scholarly journals A Novel Hybrid Boundary-Type Meshless Method for Solving Heat Conduction Problems in Layered Materials

2018 ◽  
Vol 8 (10) ◽  
pp. 1887 ◽  
Author(s):  
Jing-En Xiao ◽  
Cheng-Yu Ku ◽  
Wei-Po Huang ◽  
Yan Su ◽  
Yung-Hsien Tsai
Keyword(s):  
Heat Conduction ◽  
Composite Materials ◽  
Meshless Method ◽  
Domain Decomposition Method ◽  
Superposition Principle ◽  
Layered Materials ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Boundary Type ◽  
Layer Composite

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.

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

Geofluids ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Jing-En Xiao ◽  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Wei-Chung Yeih
Keyword(s):  
Free Surface ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Domain Decomposition Method ◽  
Subsurface Flow ◽  
Test Problems ◽  
Pressure Potential ◽  
Trefftz Method ◽  
Flow Problems ◽  
Boundary Type

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.

scholarly journals A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary

Water ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 2595 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Weichung Yeih ◽  
Chia-Ming Fan
Keyword(s):  
Diffusion Equation ◽  
Meshless Method ◽  
Moving Boundary ◽  
Iterative Scheme ◽  
Separation Of Variables ◽  
Subsurface Flow ◽  
Basis Functions ◽  
Method Of Fundamental Solutions ◽  
Trefftz Method ◽  
Flow Problems

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.

scholarly journals On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method

2019 ◽  
Vol 9 (13) ◽  
pp. 2629 ◽  
Author(s):  
Ku ◽  
Xiao ◽  
Huang ◽  
Yeih ◽  
Liu
Keyword(s):  
Heat Conduction ◽  
Meshless Method ◽  
Domain Decomposition Method ◽  
Domain Boundary ◽  
Addition Theorem ◽  
Two Dimensions ◽  
Multiple Source ◽  
Inverse Heat Conduction ◽  
Trefftz Method ◽  
Inverse Heat Conduction Problems

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.

scholarly journals On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1114 ◽  
Author(s):  
Ku ◽  
Xiao ◽  
Yeih ◽  
Liu
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition Method ◽  
Domain Boundary ◽  
Layered Materials ◽  
Fundamental Solutions ◽  
Multiple Source ◽  
Method Of Fundamental Solutions ◽  
Multiple Sources ◽  
Trefftz Method ◽  
Modified Helmholtz Equation

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.

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