National Taiwan Ocean University Research Hubhttps://scholars.ntou.edu.twThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 19 Jun 2024 03:02:38 GMT2024-06-19T03:02:38Z503911Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element methodhttp://scholars.ntou.edu.tw/handle/123456789/24588Title: Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element method
Authors: Kao, Jeng-Hong; Lee, Jia-Wei; Yang, Chia-Ying; Chen, Jeng-Tzong
Abstract: In this paper, we employed the boundary element method (BEM) to determine the heat conduction shape factor (CSF) for the steady-state heat conduction problem in the heat exchanger tube. Various doubly-connected geometric shapes were considered. Numerical results of the conduction shape factor matched well with the theoretical data. Numerical instability was found for a special size. Degenerate scale occurred for all cases of regular N-gons in determining the conduction shape factor and matched the data as analytically predicted. Numerical results showed that a degenerate scale is only dependent on the outer boundary instead of the inner boundary. Three regularization techniques, the dual BEM, the CLEEF(UT), and the CLEEF(LM), were provided to avoid numerical instability. It was found that the larger number of sides for the N-gon shape is given, the lower the CSF is obtained. As N approaches infinity, it converges to the minimum value of the solution for an annulus.
Thu, 02 Nov 2023 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245882023-11-02T00:00:00ZSteady State Heat Conduction in Exchanger Tubes by Using the Meshfree Boundary Integral Equation Method: Conduction Shape Factor and Degenerate Scalehttp://scholars.ntou.edu.tw/handle/123456789/24531Title: Steady State Heat Conduction in Exchanger Tubes by Using the Meshfree Boundary Integral Equation Method: Conduction Shape Factor and Degenerate Scale
Authors: Lee, Jia-Wei; Yang, Hung-Wen; Chen, Jeng-Tzong
Abstract: Regarding the steady-state heat conduction problem in exchanger tubes, the meshfree boundary integral equation method is employed to determine the conduction shape factor in this paper. Different from the conventional boundary element method, the present method is free of mesh generation. After using the parametric function to represent the boundary contour and adopting the Gaussian quadrature, only collocating points on the boundary is required to obtain the linear algebraic equations. By introducing the local exact solution, the singular integral in the sense of the Cauchy principal value can be novelly determined. In addition, the boundary layer effect due to the nearly singular integral in the boundary integral equation can be dealt with. Two cases of different boundary conditions are considered. One is the isothermal condition on both the inner and outer surfaces. The other is the isothermal condition on the inner surface and the convection condition on the outer surface. Besides, numerical instability is found and the nonuniqueness solution due to the degenerate scale is examined by calculating the conduction shape factor and the temperature on the outer surface.
Wed, 01 Nov 2023 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245312023-11-01T00:00:00ZAre the direct and indirect BEM/BIEMs equivalent?http://scholars.ntou.edu.tw/handle/123456789/24547Title: Are the direct and indirect BEM/BIEMs equivalent?
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Kao, Shing-Kai; Wu, Ting-Ann
Abstract: For a long time, direct and indirect BEM/BIEMs were always unified and seemed equivalent in real implementation for ensuring the solution. Regarding the solution space and the numerical aspects in linear algebra, it is interesting to find that they are not fully equivalent. To demonstrate this finding, antiplane shear problems containing the elliptical rigid inclusion and hole as well as rigid-line inclusion and line crack were given to demonstrate the non-equivalence between the direct and indirect BEM/BIEMs. It is interesting to find that nonunique boundary densities may yield the unique field solution in both analytical derivation and numerical implementation. The mechanism was analytically studied and numerically performed by using the degenerate kernel and the singular value decomposition technique, respectively. Besides, the Fredholm alternative theorem was employed to examine the solvability condition in both the indirect boundary integral equation method and indirect boundary element method. For continuous system and discrete system, we can explain why the boundary density does not match the exact solution but the field solution is always acceptable when setting xi 0 = 0 in the beginning of the derivation of BIE. Two approaches, direct and indirect BEM/BIEMs, show the non-equivalence of solution in case of degenerate boundary and degenerate scale.
Sun, 01 Oct 2023 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245472023-10-01T00:00:00ZAnalytical solutions for the Laplace problem of an eccentric annular domainhttp://scholars.ntou.edu.tw/handle/123456789/23691Title: Analytical solutions for the Laplace problem of an eccentric annular domain
Authors: Chen, Jeng-Tzong; Lee, Ying-Te; Tai, Wei -Chen; Tsao, Mei-Na
Abstract: Boundary value problems including a source and no source for an eccentric annular domain are both revisited in this paper. Instead of using complex variables, the null-field boundary integral equation in conjunction with the degenerate kernel is employed to analytically solve the problem. Due to the geometry of the eccentric domain, the fundamental solution is expanded to the degenerate kernel under the bipolar coordinates. Once the degen-erate kernel is found, the boundary integral equation is nothing more than a linear algebraic system. The so-lutions derived by using the present method have been compared with those done by using the complex variables. The potential field of the problems are plotted to show the validity of the present method.
Sun, 01 Jan 2023 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/236912023-01-01T00:00:00ZSupport motion of a finite bar with a viscously damped boundaryhttp://scholars.ntou.edu.tw/handle/123456789/23630Title: Support motion of a finite bar with a viscously damped boundary
Authors: Chen, Jeng-Tzong; Kao, Hao-Chen; Lee, Jia-Wei; Lee, Ying-Te
Abstract: In this paper, we extended the previous experience to solve the vibration problem of a finite bar with a viscously damped boundary and the support motion on the other side. Two analytical methods, the mode superposition method in conjunction with the quasi-static decomposition method and the method of diamond rule based on the method of characteristics, were employed to derive two analytical solutions. One is a series solution by using the mode superposition method. The other is an exact solution by using the method of diamond rule. The non-conservative system with an external damper is solved straightforward by using the method of diamond rule to avoid the complex-valued eigen system. Agreement is made well. Both advantages and disadvantages of two methods were discussed.
Thu, 17 Nov 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/236302022-11-17T00:00:00ZConstruction of dynamic Green's function for an infinite acoustic field with multiple prolate spheroids (September, 10.1007/s00707-022-03301-8, 2022)http://scholars.ntou.edu.tw/handle/123456789/23137Title: Construction of dynamic Green's function for an infinite acoustic field with multiple prolate spheroids (September, 10.1007/s00707-022-03301-8, 2022)
Authors: Lee, W. M.; Chen, J. T.
Thu, 13 Oct 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/231372022-10-13T00:00:00ZAnalytical solution for potential flow across two circular cylinders using the BIE in conjunction with degenerate kernels of bipolar coordinateshttp://scholars.ntou.edu.tw/handle/123456789/22082Title: Analytical solution for potential flow across two circular cylinders using the BIE in conjunction with degenerate kernels of bipolar coordinates
Authors: Chen, Jeng-Tzong; Chou, Yen-Ting; Kao, Jeng-Hong; Lee, Jia-Wei
Abstract: In this paper, we employ the bipolar degenerate-kernel in the boundary integral equation (BIE) to derive the analytical solution of potential flow across two identical circular cylinders. Analytical results are compared with those in Lebedev et al.. Discussions of the difference between our results and Lebedev's solution are addressed. Velocity distribution is also plotted. Both analytical solution and velocity plots are given. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using the bipolar coordinates for a case of two cylinders. The orientation, or so called the angle of attack is considered. The result shows that the degenerate-kernel approach can be an alternative tool for analytically solving some boundary value problems, although the complex variables is always employed. The extension to 3-D problems is promising and straightforward once the degenerate kernel is available since complex variable may have difficulty. (C) 2022 Elsevier Ltd. All rights reserved.
