Geoengineering & Scientific Computation Lab

Abstract: 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. Keywords: shape parameter; multiquadric; radial basis function; fictitious source point; meshless method Full paper download : https://www.mdpi.com/2073-8994/12/11/1813 Abstract: https://www.mdpi.com/2073-8994/12/11/1813 HTML Version: https://www.mdpi.com/2073-8994/12/11/1813/htm PDF Version: https://www.mdpi.com/2073-8994/12/11/1813/pdf]]>

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Abstract: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. Keywords: space–time; transient radial basis function; convection–diffusion equation; meshless; radial polynomials            

Abstract: https://www.mdpi.com/2227-7390/8/10/1735
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Abstract: 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. Keywords: multiquadric; radial basis function; radial polynomials; the shape parameter; meshless; Kansa method

Abstract: https://www.mdpi.com/2073-8994/12/9/1419
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會議聯絡人資訊

召集人：    顧承宇    特聘教授     chkst26@mail.ntou.edu.tw 協同召集人：張良正    教授         lcchang@g2.nctu.edu.tw 許世孟    副教授       shihmeng@mail.ntou.edu.tw 邱永嘉    副教授       ycchiu@mail.ntou.edu.tw 執行秘書：  劉芷妤    博士         20452003@email.ntou.edu.tw 『第十二屆地下水資源及水質保護研討會』專用電郵：hre2514@email.ntou.edu.tw

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不連續變形分析法在落石問題之應用

https://doi.org/10.1016/S0148-9062(97)00319-7

Abstract

This paper presents the development of the rigid block version of discontinuous deformation analysis and its preliminary application in rockfall simulation. Modifications, including the reduction of the error due to large rigid body rotation by using post-correction at each time step of calculation and the incorporation of mass proportional damping to account for energy losses, are made to meet the requirement for large block translation as well as rotation. To improve the program efficiency, rigid body displacement function and preconditioned conjugate gradient solver are adopted. The results obtained show that the newly developed code, RIG-DDA, can appropriately and efficiently model various modes of rockfall movement and predict the trajectory of falling rock debris, which is helpful in the design of protection measures.

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NUMERICAL ANALYSES OF PILE FOUNDATION FOR SUPPORT STRUCTURE OF OFFSHORE WIND TURBINE AT CHANGHUA COAST IN TAIWAN

This study investigates the bearing capacities of group pile foundation (jacket foundation) installed on the seabed of offshore wind farm (OWF) at the Changhua coast of Western Taiwan for the jacket support structure of offshore wind turbine (OWT) using three-dimensional (3-D) finite element method (FEM). The jacket foundations are subjected to a combined Vertical-Horizontal-Moment (V-H-M) loading for the operational period. The responses of installed group pile foundations are investigated under the combined loading in marine silty sand-low plasticity silt & clay (SM-ML-CL) layers determined by 19 offshore boring logs. The validity of numerical procedures was verified by a large-scale lateral loading test of steel tubular model pile in laboratory. A systematic parametric study was performed to investigate the effects of the pile length L, pile diameter D, and pile spacing S on the ultimate bearing (or load) capacity behavior of the foundation. The effect of pile length is significant on the vertical bearing capacity (Vult) whereas pile diameter and pile spacing on the ultimate horizontal and moment loads bearing capacities (Hult and Mult). The normalized V-H and V-H-M failure envelopes of bearing capacity for the jacket foundations subjected to combined loadings can be expressed as functions of L, D, and S and fitted by elliptical shape curves. The V-H-M failure envelopes and approximated expressions are proposed to evaluate the mechanical stability of the group pile foundations for the jacket support structure of OWT under the combined loading condition.

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https://jmst.ntou.edu.tw/marine/28-3/179-199.pdf

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Abstract: 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. Dierent 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 dierentiable 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.

 

Figure 1. Configuration of the collocation points for the direct heat conduction problem.

Figure 2. Illustration of the space–time collocation scheme: (a) The DHCP; (b) The BHCP.

Keywords: collocation method; space–time; radial polynomial; basis function; heat conduction

doi:10.3390/app10093215

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Abstract: https://www.mdpi.com/2076-3417/10/9/3215

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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.

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On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Abstract: 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.
Keywords: free surface; nonlinear; heterogeneity; the collocation Trefftz method; nonlinear boundary condition

doi:10.3390/w11040835

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