National Taiwan Ocean University Research Hubhttps://scholars.ntou.edu.twThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 15 Jun 2024 17:38:30 GMT2024-06-15T17:38:30Z50341Study 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/123456789/210552022-03-01T00:00:00ZSemi-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/123456789/10802019-10-01T00:00:00ZLocalized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equationshttp://scholars.ntou.edu.tw/handle/123456789/1163Title: Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations
Authors: C.M. Fan; Y.K. Huang; C.S. Chen; S.R. Kuo
Abstract: The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS.
Mon, 01 Apr 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11632019-04-01T00:00:00ZA qualified plate theory for rigid rotation in postcritical nonlinear analysishttp://scholars.ntou.edu.tw/handle/123456789/1521Title: A qualified plate theory for rigid rotation in postcritical nonlinear analysis
Authors: Shyh-Rong Kuo; Yang, J. P.; Yang, Y. B.
Sat, 01 Dec 2018 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15212018-12-01T00:00:00ZDegenerate-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/123456789/15062017-07-03T00:00:00ZRevisit 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/123456789/10472017-01-01T00:00:00ZA Novel Approach for Buckling Analysis of Pretwisted Spatially Curved Beams by State Equationshttp://scholars.ntou.edu.tw/handle/123456789/1520Title: A Novel Approach for Buckling Analysis of Pretwisted Spatially Curved Beams by State Equations
Authors: Shyh-Rong Kuo; Yang, J. P.; Yang, Y. B.
Wed, 01 Jun 2016 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15202016-06-01T00:00:00ZA nonsingular boundary collocation method for the inverse problems in elasticityhttp://scholars.ntou.edu.tw/handle/123456789/18370Title: A nonsingular boundary collocation method for the inverse problems in elasticity
Authors: Ying-Te Lee; Chia-Ming Fan; Shyh-Rong Kuo
Thu, 01 Oct 2015 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/183702015-10-01T00:00:00ZRevisit of a degenerate scale: A semi-circular dischttp://scholars.ntou.edu.tw/handle/123456789/1509Title: Revisit of a degenerate scale: A semi-circular disc
Authors: Jeng-Tzong Chen; Shyh-Rong Kuo; Shing-Kai Kao; Jie Jian
Abstract: Boundary element method (BEM) has been employed in engineering analysis since 1956, it has been widely applied in the engineering. However, the BEM/BIEM may result in an ill-conditioned system in some special situations, such as the degenerate scale. The degenerate scale also relates to the logarithmic capacity in the modern potential theory. In this paper, three indexes to detect the degenerate scale and five regularization techniques to circumvent the degenerate scale are reviewed and a new self-regularization technique by using the bordered matrix is proposed. Both the analytical study and the BEM implementation are addressed. For the analytical study, we employ the Riemann conformal mapping of complex variables to derive the unit logarithmic capacity. The degenerate scale can be analytically derived by using the conformal mapping as well as numerical detection by using the BEM. In the theoretical aspect, we prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. We revisit the Fredholm alternative theorem by using the singular value decomposition (SVD, the discrete system) and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. Besides, a zero index by using the free constant in Fichera’s approach is also proposed to examine the degenerate scale. According to the relation between the SVD structure and Fichera’s technique, we numerically provide a new self-regularization method in the matrix level. Finally, a semi-circular case and a special-shape case are designed to demonstrate the validity of six regularization techniques.
Sat, 01 Aug 2015 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15092015-08-01T00:00:00ZRevisit of the degenerate scale for plate problemshttp://scholars.ntou.edu.tw/handle/123456789/16773Title: Revisit of the degenerate scale for plate problems
Authors: Shyh-Rong Kuo; Jeng-Tzong Chen; Yu-Lung Chang; Shing-Kai Kao
Abstract: Degenerate scale occurs once the boundary integral equation (BIE) is employed to solve plate problems. There are two ways to be considered. First, the degenerate kernel for expressing the closed-form fundamental solution is utilized to study the occurring mechanism for circular and elliptical plates using polar and elliptical coordinates, respectively. Second, the BIE is discretized and reformulated to a 4 by 4 eigenproblem to obtain degenerate scales instead of the direct-searching scheme. Analytical degenerate scales for the circular and elliptical shapes are obtained while numerical results for general shapes are also considered.