Sat, 01 Oct 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/220822022-10-01T00:00:00ZInteraction between a screw dislocation and an elastic elliptical inhomogeneity by using the angular basis functionhttp://scholars.ntou.edu.tw/handle/123456789/23134Title: Interaction between a screw dislocation and an elastic elliptical inhomogeneity by using the angular basis function
Authors: Chen, J. T.; Lee, J. W.; Kao, S. K.
Abstract: The behavior of displacement field due to a screw dislocation is similar to the angular basis function (ABF) Arg(z). It is different from the radial basis function (RBF) In(r) that is used to describe the velocity potential of a sink or source. Nevertheless, the complex-valued fundamental solution In(z) contains the two parts of RBF In(r) and ABF Arg(z). In this paper, not only the RBF in the null-field boundary integral equation (BIE) but also the ABF for the screw dislocation are employed to study the interaction between a screw dislocation and an elastic elliptical inhomogeneity. This problem is decomposed into a free field with a screw dislocation and a boundary value problem containing an elliptical inhomogeneity. The boundary value problem is solved by using the RBF and the null-field BIE. Since the geometric shape is an ellipse, the degenerate kernel is expanded to a series form under the elliptical coordinates, while the unknown boundary densities are expanded to eigenfunctions. By combining the degenerate kernel and the null-field BIE, the boundary value problem can be easily solved. The inconsistency between Sendeckyj (In: Simmons JA, et al (eds) Fundamental aspects of dislocation theory. US National Bureau of Standards, Gaithersburg, pp 57-69, 1970) and Gong and Meguid (Int J Eng Sci 32(8):1221-1228, 1994) for the problem was also found by using the present approach. The error in Gong and Meguid (Int J Eng Sci 32(8):1221-1228, 1994) was also printed out. Finally, some examples are demonstrated to verify the validity of the present approach.
Sat, 01 Oct 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/231342022-10-01T00:00:00ZConstruction of dynamic Green's function for an infinite acoustic field with multiple prolate spheroidshttp://scholars.ntou.edu.tw/handle/123456789/22200Title: Construction of dynamic Green's function for an infinite acoustic field with multiple prolate spheroids
Authors: Leem, W. M.; Chen, J. T.
Abstract: The acoustic pressure of an unbounded acoustic field with multiple prolate spheroids with the Robin boundary conditions subjected to a time-harmonic point source located at an arbitrary location is solved semi-analytically in this work. This resultant solution is the so-called dynamic Green's function, which is important for acoustic problems such as sound scattering and noise control. It can be obtained by combining the fundamental solution with a homogenous solution, which is determined by using the collocation multipole procedure to satisfy the required Robin boundary conditions. To consider the geometries as described herein, the regular solution is expanded with angular and radial prolate spheroidal wave functions. As an alternate to the complex addition theorem applied to problems in multiply connected domains, by the directional derivative, the multipole expansion is computed in a straightforward manner among different local prolate spheroidal coordinate systems. By taking the finite terms of the multipole expansion at all collocating points, an algebraic system is acquired, and then the unknown coefficients are determined to complete the proposed dynamic Green's function by the Robin boundary conditions. The present results of one spheroid agree with the available analytical solutions. For the case of more than one spheroid, the proposed results are verified by comparison with the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. In the end, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheroids, and the aspect ratio of spheroid on the dynamic Green's functions. The proposed results can be applied to solve the time-harmonic problems for an unbounded acoustic field containing multiple spheroids. In the form of numerical Green's functions, they can improve the computational efficiency and increase the application of the boundary integral equation method.
Thu, 01 Sep 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/222002022-09-01T00:00:00ZA systematic approach for potentials on closely packed cells using the null-field boundary integral equation in conjunction with the degenerate kernel and eigenfunction expansionhttp://scholars.ntou.edu.tw/handle/123456789/22010Title: A systematic approach for potentials on closely packed cells using the null-field boundary integral equation in conjunction with the degenerate kernel and eigenfunction expansion
Authors: Lee, Ying-Te; Kao, Jeng-Hong; Chou, Yen-Ting; Chen, Jeng-Tzong
Abstract: A systematic approach based on the null-field integral formula is used to determine the electric potential of a tissue with many cells stimulated by remote electric fields. When the cells are very close to each other, the problem becomes nearly singular and the accuracy of the solution deteriorates. However, in the proposed approach, the highly accurate results are obtained because the separable kernel (degenerate kernel) and eigenfunction expansion are introduced to capture the geometry property in the integral formulation. Only boundary nodes are required instead of boundary elements to satisfy the boundary conditions or interface conditions. The proposed approach could be seen as one kind of meshless and semi-analytical methods. In addition, the error just stems from the number of truncation terms of the eigenfuntion expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. For the problem of closely packed cells, the boundary density of sharp variation could be accurately simulated or captured by increasing the number of terms of eigenfunctions. Finally, the acceptable results are shown to see the efficiency and accuracy of the proposed approach by the given numerical examples including one, three and twenty cells.
Fri, 01 Jul 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/220102022-07-01T00:00:00ZInteraction between a screw dislocation and an elliptical hole or rigid inclusion by using the angular basis functionhttp://scholars.ntou.edu.tw/handle/123456789/21853Title: Interaction between a screw dislocation and an elliptical hole or rigid inclusion by using the angular basis function
Authors: Chen, Jeng-Tzong; Lee, Jia-Wei; Kao, Shing-Kai; Tai, Wei-Chen
Abstract: The complex-valued fundamental solution ln(z) can be decomposed into the radial basis function (RBF) and the angular basis function (ABF). In this paper, not only the RBF but also the ABF are employed to solve the problem of interaction between a screw dislocation and an elliptical hole or rigid inclusion. The problem is decomposed into a free field with a screw dislocation and a boundary value problem subject to a specific boundary condition. The boundary value problem is solved by using the RBF and the boundary integral equation. Since the geometric shape is an ellipse, the degenerate kernel is expanded to a series form under the elliptic coordinates, while the unknown boundary densities are expanded to Fourier series. By combining the degenerate kernel and the null-field integral equation, the boundary value problem can be easily solved. Finally, two examples are demonstrated to verify the validity of the present approach.
Wed, 18 May 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/218532022-05-18T00:00:00ZStudy on the interaction between a screw dislocation and circular holes or rigid inclusions by using the angular basis function in conjunction with bipolar coordinateshttp://scholars.ntou.edu.tw/handle/123456789/21855Title: Study on the interaction between a screw dislocation and circular holes or rigid inclusions by using the angular basis function in conjunction with bipolar coordinates
Authors: Chen, Jeng-Tzong; Kao, Shing-Kai; Tai, Wei-Chen; Lee, Ying-Te; Lee, Jia-Wei; Chou, Yen-Ting
Abstract: In this paper, the degenerate kernel in conjunction with the bipolar coordinates is employed to solve the anti-plane problems of interaction between a screw dislocation and circular holes or rigid inclusions. Once the degenerate kernel of the angular basis function (ABF) is provided in terms of the bipolar coordinates, the analytical solution for cases of one or two circular holes and rigid inclusions can be derived. Not only the radial basis function (RBF) but also the ABF is used. First, the observer objectivity of the degenerate kernel in terms of the bipolar coordinates is examined numerically. A special case, one circular hole or rigid inclusion, is considered to demonstrate the validity of the present approach. Finally, the cases containing two circular holes and two circular rigid inclusions were examined. The comparison between available results and ours is well done. Besides, for the solutions of two holes and rigid inclusions, it is interesting to find that the present method provides an analytical solution with a series form of explicitly determined coefficients, while the coefficients provided by the complex variable need to be determined using the recursive formulae.
