. 2020; 185 ():153-173.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu; Der-Guey Lin. 2020. "On solving elliptic boundary value problems using a meshless method with radial polynomials." Mathematics and Computers in Simulation 185, no. : 153-173.

Journal article
Published: 02 November 2020 in Symmetry
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In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

ACS Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Shih-Meng Hsu. Multiquadrics without the Shape Parameter for Solving Partial Differential Equations. Symmetry 2020, 12, 1813 .

AMA Style

Cheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao, Shih-Meng Hsu. Multiquadrics without the Shape Parameter for Solving Partial Differential Equations. Symmetry. 2020; 12 (11):1813.

Chicago/Turabian Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Shih-Meng Hsu. 2020. "Multiquadrics without the Shape Parameter for Solving Partial Differential Equations." Symmetry 12, no. 11: 1813.

Journal article
Published: 10 October 2020 in Mathematics
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This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of n–dimensional Euclidean space and time into an n + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations. Mathematics 2020, 8, 1735 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Chih-Yu Liu. Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations. Mathematics. 2020; 8 (10):1735.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. 2020. "Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations." Mathematics 8, no. 10: 1735.

Journal article
Published: 26 August 2020 in Symmetry
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In this article, a collocation method using radial polynomials (RPs) based on the multiquadric (MQ) radial basis function (RBF) for solving partial differential equations (PDEs) is proposed. The new global RPs include only even order radial terms formulated from the binomial series using the Taylor series expansion of the MQ RBF. Similar to the MQ RBF, the RPs is infinitely smooth and differentiable. The proposed RPs may be regarded as the equivalent expression of the MQ RBF in series form in which no any extra shape parameter is required. Accordingly, the challenging task for finding the optimal shape parameter in the Kansa method is avoided. Several numerical implementations, including problems in two and three dimensions, are conducted to demonstrate the accuracy and robustness of the proposed method. The results depict that the method may find solutions with high accuracy, while the radial polynomial terms is greater than 6. Finally, our method may obtain more accurate results than the Kansa method.

ACS Style

Cheng-Yu Ku; Jing-En Xiao. A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations. Symmetry 2020, 12, 1419 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao. A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations. Symmetry. 2020; 12 (9):1419.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao. 2020. "A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations." Symmetry 12, no. 9: 1419.

Journal article
Published: 05 May 2020 in Applied Sciences
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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. Different 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 differentiable 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.

ACS Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Ming-Ren Chen. Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method. Applied Sciences 2020, 10, 3215 .

AMA Style

Cheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao, Ming-Ren Chen. Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method. Applied Sciences. 2020; 10 (9):3215.

Chicago/Turabian Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Ming-Ren Chen. 2020. "Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method." Applied Sciences 10, no. 9: 3215.

Journal article
Published: 18 February 2020 in Mathematics
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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.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations. Mathematics 2020, 8, 270 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Chih-Yu Liu. A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations. Mathematics. 2020; 8 (2):270.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. 2020. "A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations." Mathematics 8, no. 2: 270.

Journal article
Published: 09 December 2019 in Water
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In this paper, a spacetime meshless method utilizing Trefftz functions for modeling subsurface flow problems with a transient moving boundary is proposed. The subsurface flow problem with a transient moving boundary is governed by the two-dimensional diffusion equation, where the position of the moving boundary is previously unknown. We solve the subsurface flow problems based on the Trefftz method, in which the Trefftz basis functions are obtained from the general solutions using the separation of variables. The solutions of the governing equation are then approximated numerically by the superposition theorem using the basis functions, which match the data at the spacetime boundary collocation points. Because the proposed basis functions fully satisfy the diffusion equation, arbitrary nodes are collocated only on the spacetime boundaries for the discretization of the domain. The iterative scheme has to be used for solving the moving boundaries because the transient moving boundary problems exhibit nonlinear characteristics. Numerical examples, including harmonic and non-harmonic boundary conditions, are carried out to validate the method. Results illustrate that our method may acquire field solutions with high accuracy. It is also found that the method is advantageous for solving inverse problems as well. Finally, comparing with those obtained from the method of fundamental solutions, we may obtain the accurate location of the nonlinear moving boundary for transient problems using the spacetime meshless method with the iterative scheme.

ACS Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Weichung Yeih; Chia-Ming Fan. A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary. Water 2019, 11, 2595 .

AMA Style

Cheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao, Weichung Yeih, Chia-Ming Fan. A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary. Water. 2019; 11 (12):2595.

Chicago/Turabian Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Weichung Yeih; Chia-Ming Fan. 2019. "A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary." Water 11, no. 12: 2595.

