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Chih-Yu Liu currently works at the Department of Harbor and River Engineering, National Taiwan Ocean University. Chih-Yu Liu does research in Hydrogeology, Groundwater and Numerical Modeling.
In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.
Chih-Yu Liu; Cheng-Yu Ku; Li-Dan Hong; Shih-Meng Hsu. Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs. Mathematics 2021, 9, 1535 .
AMA StyleChih-Yu Liu, Cheng-Yu Ku, Li-Dan Hong, Shih-Meng Hsu. Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs. Mathematics. 2021; 9 (13):1535.
Chicago/Turabian StyleChih-Yu Liu; Cheng-Yu Ku; Li-Dan Hong; Shih-Meng Hsu. 2021. "Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs." Mathematics 9, no. 13: 1535.
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.
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 StyleCheng-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 StyleCheng-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.
This article presents a geographic information system (GIS)-based artificial neural network (GANN) model for flood susceptibility assessment of Keelung City, Taiwan. Various factors, including elevation, slope angle, slope aspect, flow accumulation, flow direction, topographic wetness index (TWI), drainage density, rainfall, and normalized difference vegetation index, were generated using a digital elevation model and LANDSAT 8 imagery. Historical flood data from 2015 to 2019, including 307 flood events, were adopted for a comparison of flood susceptibility. Using these factors, the GANN model, based on the back-propagation neural network (BPNN), was employed to provide flood susceptibility. The validation results indicate that a satisfactory result, with a correlation coefficient of 0.814, was obtained. A comparison of the GANN model with those from the SOBEK model was conducted. The comparative results demonstrated that the proposed method can provide good accuracy in predicting flood susceptibility. The results of flood susceptibility are categorized into five classes: Very low, low, moderate, high, and very high, with coverage areas of 60.5%, 27.4%, 8.6%, 2.5%, and 1%, respectively. The results demonstrate that nearly 3.5% of the study area, including the core district of the city and an exceedingly populated area including the financial center of the city, can be categorized as high to very high flood susceptibility zones.
Nanda Khoirunisa; Cheng-Yu Ku; Chih-Yu Liu. A GIS-Based Artificial Neural Network Model for Flood Susceptibility Assessment. International Journal of Environmental Research and Public Health 2021, 18, 1072 .
AMA StyleNanda Khoirunisa, Cheng-Yu Ku, Chih-Yu Liu. A GIS-Based Artificial Neural Network Model for Flood Susceptibility Assessment. International Journal of Environmental Research and Public Health. 2021; 18 (3):1072.
Chicago/Turabian StyleNanda Khoirunisa; Cheng-Yu Ku; Chih-Yu Liu. 2021. "A GIS-Based Artificial Neural Network Model for Flood Susceptibility Assessment." International Journal of Environmental Research and Public Health 18, no. 3: 1072.
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.
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 StyleCheng-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 StyleCheng-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.
This paper presents a space–time meshfree method for solving transient inverse problems in subsurface flow. Based on the transient groundwater equation, we derived the Trefftz basis functions utilizing the method of separation of variables. Due to the basis functions completely satisfying the equation to be solved, collocation points are placed on the space–time boundaries. Numerical solutions are approximated based on the superposition theorem. Accordingly, the initial and boundary conditions are both regarded as space–time boundary conditions. Forward and inverse examples are conducted to validate the proposed approach. Emphasis is placed on the two-dimensional boundary detection problem in which the nonlinearity is solved using the fictitious time integration method. Results demonstrate that approximations with high accuracy are acquired in which the boundary data on the absent boundary may be efficiently recovered even when inaccessible partial data are provided.
Cheng-Yu Ku; Li-Dan Hong; Chih-Yu Liu. Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method. Water 2020, 12, 3580 .
AMA StyleCheng-Yu Ku, Li-Dan Hong, Chih-Yu Liu. Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method. Water. 2020; 12 (12):3580.
Chicago/Turabian StyleCheng-Yu Ku; Li-Dan Hong; Chih-Yu Liu. 2020. "Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method." Water 12, no. 12: 3580.
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.
