A meshless procedure for analysis of fluid flow and heat transfer in an internally finned square duct

et al. 2021

Applied Sciences

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.

Section: Introductionmentioning

confidence: 99%

Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

Hong

et al. 2021

Applied Sciences

“…The collocation Trefftz method (CTM) is one of meshfree approaches for dealing with boundary value problem (BVP) where solutions can be described as a combination of general solutions which satisfy the governing equation [23][24][25]. The CTM may release the difficulties for finding exact solutions of governing equations and remains the characteristics of high accuracy due to the adoption of the general solutions [26][27][28]. The original CTM is limited to homogeneous and stationary BVPs.…”

Section: Introductionmentioning

confidence: 99%

Modeling Tide–Induced Groundwater Response in a Coastal Confined Aquifer Using the Spacetime Collocation Approach

et al. 2020

Applied Sciences

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.

“…Various numerical approaches [12], such as the boundary element method [13], the interpolation finite difference method [14], the finite element method [15], the finite volume method [16], the local radial basis function collocation method [17], the method of approximate particular solutions [18], and the method of fundamental solutions (MFS) [19,20], have been utilized for dealing with moving boundary problems. The collocation method can be categorized into one of the meshless methods [21,22].…”

Section: Introductionmentioning

confidence: 99%

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

Xiao

et al. 2019

Water

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.