Tue, 26 Apr 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/218552022-04-26T00:00:00ZSupport motion of a finite bar with an external springhttp://scholars.ntou.edu.tw/handle/123456789/21554Title: Support motion of a finite bar with an external spring
Authors: Chen, Jeng-Tzong; Kao, Hao-Chen; Lee, Ying-Te; Lee, Jia-Wei
Abstract: In this paper, we gave the vibration analysis of a finite bar with an external spring on one side and the support motion on the other side. Two analytical methods, the mode superposition method in conjunction with the quasi-static decomposition method and the method of characteristics using the diamond rule, were employed to solve this problem. Both advantages and disadvantages of two methods were discussed. It is interesting to find that the mode superposition method can capture the silent area in terms of sum of an infinite series while the method of characteristics using the diamond rule can exactly derive the dead zone. Besides, it is found that discontinuities always occur at the location on the characteristic lines. Discussions of direct and inverse problems are also addressed.
Fri, 22 Apr 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/215542022-04-22T00:00:00ZStudy on the double-degeneracy mechanism of BEM/BIEM for a plane elasticity problem with line segmentshttp://scholars.ntou.edu.tw/handle/123456789/21055Title: Study on the double-degeneracy mechanism of BEM/BIEM for a plane elasticity problem with line segments
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Kao, Shing-Kai; Lee, Ying-Te; Kuo, Shyh-Rong
Abstract: There are four degenerate problems by using the BEM/BIEM. Only the degenerate scale and degenerate boundary may appear at the same time. This is called double degeneracy. A double-degeneracy mechanism of BEM/BIEM for the plane elasticity problem with line segments is studied, and the degenerate scale is analytically examined and numerically performed. Different from the past result with complex variables, we propose a new idea to deal with the problem, and obtain the analytical solution of the degenerate scale. This analytical derivation can clearly show why the BEM/BIEM suffer the degenerate scale in the plane elasticity problem with the line segment. Only rigid inclusion faces this problem instead of the crack due to the use of single-layer kernel. Double degeneracy of degenerate boundary and degenerate scale in the BEM are numerically examined. The double degeneracy mechanism is clearly displayed through numerical results by showing the number of zero singular values in the influence matrix. Following the result of single line segment, we can extend to multiple line segments. Finally, the analytical and numerical results show consistency.
Tue, 01 Mar 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/210552022-03-01T00:00:00ZOn the role of singular and hypersingular BIEs for the BVPs containing a degenerate boundaryhttp://scholars.ntou.edu.tw/handle/123456789/18201Title: On the role of singular and hypersingular BIEs for the BVPs containing a degenerate boundary
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Kao, Shing-Kai; Shao, Cheng-Hsiang; Tai, Wei-Chen
Abstract: ABSTR A C T In this paper, we find that using either the singular boundary integral equation (BIE) alone or the hypersingular BIE alone may solve the symmetric and anti-symmetric problems containing a degenerate boundary subject to the remote anti-plane shear, respectively. Although the rank of the influence matrix is deficient due to the degenerate boundary, it is still solvable. The mechanism was analytically demonstrated by using the degenerate kernel and numerically performed by using the singular value decomposition (SVD) technique. Besides, solv-ability was also discussed by using the Fredholm alternative theorem in both the boundary integral equation and boundary element method (BEM). We can analytically prove and numerically explain that the boundary density solved by using the singular BIE alone is different from the exact solution but the field solution is acceptable. The solvability of the linear algebraic system was discussed from the mapping table of the vector space of influence matrix where the domain is composed by the right singular vectors and the vector space of the range is composed of left singular vectors. The corresponding null space and the left null space due to the zero singular value in-dicates the homogeneous solution and the range deficiency, respectively. Several approaches, the singular or hypersingular BIE alone, adding symmetric or anti-symmetric boundary constraints, the self-regularized approach provided us alternative ways to solve the degenerate-boundary problem without using the dual BEM.
Wed, 01 Dec 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/182012021-12-01T00:00:00ZDual BEM for wave scattering by an H-type porous barrier with nonlinear pressure drophttp://scholars.ntou.edu.tw/handle/123456789/17813Title: Dual BEM for wave scattering by an H-type porous barrier with nonlinear pressure drop
Authors: Nishad, C. S.; Vijay, K. G.; Neelamani, S.; Chen, J. T.
Abstract: In this paper, wave scattering by an H-type porous barrier having nonlinear pressure drop boundary condition is analysed within the framework of small amplitude water wave theory. The H-type barrier is constructed using multiple thin (near zero thickness) rigid porous plates which are termed degenerate boundaries. The boundary value problem is solved using an iterative dual boundary element method. Different barrier configurations are analysed and compared to demonstrate the improved hydrodynamic performance of the H-type barrier. Further, the effect of porosity, relative spacing, the relative depth of submergence of the horizontal plate, wave steepness, and the rotation of the horizontal plate is investigated parametrically. Several results such as scattering coefficients (reflection, transmission, and energy-loss) and force coefficients (horizontal, vertical, and moment) are presented to understand the feasibility of the H-type barrier in real field applications. It is revealed that the increase in the structure porosity consistently increases the wave transmission, but reduces wave reflection, and force coefficients. Further, an increase in the relative spacing between the vertical barriers reduces the wave transmission by 20% without increasing the horizontal wave force but at the expense of increasing the vertical force coefficient in the shallow water regime. The results of this study are expected to be useful for the appropriate selection of different structure parameters to optimize the design.
Fri, 01 Oct 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/178132021-10-01T00:00:00ZApplications of degenerate kernels to potential flow across circular, elliptical cylinders and a thin airfoilhttp://scholars.ntou.edu.tw/handle/123456789/24507Title: Applications of degenerate kernels to potential flow across circular, elliptical cylinders and a thin airfoil
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Huang, Yi-Ling; Kao, Shing-Kai
Abstract: In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available. (C) 2021 Elsevier Masson SAS. All rights reserved.
Wed, 01 Sep 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245072021-09-01T00:00:00ZAN INDIRECT BIE FREE OF DEGENERATE SCALEShttp://scholars.ntou.edu.tw/handle/123456789/19099Title: AN INDIRECT BIE FREE OF DEGENERATE SCALES
Authors: Chen, Jeng-Tzong; Kao, Shing-Kai; Kao, Jeng-Hong; Tai, Wei-Chen
Abstract: Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.
Sat, 21 Aug 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/190992021-08-21T00:00:00ZConstruction of a curve by using the state equation of Frenet formulahttp://scholars.ntou.edu.tw/handle/123456789/18157Title: Construction of a curve by using the state equation of Frenet formula
Authors: Chen, J. T.; Lee, J. W.; Kao, S. K.; Chou, Y. T.
Abstract: In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley-Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.
Wed, 30 Jun 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/181572021-06-30T00:00:00ZOn the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domainhttp://scholars.ntou.edu.tw/handle/123456789/24506Title: On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domain
Authors: Chen, Jeng-Tzong; Lee, Jia-Wei; Huang, Yi-Ling; Shao, Cheng-Hsiang; Lu, Cheng-Hsuan
Abstract: In this paper, we proposed two ways to understand the rank deficiency in the continuous system (boundary integral equation method, BIEM) and discrete system(boundary element method, BEM) for a circular case. The infinite-dimensional degree of freedom for the continuous system can be reduced to finite-dimensional space using the generalized Fourier coordinates. The property of the second-order tensor for the influence matrix under different observers is also examined. On the other hand, the discrete system in the BEM can be analytically studied, thanks to the spectral property of circulant matrix. We adopt the circulant matrix of odd dimension, (2N + 1) by (2N + 1), instead of the previous even one of 2N by 2N to connect the continuous system by using the Fourier bases. Finally, the linkage of influence matrix in the continuous system (BIE) and discrete system (BEM) is constructed. The equivalence of the influence matrix derived by using the generalized coordinates and the circulant matrix are proved by using the eigen systems (eigenvalue and eigenvector). The mechanism of degenerate scale for a circular domain can be analytically explained in the discrete system.