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 StyleCheng-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 StyleCheng-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.
A collocation method with space–time radial polynomials for solving two–dimensional inverse heat conduction problems (IHCPs) is presented. The space–time radial polynomial series function is developed for spatial and temporal discretization of the government equation within the space–time domain. Because boundary and initial data are assigned on the space–time boundaries, the numerical solution of the IHCP can be approximated by solving the inverse boundary value problem in the space–time domain without using the time–marching scheme. The inner, source, and boundary points are uniformly distributed using the proposed outer source space–time collocation scheme. Since all partial derivatives up to order of the problem's operator of the proposed basis functions are a series of continuous functions, which are nonsingular and smooth, the numerical solutions are obtained without using the shape parameter. Numerical examples for solving IHCPs with missing both parts of initial and boundary data are carried out. The results of our study are then compared with those of other collocation methods using multiquadric basis function. Results illustrate that highly accurate recovered temperatures are acquired. Additionally, the recovered temperatures on inaccessible boundaries with high accuracy can be acquired even 1/5 portion of the entire space–time boundaries are inaccessible.
Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Shih-Meng Hsu; Weichung Yeih. A collocation method with space–time radial polynomials for inverse heat conduction problems. Engineering Analysis with Boundary Elements 2020, 122, 117 -131.
AMA StyleCheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao, Shih-Meng Hsu, Weichung Yeih. A collocation method with space–time radial polynomials for inverse heat conduction problems. Engineering Analysis with Boundary Elements. 2020; 122 ():117-131.
Chicago/Turabian StyleCheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Shih-Meng Hsu; Weichung Yeih. 2020. "A collocation method with space–time radial polynomials for inverse heat conduction problems." Engineering Analysis with Boundary Elements 122, no. : 117-131.
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.
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 StyleCheng-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 StyleCheng-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.
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.
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 StyleCheng-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 StyleCheng-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.
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.
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 StyleCheng-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 StyleCheng-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.
This paper presents the modeling of tide–induced groundwater response using the spacetime collocation approach (SCA). The newly developed SCA begins with the consideration of Trefftz basis functions which are general solutions of the governing equation deriving from the separation of variables. The solution of the groundwater response in a coastal confined aquifer with an estuary boundary where the phase and amplitude of tide can vary with time and position is then approximated by the linear combination of Trefftz basis functions using the superposition theorem. The SCA is validated for several numerical examples with analytical solutions. The comparison of the results and accuracy for the SCA with the time–marching finite difference method is carried out. In addition, the SCA is adopted to examine the tidal and groundwater piezometer data at the Xing–Da port, Kaohsiung, Taiwan. The results demonstrate the SCA may obtain highly accurate results. Moreover, it shows the advantages of the SCA such that we only discretize by a set of points on the spacetime boundary without tedious mesh generation and thus significantly enhance the applicability.
Cheng-Yu Ku; Chih-Yu Liu; Yan Su; Luxi Yang; Wei-Po Huang. Modeling Tide–Induced Groundwater Response in a Coastal Confined Aquifer Using the Spacetime Collocation Approach. Applied Sciences 2020, 10, 439 .
AMA StyleCheng-Yu Ku, Chih-Yu Liu, Yan Su, Luxi Yang, Wei-Po Huang. Modeling Tide–Induced Groundwater Response in a Coastal Confined Aquifer Using the Spacetime Collocation Approach. Applied Sciences. 2020; 10 (2):439.
Chicago/Turabian StyleCheng-Yu Ku; Chih-Yu Liu; Yan Su; Luxi Yang; Wei-Po Huang. 2020. "Modeling Tide–Induced Groundwater Response in a Coastal Confined Aquifer Using the Spacetime Collocation Approach." Applied Sciences 10, no. 2: 439.
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.
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 StyleCheng-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 StyleCheng-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.
