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

Ku, Cheng-Yu; Xiao, Jing-En; Liu, Chih-Yu

doi:10.3390/app9081715

Cited by 2 publications

(3 citation statements)

References 40 publications

(58 reference statements)

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“…Previous studies have found it difficult to calculate the Jacobian matrix using Newton's method. Thus, the Picard iterative method is used in this study [5]. The Picard iteration first begins from the initial guess of the location for the moving boundary.…”

Section: The Iterative Scheme For Modeling Transient Moving Boundarymentioning

confidence: 99%

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

Liu

Xiao

et al. 2019

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.

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Section: The Iterative Scheme For Modeling Transient Moving Boundarymentioning

confidence: 99%

“…The phreatic line is located between the fluid phase and the air phase of the soil. It is sometimes regarded as the phase change problem [5][6][7]. Phase change problems are often encountered in engineering, industry, and problems such as the design of roadways in cold regions [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

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

Liu

Xiao

et al. 2019

Water

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“…The other non-linear problem is related to the flow in a porous medium with a free surface. The 3D version of such a flow was considered in [ 36 ]. The problem is the BVP with the Laplace equation and non-linear boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations Based on a Meshfree Method for Nickel-Steel Welded Joint Manufactured by Micro-Jet Cooling

et al. 2022

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The article presents a numerical–experimental approach to the weldability and mechanical resistance of the joint of Alloy 59 (2.4605, nickel-chromium-molybdenum) and S355J2W (1.8965) structural steel manufactured by the MIG process with the use of micro-jet cooling. This research was considered because the standard MIG process does not guarantee the procurement of a mixed hard-rusting structural steel superalloy weld of a repeatable and acceptable quality. Welds made through the classic MIG process express cracks that result from their unfavorable metallographic microstructure, while the joint supported by micro-jet cooling does not reflect any cracks and has a high strength with good flexibility. This was achieved by the application of helium for cooling. The joining technology was also considered in the numerical stage, represented by calculations in situ. For this purpose, the fundamental solution method (FSM) for the simulation of heat transfer during the process of welding with micro-jet cooling was implemented according to the initial boundary value problem (IBVP). The problem was solved employing the method of combining the finite difference method, Picard iterations, approximation by the radial basis function, and the fundamental solution method so as to solve the IVBP. The proposed method was validated by the data and results obtained during in situ experiments. The numerical approach enabled us to obtain variations in the temperature distribution values in HAZ with its different dimensional variants, ranging between 600 °C and 1400 °C.

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

Cited by 2 publications

References 40 publications

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

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

Numerical Simulations Based on a Meshfree Method for Nickel-Steel Welded Joint Manufactured by Micro-Jet Cooling

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