Fri, 01 Jan 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245062021-01-01T00:00:00ZOn the path independence and invariant of the J-integral for a slant crack and rigid-line inclusion using degenerate kernels and the dual BEMhttp://scholars.ntou.edu.tw/handle/123456789/24505Title: On the path independence and invariant of the J-integral for a slant crack and rigid-line inclusion using degenerate kernels and the dual BEM
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Kao, Shing-Kai; Huang, Yi-Ling; Chou, Yen-Ting
Abstract: The J-integral and stress intensity factor (SIF) are two major parameters in linear elastic fracture mechanics (LEFM) for the fracture criterion. In this paper, we focus on the J-integral of the slant crack and the slant rigid line inclusion under the remote anti-plane shear. By employing the degenerate kernel, the path independence of J-integral is analytically demonstrated by using the elliptic coordinates. The positive and negative J-integrals are also analytically derived and numerically implemented by using the dual BEM for the crack and the rigid-line inclusion, respectively. It is interesting to find that the J-integral is not an invariant by using different observer systems but is one component of the vector of the first order tensor. Transformation law of the J-integral with respect to different observers is analytically proved and numerically demonstrated. Finally, the tensor property of order one is examined.
Fri, 01 Jan 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245052021-01-01T00:00:00ZStudy on the stress intensity factor and the double-degeneracy mechanism in the BEM/BIEM for anti-plane shear problemshttp://scholars.ntou.edu.tw/handle/123456789/24503Title: Study on the stress intensity factor and the double-degeneracy mechanism in the BEM/BIEM for anti-plane shear problems
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Huang, Yi-Ling; Kao, Shing-Kai
Abstract: The crack and the rigid-line inclusion problems under the anti-plane shear are the special case of the elliptic hole and elliptic rigid inclusion, respectively. We revisit this kind of problem by using the degenerate kernels in terms of the elliptic coordinates. The stress intensity factor (SIF) is also addressed. Three ways are employed to determine the SIF. One is the extrapolation approach for the boundary or interior displacement near the tip. Another is the extrapolation approach for the boundary stress or interior stress near the tip. The other is the J-integral enclosing the crack tip. It is interesting to find that a rigid-line inclusion case yields a singular influence matrix due to the degenerate scale of length four in the log kernel. However, double-degeneracy including degenerate scale and degenerate boundary may still result in a singular matrix even though the dual BEM/BIEM is employed. The mechanism is well explained thanks to the introduction of degenerate kernel. Without the introduction of degenerate kernel, the mechanism of the double-degeneracy problem in the BIEM can't be clearly examined. By using the degenerate kernel, the SIF can be easily determined for the crack or the rigid-line inclusion under the anti-plane shear. The path independence of the J-integral can be derived analytically. The reciprocal relation for the SIF between a crack and a rigid-line inclusion with respect to opposite loading is also addressed. In the numerical implementation, the SIF can be determined using the dual BEM. It is worth noting that BEM shows the advantage that the obtained boundary displacement or boundary stress can be directly used to obtain the more accurate SIF for the crack or the rigid inclusion, respectively.
Fri, 01 Jan 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245032021-01-01T00:00:00ZOn the stress concentration factor of circular/elliptic hole and rigid inclusion under the remote anti-plane shear by using degenerate kernelshttp://scholars.ntou.edu.tw/handle/123456789/24502Title: On the stress concentration factor of circular/elliptic hole and rigid inclusion under the remote anti-plane shear by using degenerate kernels
Authors: Chen, Jeng-Tzong; Kao, Jeng-Hong; Huang, Yi-Ling; Kao, Shing-Kai
Abstract: The stress concentration factor (SCF) along the boundary of a hole and a rigid inclusion in an infinite isotropic solid under the anti-plane shear is revisited by using degenerate kernels in the boundary integral equation (BIE) although this result was obtained by invoking the extended circle theorem of Milne-Thomson as well as the complex variable approach. The degenerate kernel of series form for the closed-form fundamental solution is used for the circle and the ellipse in terms of polar and elliptic coordinates, respectively. The slender ratio of the ellipse and the orientation are two parameters for our study. The strain energy density along the boundary is increased or decreased due to the different types of loading and various aspect ratios of the ellipse. An analytical solution for the SCF is then derived for any orientation of the ellipse relative to the applied load. The reciprocal relation for the SCF between a hole and a rigid inclusion with respect to different loading is also addressed. Besides, this analytical derivation can clearly show the appearing mechanism why the BEM/BIEM suffers the degenerate scale in the rigid inclusion.
Mon, 01 Feb 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245022021-02-01T00:00:00ZDynamic Green's functions for multiple elliptical inclusions with imperfect interfaceshttp://scholars.ntou.edu.tw/handle/123456789/2508Title: Dynamic Green's functions for multiple elliptical inclusions with imperfect interfaces
Authors: Lee, W. M.; Jeng-Tzong Chen
Abstract: The problem of an unbounded elastic solid with multiple elliptical inclusions subjected to a time-harmonic anti-plane concentrated force is semi-analytically solved by using the collocation multipole method. The displacement of matrix and inclusion are represented by angular and radial Mathieu functions. The imperfect interface between the matrix and the inclusion is characterized as a linear spring model with vanishing thickness. It is the derivative that the imperfect condition is involved. The addition theorem of Mathieu function is frequently used to solve multiply-connected domain problems in the traditional multipole method. An alternate here is a direct computation. The associated normal derivative with respect to a non-local elliptical coordinate system is developed by means of directional derivative. Besides simple computation, no truncation error is caused. The displacement field is determined by using the imperfect interface conditions through collocating points along the interface. Several numerical experiments are done to investigate the effects of the driving frequency of the concentrated force, imperfect interface and the convexity of elliptical inclusions on the dynamic Green's functions.
Tue, 01 Sep 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/25082020-09-01T00:00:00ZA new error estimation technique for solving torsion bar problem with inclusion by using BEMhttp://scholars.ntou.edu.tw/handle/123456789/2476Title: A new error estimation technique for solving torsion bar problem with inclusion by using BEM
Authors: Chen, K. H.; Kao, J.H.; Jeng-Tzong Chen; Liau, J. F.
Abstract: In this paper, the torsion problem is analyzed by boundary element method (BEM). After applying a new error estimation technique in the BEM, we can derive the numerical error of BEM. We extend the research of previous literature by the authors Chen and Chen [1], to the real engineering problem. This paper estimates the discretizing error caused by using BEM for solving the torsion problem with inclusions. The main characteristic of this technique is that the exact solution is not known in prior. In the technique, we need to create an auxiliary problem that the government equation, domain shape, and boundary condition type are the same as the given real problem. Besides, it has an analytical solution that satisfies the governing equation. We can derive the suitable number of elements by solving the auxiliary problem. Subsequently, by using the suitable number of elements in the BEM, we can obtain the appropriate solution for the real problem. Finally, several cases in the literature are given to illustrate the validity of the novel approach applied in the BEM to solve the real problem.