The novel meshfree approach based on the space-time collocation scheme for dealing with the two-dimensional inverse heat conduction problem (IHCP) is presented in this study. Numerical solutions obtained in the space-time collocation scheme are approximated by linear combinations of basis functions exactly satisfying the two-dimensional heat equation. We can then describe the approximated solutions of the two-dimensional heat conduction problems as a series by utilizing the addition theorem. Several numerical implementations including three cases with the consideration of the different combinations of absent initial or boundary conditions are carried out to verify the proposed numerical approach. The results reveal that highly accurate initial and boundary heat distribution with an accuracy of the order of 10−6 can be recovered, even when the data are absent on the initial as well as boundary condition and only final time data are specified. It may be concluded that the proposed numerical approach is able to provide promising approximations for solving two-dimensional IHCP.
Chih-Yu Liu; Cheng-Yu Ku. Recovering the Initial and Boundary Data in the Two-Dimensional Inverse Heat Conduction Problems Using the Novel Space-Time Collocation Meshfree Approach. Mechanical Engineering and Materials 2019, 853 -859.
AMA StyleChih-Yu Liu, Cheng-Yu Ku. Recovering the Initial and Boundary Data in the Two-Dimensional Inverse Heat Conduction Problems Using the Novel Space-Time Collocation Meshfree Approach. Mechanical Engineering and Materials. 2019; ():853-859.
Chicago/Turabian StyleChih-Yu Liu; Cheng-Yu Ku. 2019. "Recovering the Initial and Boundary Data in the Two-Dimensional Inverse Heat Conduction Problems Using the Novel Space-Time Collocation Meshfree Approach." Mechanical Engineering and Materials , no. : 853-859.
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.
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 StyleCheng-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 StyleCheng-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.
In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.
Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Wei-Po Huang; Yan Su. A spacetime collocation Trefftz method for solving the inverse heat conduction problem. Advances in Mechanical Engineering 2019, 11, 1 .
AMA StyleCheng-Yu Ku, Chih-Yu Liu, Jing-En Xiao, Wei-Po Huang, Yan Su. A spacetime collocation Trefftz method for solving the inverse heat conduction problem. Advances in Mechanical Engineering. 2019; 11 (7):1.
Chicago/Turabian StyleCheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Wei-Po Huang; Yan Su. 2019. "A spacetime collocation Trefftz method for solving the inverse heat conduction problem." Advances in Mechanical Engineering 11, no. 7: 1.
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.
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 StyleCheng-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 StyleCheng-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.
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.
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 StyleCheng-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 StyleCheng-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.
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.
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 StyleCheng-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 StyleCheng-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.
Cheng-Yu Ku; Jing-En Xiao; Liu Chih-Yu; Chia-Ming Fan. On modeling subsurface flow using a novel hybrid Trefftz–MFS method. Engineering Analysis with Boundary Elements 2019, 100, 225 -236.
AMA StyleCheng-Yu Ku, Jing-En Xiao, Liu Chih-Yu, Chia-Ming Fan. On modeling subsurface flow using a novel hybrid Trefftz–MFS method. Engineering Analysis with Boundary Elements. 2019; 100 ():225-236.
Chicago/Turabian StyleCheng-Yu Ku; Jing-En Xiao; Liu Chih-Yu; Chia-Ming Fan. 2019. "On modeling subsurface flow using a novel hybrid Trefftz–MFS method." Engineering Analysis with Boundary Elements 100, no. : 225-236.
This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent.
Cheng-Yu Ku; Chih-Yu Liu; Weichung Yeih; Chein-Shan Liu; Chia-Ming Fan. A novel space–time meshless method for solving the backward heat conduction problem. International Journal of Heat and Mass Transfer 2018, 130, 109 -122.
AMA StyleCheng-Yu Ku, Chih-Yu Liu, Weichung Yeih, Chein-Shan Liu, Chia-Ming Fan. A novel space–time meshless method for solving the backward heat conduction problem. International Journal of Heat and Mass Transfer. 2018; 130 ():109-122.
Chicago/Turabian StyleCheng-Yu Ku; Chih-Yu Liu; Weichung Yeih; Chein-Shan Liu; Chia-Ming Fan. 2018. "A novel space–time meshless method for solving the backward heat conduction problem." International Journal of Heat and Mass Transfer 130, no. : 109-122.