Journal article
Published: 15 November 2019 in Mathematics
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This paper presents a study for solving the modified Helmholtz equation in layered materials using the multiple source meshfree approach (MSMA). The key idea of the MSMA starts with the method of fundamental solutions (MFS) as well as the collocation Trefftz method (CTM). The multiple source collocation scheme in the MSMA stems from the MFS and the basis functions are formulated using the CTM. The solution of the modified Helmholtz equation is therefore approximated by the superposition theorem using particular nonsingular functions by means of multiple sources located within the domain. To deal with the two-dimensional modified Helmholtz equation in layered materials, the domain decomposition method was adopted. Numerical examples were carried out to validate the method. The results illustrate that the MSMA is relatively simple because it avoids a complicated procedure for finding the appropriate position of the sources. Additionally, the MSMA for solving the modified Helmholtz equation is advantageous because the source points can be collocated on or within the domain boundary and the results are not sensitive to the location of source points. Finally, compared with other methods, highly accurate solutions can be obtained using the proposed method.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Weichung Yeih; Chih-Yu Liu; Ku; Xiao; Yeih; Liu. On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach. Mathematics 2019, 7, 1114 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Weichung Yeih, Chih-Yu Liu, Ku, Xiao, Yeih, Liu. On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach. Mathematics. 2019; 7 (11):1114.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Weichung Yeih; Chih-Yu Liu; Ku; Xiao; Yeih; Liu. 2019. "On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach." Mathematics 7, no. 11: 1114.

Journal article
Published: 28 June 2019 in Applied Sciences
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In this article, a newly developed multiple-source meshless method (MSMM) capable of solving inverse heat conduction problems in two dimensions is presented. Evolved from the collocation Trefftz method (CTM), the MSMM approximates the solution by using many source points through the addition theorem such that the ill-posedness is greatly reduced. The MSMM has the same superiorities as the CTM, such as the boundary discretization only, and is advantageous for solving inverse problems. Several numerical examples are conducted to validate the accuracy of solving inverse heat conduction problems using boundary conditions with different levels of noise. Moreover, the domain decomposition method is adopted for problems in the doubly-connected domain. The results demonstrate that the proposed method may recover the unknown data with remarkably high accuracy, even though the over-specified boundary data are assigned a portion that is less than 1/10 of the overall domain boundary.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Wei-Po Huang; Weichung Yeih; Chih-Yu Liu. On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method. Applied Sciences 2019, 9, 2629 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Wei-Po Huang, Weichung Yeih, Chih-Yu Liu. On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method. Applied Sciences. 2019; 9 (13):2629.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Wei-Po Huang; Weichung Yeih; Chih-Yu Liu. 2019. "On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method." Applied Sciences 9, no. 13: 2629.

Journal article
Published: 25 April 2019 in Applied Sciences
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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.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme. Applied Sciences 2019, 9, 1715 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Chih-Yu Liu. The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme. Applied Sciences. 2019; 9 (8):1715.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. 2019. "The Method of Fundamental Solutions for Three-Dimensional Nonlinear Free Surface Flows Using the Iterative Scheme." Applied Sciences 9, no. 8: 1715.

Journal article
Published: 20 April 2019 in Water
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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.

ACS Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method. Water 2019, 11, 835 .

AMA Style

Cheng-Yu Ku, Jing-En Xiao, Chih-Yu Liu. On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method. Water. 2019; 11 (4):835.

Chicago/Turabian Style

Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu. 2019. "On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method." Water 11, no. 4: 835.

Journal article
Published: 11 October 2018 in Applied Sciences
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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.

ACS Style

Jing-En Xiao; Cheng-Yu Ku; Wei-Po Huang; Yan Su; Yung-Hsien Tsai. A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials. Applied Sciences 2018, 8, 1887 .

AMA Style

Jing-En Xiao, Cheng-Yu Ku, Wei-Po Huang, Yan Su, Yung-Hsien Tsai. A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials. Applied Sciences. 2018; 8 (10):1887.

Chicago/Turabian Style

Jing-En Xiao; Cheng-Yu Ku; Wei-Po Huang; Yan Su; Yung-Hsien Tsai. 2018. "A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials." Applied Sciences 8, no. 10: 1887.

Journal article
Published: 07 December 2017 in Water
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In this paper, a novel meshless method for the transient modeling of subsurface flow in unsaturated soils was developed. A linearization process for the nonlinear Richards equation using the Gardner exponential model to analyze the transient flow in the unsaturated zone was adopted. For the transient modeling, we proposed a pioneering work using the collocation Trefftz method and utilized the coordinate system in Minkowski spacetime instead of that in the original Euclidean space. The initial value problem for transient modeling of subsurface flow in unsaturated soils can then be transformed into the inverse boundary value problem. A numerical solution obtained in the spacetime coordinate system was approximated by superpositioning Trefftz basis functions satisfying the governing equation for boundary collocation points on partial problem domain boundary in the spacetime coordinate system. As a result, the transient problems can be solved without using the traditional time-marching scheme. The validity of the proposed method is established for several test problems. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for the transient modeling of subsurface flow in unsaturated soils.

ACS Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Weichung Yeih. Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method. Water 2017, 9, 954 .

AMA Style

Cheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao, Weichung Yeih. Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method. Water. 2017; 9 (12):954.

Chicago/Turabian Style

Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Weichung Yeih. 2017. "Transient Modeling of Flow in Unsaturated Soils Using a Novel Collocation Meshless Method." Water 9, no. 12: 954.