Mon, 01 Jun 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24762020-06-01T00:00:00ZAnalytical and numerical studies for solving Steklov eigenproblems by using the boundary integral equation method/boundary element methodhttp://scholars.ntou.edu.tw/handle/123456789/2409Title: Analytical and numerical studies for solving Steklov eigenproblems by using the boundary integral equation method/boundary element method
Authors: Jeng-Tzong Chen; Jia-Wei Lee; Kuen-Ting Lien
Abstract: The theory of boundary eigensolutions is developed for boundary value problems. It is general for boundary value problem. Steklov-Poincaré operator maps the values of a boundary condition of the solution of the Laplace equation in a domain to the values of another boundary condition. The eigenvalue is imbedded in the Dirichlet to Neumann (DtN) map. The DtN operator is called the Steklov operator. In this paper, we study the Steklov eigenproblems by using the dual boundary element method/boundary integral equation method (BEM/BIEM). First, we consider a circular domain. To analytically derive the eigensolution of the above shape, the closed-form fundamental solution of the 2D Laplace equation, ln(r), is expanded into degenerate kernel by using the polar coordinate system. After the boundary element discretization of the BIE for the Steklov eigenproblem, it can be transformed to a standard linear eigenequation. Problems can be effectively solved by using the dual BEM. Finally, we consider the annulus. Not only the Steklov problem but also the mixed Steklov eigenproblem for an annular domain has been considered.
Fri, 01 May 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24092020-05-01T00:00:00ZA study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problemshttp://scholars.ntou.edu.tw/handle/123456789/1059Title: A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problems
Authors: Jeng-Tzong Chen; Ying-Te Lee; Jia-Wei Lee; Sheng-Kuang Chen
Abstract: The influence matrix may be of deficient rank in the specified scale when we have solved the 2D elasticity problem by using the boundary element method (BEM). This problem stems from lnr in the 2D Kelvin solution. On the other hand, the single-layer integral operator can not represent the constant term for the degenerate scale in the boundary integral equation method (BIEM). To overcome this problem, we have proposed the enriched fundamental solution containing an adaptive characteristic length to ensure that the argument in the logarithmic function is dimensionless. The adaptive characteristic length, depending on the domain, differs from the constant base by adding a rigid body mode. In the analytical study, the degenerate kernel for the fundamental solution in polar coordinates is revisited. An adaptive characteristic length analytically provides the deficient constant term of the ordinary 2D Kelvin solution. In numerical implementation, adaptive characteristic lengths of the circular boundary, the regular triangular boundary and the elliptical boundary demonstrate the feasibility of the method. By employing the enriched fundamental solution in the BEM/BIEM, the results show the degenerate scale free.
Fri, 01 May 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10592020-05-01T00:00:00Z莫耳圓與二階張量關係之研究及其 Mathematica 動畫模擬 (下)http://scholars.ntou.edu.tw/handle/123456789/16701Title: 莫耳圓與二階張量關係之研究及其 Mathematica 動畫模擬 (下)
Authors: 陳正宗; 李家瑋; 凃雅瀞
Abstract: 本文第一作者係台灣工業與應用數學會(TWSIAM)副理事長, 從事工程數學教育二十餘年, 發現工程師不甚了解張量, 數學家不熟悉莫耳(Mohr)圓, 而少了跨領域的交流, 本文即在此前提動機下寫成。 本文係探討莫耳圓與二階張量關係。 為打破張量艱澀難懂之迷思, 不落窠臼, 加強學子學習成效, 將莫耳圓與二階張量在力學與數學兩方面建立系統性思考。 從張量角度出發, 輔以莫耳圓、分量式與矩陣式三種形式推導, 並探討二階張量與莫耳圓間之等價關係。 再引至不變量與固有值問題探討, 建立材料力學中樑受彎曲時斷面強軸、弱軸與主應力、 主應變概念, 並可說明為何工字樑或 I 型樑在土木工程上相當普遍。 本文將物理量分析(慣性矩、 應力與應變)、 幾何圖示(莫耳圓)與數學思維(張量)相融合, 引導學生將材料力學與工程數學融會貫通。 希冀無論在教師教學或學生學習上, 都能有實質幫助, 使張量不再令人望而卻步。 在此透過連結數學張量與工程力學架構, 讓讀者能有溫故知新的收穫。 最後, 以 Mathematica 符號運算軟體的動畫模擬功能為輔助, 以動畫呈現觀察座標系統改變情況下, 各物理量分量之變化與莫耳圓中不變量之驗核。
Sun, 01 Mar 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167012020-03-01T00:00:00ZIsogeometric Analysis of the Dual Boundary Element Method for the Laplace Problem With a Degenerate Boundaryhttp://scholars.ntou.edu.tw/handle/123456789/2487Title: Isogeometric Analysis of the Dual Boundary Element Method for the Laplace Problem With a Degenerate Boundary
Authors: Kao, J. H.; Chen, K. H.; Jeng-Tzong Chen; Kuo, S. R.
Abstract: In this paper, we develop the isogeometric analysis of the dual boundary element method (IGA-DBEM) to solve the potential problem with a degenerate boundary. The non-uniform rational B-Spline (NURBS) based functions are employed to interpolate the geometry and physical function. To deal with the rank-deficiency problem due to the degenerate boundary, the hypersingular integral equation is introduced to promote the full rank for the influence matrix in the dual BEM. Finally, three numerical examples are given to verify the accuracy of our proposed method. Both circular and square domains subjected to the Dirichlet boundary condition are considered. The engineering problem containing a degenerate boundary is considered, e.g., a seepage flow problem with a sheet pile. Numerical results of the IGA-DBEM agree well with these of the exact solution and the original dual boundary element method.
Sat, 01 Feb 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24872020-02-01T00:00:00Z莫耳圓與二階張量關係之研究及其 Mathematica 動畫模擬 (上)http://scholars.ntou.edu.tw/handle/123456789/16700Title: 莫耳圓與二階張量關係之研究及其 Mathematica 動畫模擬 (上)
Authors: 陳正宗; 李家瑋; 凃雅瀞
Abstract: 本文第一作者係台灣工業與應用數學會(TWSIAM)副理事長, 從事工程數學教育二十餘年, 發現工程師不甚了解張量, 數學家不熟悉莫耳(Mohr)圓, 而少了跨領域的交流, 本文即在此前提動機下寫成。 本文係探討莫耳圓與二階張量關係。 為打破張量艱澀難懂之迷思, 不落窠臼, 加強學子學習成效, 將莫耳圓與二階張量在力學與數學兩方面建立系統性思考。 從張量角度出發, 輔以莫耳圓、分量式與矩陣式三種形式推導, 並探討二階張量與莫耳圓間之等價關係。 再引至不變量與固有值問題探討, 建立材料力學中樑受彎曲時斷面強軸、弱軸與主應力、 主應變概念, 並可說明為何工字樑或 I 型樑在土木工程上相當普遍。 本文將物理量分析(慣性矩、 應力與應變)、 幾何圖示(莫耳圓)與數學思維(張量)相融合, 引導學生將材料力學與工程數學融會貫通。 希冀無論在教師教學或學生學習上, 都能有實質幫助, 使張量不再令人望而卻步。 在此透過連結數學張量與工程力學架構, 讓讀者能有溫故知新的收穫。 最後, 以 Mathematica 符號運算軟體的動畫模擬功能為輔助, 以動畫呈現觀察座標系統改變情況下, 各物理量分量之變化與莫耳圓中不變量之驗核。
Sun, 01 Dec 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167002019-12-01T00:00:00ZSemi-analytical approach for torsion problems of a circular bar containing multiple holes and/or crackshttp://scholars.ntou.edu.tw/handle/123456789/1080Title: Semi-analytical approach for torsion problems of a circular bar containing multiple holes and/or cracks
Authors: Ying-Te Lee; Jeng-Tzong Chen; Shyh-Rong Kuo
Abstract: A semi-analytical approach of the null-field integral equation in conjunction containing the degenerate kernels is used to deal with the torsion problems of a circular bar with circular or elliptic holes and/or line cracks. In order to fully capture the elliptic geometry, the use of the addition theorem in terms of the elliptic coordinates plays an important role to expand the fundamental solution into the degenerate form. The boundary densities are expressed by using the eigenfunction expansion for the elliptic boundary. It is worthy of noting that the model of elliptic hole in companion with the limiting process of approaching the length of the semi-minor axis to zero is adopted to simulate the line crack. Besides, we also make the length of the semi-major axis close to the length of the semi-minor axis to approximate the circular boundary. By collocating the observation point exactly on the real boundary and matching the boundary conditions, a linear algebraic system is easily constructed to determine the unknown eigenfucntion coefficients. This approach can be seen as a semi-analytical manner since error purely attributes to the truncation of eigenfunction expansions and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. Finally, several numerical examples of a circular bar with circular or elliptic holes and/or line cracks are employed to show the validity of the proposed approach. Not only the torsional rigidity but also the stress intensity factors are calculated to compare with the available results in the literature.
Tue, 01 Oct 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10802019-10-01T00:00:00ZIndirect boundary element method combining extra fundamental solutions for solving exterior acoustic problems with fictitious frequencieshttp://scholars.ntou.edu.tw/handle/123456789/2494Title: Indirect boundary element method combining extra fundamental solutions for solving exterior acoustic problems with fictitious frequencies
Authors: Jia-Wei Lee; Jeng-Tzong Chen; Chi-Feng Nien
Abstract: The authors propose an alternative approach to solve the problem of fictitious frequencies. It is different from the mixed potential approach in the indirect method as well as the Burton and Miller approach in the direct boundary element method (BEM). The authors add some fundamental solutions with unknown strength in the solution representation to complete the solution space. From the viewpoint of the adding source, the present idea is similar to the combined Helmholtz interior integral equation formulation (CHIEF) method. The difference between the added source point and null-field point of CHIEF is their role. The former supplies the deficient basis due to the fictitious frequency while the latter provides the extra constraint equation. It can be alternatively found by adding the right unitary vectors of zero singular value. The bordered matrix is invertible for the fictitious frequency if the extra source points do not locate at the failure position. This is the reason why the property is analogous to the CHIEF method in the direct BEM. Therefore, it can fill in the blank area of why there is no CHIEF method in the indirect method. The authors also analytically derive the locations of possible failure source points by using the degenerate kernel.
Wed, 29 May 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24942019-05-29T00:00:00ZDynamic Green's functions for multiple circular inclusions with imperfect interfaces using the collocation multipole methodhttp://scholars.ntou.edu.tw/handle/123456789/2510Title: Dynamic Green's functions for multiple circular inclusions with imperfect interfaces using the collocation multipole method
Authors: Lee, W. M.; Jeng-Tzong Chen; Young, W. M.
Abstract: This paper presents a semi-analytical approach to solve anti-plane dynamic Green's functions for an elastic infinitely extended isotropic solid (matrix) containing multiple circular inclusions with imperfect interfaces. A linear spring model with vanishing thickness is employed to character the imperfect interface. The multipole expansions of anti-plane displacement of the matrix and inclusion, induced by a time-harmonic anti-plane line force located in the matrix or in the inclusion, are expanded by using Hankel and Bessel functions, respectively. The imperfect interface condition is satisfied by uniformly collocating points along the interface of each inclusion. For the imperfect interface condition, the normal derivative of the anti-plane displacement with respect to a non-local polar coordinate system is developed without any truncation error for multiply-connected domain problems. For the case of one circular inclusion, the proposed quasi-static stress field matches well with the analytical static solution. The proposed quasi-static stress fields containing two and three circular inclusions are critically compared with those calculated by static analysis using the finite element method. Finally, extensive studies are presented to investigate the effects of the frequency of excitation, imperfect interface and separation between inclusions on the dynamic Green's functions.
Sat, 01 Sep 2018 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/25102018-09-01T00:00:00ZAnimations and Properties of Three SDOF Damping Systemshttp://scholars.ntou.edu.tw/handle/123456789/2404Title: Animations and Properties of Three SDOF Damping Systems
Authors: Jeng-Tzong Chen; Jia-Wei Lee; Shing-Kai Kao; Sheng-Kuang Chen
Abstract: Three models—the viscous damping model, the new hysteretic damping model, and the Coulomb damping model—are studied in this paper. For the viscous damping and the Coulomb damping models, the free vibration problem is reviewed and demonstrated by animations. Regarding the new hysteretic damping model, the free vibration problem for the different range of parameters, namely 0<η<1, η=1, and η>1 are analytically derived and are also demonstrated by animations. In particular, the exact solutions of the latter two cases are derived for the first time. In animations, the trajectories for three damping models in the phase plane consist of straight lines, quarter ellipses, and hyperbolic curves. For the case of η≥1, it is interesting that permanent deformation may occur. In addition, the dead zone for the Coulomb damping model in the phase plane is also addressed. The envelope for the amplitude decay yields exponential, geometric, and linear curves for the viscous damping model, the new hysteretic damping model and the Coulomb damping model, respectively. It is also the primary focus that the same period and the same ratio of amplitude decay for the relation between the viscous coefficient and the hysteretic parameter are constructed. All animations are produced using the symbolic software Mathematica because it is easy for readers to understand the physical behavior of three damping models.
Wed, 01 Aug 2018 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24042018-08-01T00:00:00ZStatic Analysis of the Free-Free Trusses by Using a Self-Regularization Approachhttp://scholars.ntou.edu.tw/handle/123456789/2339Title: Static Analysis of the Free-Free Trusses by Using a Self-Regularization Approach
Authors: Jeng-Tzong Chen; Chang, Y. L.; Leu, S. Y.; Lee, J. W.
Abstract: Following the success of static analysis of free-free 2-D plane trusses by using a self-regularization approach uniquely, we further extend the technique to deal with 3-D problems of space trusses. The inherent singular stiffness of a free-free structure is expanded to a bordered matrix by adding r singular vectors corresponding to zero singular values, where r is the nullity of the singular stiffness matrix. Besides, r constraints are accompanied to result in a nonsingular matrix. Only the pure particular solution with nontrivial strain is then obtained but without the homogeneous solution of no deformation. To link with the Fredholm alternative theorem, the slack variables with zero values indicate the infinite solutions while those with nonzero values imply the case of no solutions. A simple space truss is used to demonstrate the validity of the proposed model. An alternative way of reasonable support system to result in a nonsingular stiffness matrix is also addressed. In addition, the finite-element commercial code ABAQUS is also implemented to check the results.
Wed, 01 Aug 2018 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/23392018-08-01T00:00:00ZQuaternion Boundary Element Method for Coupled Exterior and Interior Magnetostatic Fieldshttp://scholars.ntou.edu.tw/handle/123456789/2486Title: Quaternion Boundary Element Method for Coupled Exterior and Interior Magnetostatic Fields
Authors: Hong-Ki Hong; Yi-Chuan Kao; Jia-Wei Lee; Li-Wei Liu; Jeng-Tzong Chen
Abstract: In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic problem. The present quaternion-valued BEM is developed by discretizing the quaternion-valued boundary integral equation (BIE). The quaternion-valued BIE can be seen as an extension of the generalized complex variable BIE in 3-D space. In other words, quaternion algebra is an extension of the complex variable in 3-D space. To derive quaternion-valued BIEs, the quaternion-valued Stokes' theorem is utilized. The quaternion-valued BIEs are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). An analytical scheme is developed to evaluate the CPV by introducing a simple quaternion-valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called “numericala boundary layer” phenomena appear and are remedied by using a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. It is noted that all derivations are done in quaternion algebra. In addition, quaternion algebra is more flexible than vector algebra in solving some 3-D problems from the point view of algebraic space. In deriving BIEs for exterior fields, the conditions at infinity for the quaternion-valued functions are carefully examined. Later, a magnetic sphere in a uniform magnetic field is considered. This problem is a magnetostatic problem of coupled exterior and interior magnetostatic fields. Finally, we apply the present approach to solve the magnetostatic problem. By comparing with exact solutions, the validity of the present approach is checked.
Fri, 01 Jun 2018 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24862018-06-01T00:00:00ZScattering problems of the SH Wave by Using the Null-Field Boundary Integral Equation Methodhttp://scholars.ntou.edu.tw/handle/123456789/2384Title: Scattering problems of the SH Wave by Using the Null-Field Boundary Integral Equation Method
Authors: Jeng-Tzong Chen; Shing-Kai Kao; Yin-Hsiang Hsu; Yu Fan
Abstract: The scattering problem of seismic waves is an important issue for studying earthquake engineering. In this paper, the null-field boundary integral equation approach was used in conjunction with degenerate kernels and eigenfunction expansion to solve the SH-wave scattering problem of a circular or an elliptical-arc hill. The original problem is divided into subdomains by taking a free-body diagram. One region is an interior boundary value problem. The other is a canyon scattering problem. For the boundary value problem, not only a simply connected domain (elliptical-arc hill problem) but also a doubly connected domain (a circular-arc hill problem containing a circular tunnel or a circular inclusion) is considered. The canyon scattering problem may be addressed in an infinite domain with an artificial boundary of a full plane such that the degenerate kernel can be fully utilized. The null-field integral equation method is used to match boundary conditions. Numerical results are compared favorably with the available data.
Mon, 01 Jan 2018 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/23842018-01-01T00:00:00ZAnalytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinateshttp://scholars.ntou.edu.tw/handle/123456789/2385Title: Analytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinates
Authors: Jeng-Tzong Chen; Kao Shing-Kai; Lee Jia-Wei
Abstract: Degenerate scales of an eccentric annulus and an infinite plane with two identical circular holes in the boundary integral equation method (BIEM) are analytically derived and numerically implemented in this paper. To analytically study the degenerate scale of the BIE, the closed-form fundamental solution of the two-dimensional Laplace equation, ln r, is expanded by a degenerate (separate) kernel in terms of the bipolar coordinates. It is proved that unit radius of the outer circle dominates the degenerate scale of eccentric annulus. An analytical formula of degenerate scale for the infinite plane with two identical circular boundaries was also derived at the first time. In addition, null fields of the domain and complementary domain for the ordinary and degenerate scales are both shown, respectively. Finally, comparison with available results and the BEM data are well done.
Fri, 01 Dec 2017 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/23852017-12-01T00:00:00ZDegenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysishttp://scholars.ntou.edu.tw/handle/123456789/1506Title: Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis
Authors: Jeng-Tzong Chen; Shyh-Rong Kuo; Yu-Lung Chang; Shing-Kai Kao
Abstract: Purpose
The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically.
Design/methodology/approach
To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered.
Findings
Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function.
Originality/value
The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.
Mon, 03 Jul 2017 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15062017-07-03T00:00:00ZWhy dual boundary element method is necessary?http://scholars.ntou.edu.tw/handle/123456789/2468Title: Why dual boundary element method is necessary?
Authors: Jeng-Tzong Chen; Ching-Yun Yueh; Yu-Lung Chang; Chun-Chiang Wen
Abstract: Dual boundary integral equation (BIE) was developed for problems containing degenerate boundaries in 1988 by Hong and Chen [Journal of Engineering Mechanics-ASCE, 114, 6, 1988] and was termed the dual boundary element method (BEM) in 1992 by Portela et al. [International Journal for Numerical Methods in Engineering, 33, 6, 1992]. After near 30 years, the dual BIE/BEM for the problem containing a zero-thickness barrier was revisited mathematically to study the rank deficiency from the viewpoint of the updating term and the updating document of singular value decomposition (SVD) [Journal of Mechanics, 31, 5, 2015]. In this paper, we revisit the dual BEM from the physical point of view. Although there is no zero-thickness barrier in the real world, it is always required to simulate a finite-thickness degenerate boundary to be zero-thickness in comparison with sea, air or earth scale. For example, a sheet pile, a screen, a crack problem, a thin airfoil and a breakwater were modeled by the geometry of zero-thickness. The role of the dual BEM is evident since Lafe et al. [Journal of the Hydraulics Division-ASCE, 106, 6, 1980] used the conventional BEM to model the finite-thickness pile wall to geometrically approximate zero-thickness barrier but numerically yielding divergent solution. On the contrary, we physically model the finite-thickness breakwater as a zero-thickness barrier. The breakwater is employed as an illustrative case to demonstrate that the dual BEM simulated by a zero-thickness barrier can yield more acceptable results to match the experiment data in comparison with those of the finite thickness using the conventional BEM. Finally, a single horizontal plate and two dual horizontal plates in vertical direction and in horizontal direction are three illustrative cases to tell you why the dual BEM is necessary not only in mathematics but also in physics.
Wed, 01 Mar 2017 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24682017-03-01T00:00:00ZA self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problemshttp://scholars.ntou.edu.tw/handle/123456789/1053Title: A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems
Authors: Jeng-Tzong Chen; Ying-Te Lee; Yu-Lung Chang; Jie Jian
Abstract: The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.
Sun, 01 Jan 2017 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10532017-01-01T00:00:00ZRevisit of degenerate scales in the BIEM/BEM for 2D elasticity problemshttp://scholars.ntou.edu.tw/handle/123456789/1047Title: Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems
Authors: Jeng-Tzong Chen; Wen-Sheng Huang; Ying-Te Lee; Shyh-Rong Kuo; Shing-Kai Kao
Abstract: The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.
Sun, 01 Jan 2017 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10472017-01-01T00:00:00Z工程數學力學計算與動畫模擬http://scholars.ntou.edu.tw/handle/123456789/16703Title: 工程數學力學計算與動畫模擬
Authors: 陳正宗; 高聖凱; 李家瑋
Sun, 01 Jan 2017 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167032017-01-01T00:00:00ZComputation of scattering of a plane wave from multiple prolate spheroids using the collocation multipole methodhttp://scholars.ntou.edu.tw/handle/123456789/2507Title: Computation of scattering of a plane wave from multiple prolate spheroids using the collocation multipole method
Authors: Lee, W. M.; Jeng-Tzong Chen
Abstract: The collocation multipole method is presented to solve three-dimensional acoustic scattering problems with multiple prolate spheroids subjected to a plane sound wave. To satisfy the three-dimensional Helmholtz equation in prolate spheroidal coordinates and the radiation condition at infinity, the scattered field is formulated in terms of radial and angular prolate spheroidal wave functions. Instead of using the complicated addition theorem of prolate spheroidal wave functions, the multipole method, the directional derivative, and the collocation technique are combined to solve multiple scattering problems semi-analytically. For the sound-hard or Neumann conditions, the normal derivative of the acoustic pressure with respect to a non-local prolate spheroidal coordinate system is developed without any truncation error for multiply connected domain problems. By truncating the higher order terms of the multipole expansion, a finite linear algebraic system is obtained and the scattered field is determined from the given incident acoustic wave. Once the total field is calculated as the sum of the incident field and the scattered field, the near field acoustic pressure and the far field scattering pattern are determined. Numerical experiments for convergence are performed to provide the guide lines for the proposed method. The proposed results of acoustic scattering by one, two, and three prolate spheroids are compared with those of an available analytical method and the boundary element method to validate the proposed method. Finally, the effects of the eccentricity of a prolate spheroidal scatterer, the separation between scatterers and the incident wave number on the near-field acoustic pressure and the far-field scattering pattern are investigated.
Mon, 03 Oct 2016 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/25072016-10-03T00:00:00ZApplications of the Clifford algebra valued boundary element method to electromagnetic scattering problemshttp://scholars.ntou.edu.tw/handle/123456789/2496Title: Applications of the Clifford algebra valued boundary element method to electromagnetic scattering problems
Authors: Jia-Wei Lee; Li-Wei Liu; Hong-Ki Hong; Jeng-Tzong Chen
Abstract: Electromagnetic problems governed by Maxwell's equations are solved by using a Clifford algebra valued boundary element method (BEM). The well-known Maxwell's equations consist of eight pieces of scalar partial differential equations of the first order. They can be rewritten in terms of the language of Clifford analysis as a nonhomogeneous k-Dirac equation with a Clifford algebra valued function. It includes three-component electric fields and three-component magnetic fields. Furthermore, we derive Clifford algebra valued boundary integral equations (BIEs) with Cauchy-type kernels and then develop a Clifford algebra valued BEM to solve electromagnetic scattering problems. To deal with the problem of the Cauchy principal value, we use a simple Clifford algebra valued k-monogenic function to exactly evaluate the Cauchy principal value. Free of calculating the solid angle for the boundary point is gained. The remaining boundary integral is easily calculated by using a numerical quadrature except the part of Cauchy principal value. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. In the numerical implementation, we introduce an oriented surface element instead of the unit outward normal vector and the ordinary surface element. In addition, we adopt the Dirac matrices to express the bases of Clifford algebra . We also use an orthogonal matrix to transform global boundary densities into local boundary densities for satisfying boundary condition straightforward. Finally, two electromagnetic scattering problems with a perfect spherical conductor and a prolate spheroidal conductor are both considered to examine the validity of the Clifford algebra valued BEM with Cauchy-type kernels.
Sat, 01 Oct 2016 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24962016-10-01T00:00:00ZFocusing phenomenon and near-trapped modes of SH waveshttp://scholars.ntou.edu.tw/handle/123456789/2412Title: Focusing phenomenon and near-trapped modes of SH waves
Authors: Jeng-Tzong Chen; Jia-Wei Lee; Ya-Ching Tu
Abstract: In this study, the null-field boundary integral equation method (BIEM) and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.
Thu, 08 Sep 2016 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/24122016-09-08T00:00:00ZA necessary and sufficient BEM/BIEM for two-dimensional elasticity problemshttp://scholars.ntou.edu.tw/handle/123456789/1046Title: A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems
Authors: Jeng-Tzong Chen; Wen-Sheng Huang; Ying-Te Lee; Shing-Kai Kao
Abstract: It is well known that the patch test is required for the finite element method (FEM). We may wonder whether we need any special test for the boundary element method (BEM). A sufficient and necessary boundary integral equation method (BIEM) to ensure a unique solution is our concern. In this paper, we revisit this issue for the interior two-dimensional (2-D) elasticity problem and investigate the equivalence of the solution space between the integral equation and the partial differential equation. Based on the degenerate kernel and the eigenfunction expansion, the range deficiency of the integral operator for the solution space in the degenerate-scale problem for the 2-D elasticity in the BIEM is analytically studied. According to the Fichera׳s idea, we enrich the conventional BIEM by adding constants and corresponding constraints. In addition, we introduce the concept of modal participation factor (MPF) to examine whether the adding term of rotation is required for interior simply-connected problems. Finally, two simple examples of degenerate-scale problems containing circular and elliptical boundaries subjected to various boundary conditions of the rigid body translation and rotation for 2-D elasticity problems are demonstrated by using the necessary and sufficient BIEM.
Description: BEM/MRM 38 英國布羅肯赫斯特
Wed, 01 Jun 2016 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10462016-06-01T00:00:00ZGreen's Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equationhttp://scholars.ntou.edu.tw/handle/123456789/1038Title: Green's Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equation
Authors: Yu-Lung Chang; Ying-Te Lee; Li-Jie Jiang; Jeng-Tzong Chen
Abstract: A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.
Mon, 07 Mar 2016 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10382016-03-07T00:00:00ZOn the free terms of the dual BIE for N-dimensional Laplace problemshttp://scholars.ntou.edu.tw/handle/123456789/2379Title: On the free terms of the dual BIE for N-dimensional Laplace problems
Authors: Jeng-Tzong Chen; Wen-Sheng Huang; Jia-Wei Lee; Hong-Ki Hong
Abstract: Dual boundary integral equations for the N-dimensional Laplace problems with a smooth boundary are derived by using the contour approach surrounding the singularity. The potentials resulted from the four kernel functions in the dual formulation have different properties across the smooth boundary. For the generalization, we focus on the N-dimensional Laplace equation. The Hadamard principal value (H.P.V.) is derived naturally and is composed of two parts, the Cauchy principal value (C.P.V.) and an unbounded boundary term. The hypersingular integral is not a divergent integral since we can collect the C.P.V. and the unbounded term together. Besides, the weighting of the free term contributed by different kernels is also examined. Finally, a special case of the four-dimensional Laplace equation is implemented and the free term, for any dimension are obtained. The contributions of the free terms for the boundary normal derivative of potential due to the single (L kernel) and the double (M kernel) layer potentials are 1/N and (N-1)/N, respectively. It is an interesting phenomenon that the hypersingular kernel contributes more than the singular kernel, and, in addition, the former also yields an unbounded boundary term.
Thu, 01 Oct 2015 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/23792015-10-01T00:00:00ZRevisit of the Dual Bem using SVD Updating Techniquehttp://scholars.ntou.edu.tw/handle/123456789/2378Title: Revisit of the Dual Bem using SVD Updating Technique
Authors: Jeng-Tzong Chen; Huang, W. S.; Fan, Y.; Kao, S. K.
Abstract: The boundary element method (BEM) is easier than the finite element method (FEM) on the viewpoint of the discretization of one dimension reduction rather than the domain discretization of finite element method. The disadvantage of BEM is the rank deficiency in the influence matrix, e.g., degenerate boundary, degenerate scale, spurious eigenvalues and fictitious frequencies, which do not occur in the FEM. The conventional BEM can not be straightforward applied to solve a problem which contains a degenerate boundary without decomposing the domain to multi-regions. A hypersingular integral equation is used to ensure a unique solution for the problem containing a degenerate boundary. By combining the singular and hypersingular equations, it’s termed the dual BEM due to its dual frame. Following the successful experience on the retrieval of information using the singular value decomposition (SVD) updating term and updating document, this technique is also used to extract out the degenerate-boundary information and the rigid-body information in the dual BEM. It is interesting to find that true information due to a rigid-body mode in physics is found in the right singular vector with respect to the corresponding zero singular value while the degenerate-boundary mode (geometry degeneracy) in mathematics is imbedded in the left singular vector with respect to the corresponding zero singular value. The role of the common right and left singular vectors of SVD for the four influence matrices in the dual BEM is also discussed in this paper. Two examples, a potential flow problem across a cutoff wall and a cracked bar under torsion were demonstrated to see the mathematical SVD structure of four influence matrices in the dual BEM.
Thu, 01 Oct 2015 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/23782015-10-01T00:00:00Z