Description: ICOME 2015 中國浙江
Thu, 01 Jan 2015 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167732015-01-01T00:00:00ZNull-field integral approach for the piezoelectricity problems with multiple elliptical inhomogeneitieshttp://scholars.ntou.edu.tw/handle/123456789/1079Title: Null-field integral approach for the piezoelectricity problems with multiple elliptical inhomogeneities
Authors: Ying-Te Lee; Jeng-Tzong Chen; Shyh-Rong Kuo
Abstract: Based on the successful experience of solving anti-plane problems containing multiple elliptical inclusions, we extend to deal with the piezoelectricity problems containing arbitrary elliptical inhomogeneities. In order to fully capture the elliptical geometry, the keypoint of the addition theorem in terms of the elliptical coordinates is utilized to expand the fundamental solution to the degenerate kernel and boundary densities are simulated by the eigenfunction expansion. Only boundary nodes are required instead of boundary elements. Therefore, the proposed approach belongs to one kind of meshless and semi-analytical methods. Besides, the error stems from the number of truncation terms of the eigenfunction expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. It is worth noting that there are Jacobian terms in the degenerate kernel, boundary density and contour integral. However, they would cancel each other out in the process of the boundary contour integral. As the result, the orthogonal property of eigenfunction is preserved and the boundary integral can be easily calculated. For verifying the validity of the present method, the problem of an elliptical inhomogeneity in an infinite piezoelectric material subject to anti-plane shear and in-plane electric field is considered to compare with the analytical solution in the literature. Besides, two circular inhomogenieties can be seen as a special case to compare with the available data by approximating the major and minor axes. Finally, the problem of two elliptical inhomogeneities in an infinite piezoelectric material is also provided in this paper.
Sat, 01 Feb 2014 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10792014-02-01T00:00:00ZRegularization methods for ill-conditioned system of the integral equation of the first kind with the logarithmic kernelhttp://scholars.ntou.edu.tw/handle/123456789/1504Title: Regularization methods for ill-conditioned system of the integral equation of the first kind with the logarithmic kernel
Authors: Jeng-Tzong Chen; Houde Han; Shyh-Rong Kuo; Shing-Kai Kao
Abstract: The occurring mechanism of the ill-conditioned system due to degenerate scale in the direct boundary element method (BEM) and the indirect BEM is analytically examined by using degenerate kernels. Five regularization techniques to ensure the unique solution, namely hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium (direct BEM), CHEEF method and the Fichera’s method (indirect BEM), are analytically studied and numerically implemented. In this paper, we examine the sufficient and necessary condition of boundary integral formulation for the uniqueness solution of 2D Laplace problem subject to the Dirichlet boundary condition. Both analytical study and BEM implementation are addressed. For the analytical study, we employ the degenerate kernel in the polar and elliptic coordinates to derive the unique solution by using five regularization techniques for any size of circle and ellipse, respectively. Full rank of the influence matrix in the BEM using Fichera’s method for both ordinary scale and degenerate scale is also analytically proved. In numerical implementation, the BEM programme developed by NTOU/MSV group is employed to see the validity of the above formulation. Finally, the circular and elliptic cases are numerically demonstrated by using five regularization techniques. Besides, a general shape of a regular triangle is numerically implemented to check the uniqueness solution of BEM.
Thu, 16 Jan 2014 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15042014-01-16T00:00:00ZThree detecting indexes and five regularization techniques for degenerate scales in the BEM/BIEMhttp://scholars.ntou.edu.tw/handle/123456789/16758Title: Three detecting indexes and five regularization techniques for degenerate scales in the BEM/BIEM
Authors: Jeng-Tzong Chen; Shing-Kai Kao; Shyh-Rong Kuo
Abstract: It is well known that BEM/BIEM results in degenerate scale for a twodimensional
Laplace problem subjected to the Dirichlet boundary condition. In this paper, we reviewed three indexes for detecting the degenerate scale in BEM/BIEM and five regularization techniques to ensure the unique solution, the hypersingular formulation rank promotion by adding the boundary flux equilibrium, CHEEF method, (direct BEMs), Fichera’s method (indirect BEM) and method of adding a rigid body mode. In the numerical implementation, the BEM program developed by the NTOU/MSV group is employed to see the validity of the above formulation. Finally, a general shape of a regular triangle is numerically implemented to check the uniqueness solution of BEM.
Description: Boundary Elements and Other Mesh Reduction Methods XXXVI, October 22-24, 2013, 中國大連, 大連理工大學
Tue, 22 Oct 2013 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167582013-10-22T00:00:00ZLinkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMshttp://scholars.ntou.edu.tw/handle/123456789/1515Title: Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs
Authors: Shyh-Rong Kuo; Jeng-Tzong Chen; Shing-Kai Kao
Abstract: It is well known that BEM/BIEM results in degenerate scale for a two-dimensional Laplace interior problem subjected to the Dirichlet boundary condition. In such a case, there is nontrivial boundary normal flux even if the trivial boundary potential is specified. It is proved that the unit logarithmic capacity in the Riemann conformal mapping with respect to the unit circle results in a null field for the interior domain. The logarithmic capacity is defined as the leading coefficient of the linear term in the Riemann conformal mapping. First, the real-variable BIE is transformed to the complex variable BIE. By considering the analytical field and taking care of the path of the branch cut, we can prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. When the logarithmic capacity is equal to one, a trivial interior field can be obtained but an exterior field is derived to be nonzero using the logarithmic function. Two mapping functions, the Riemann conformal mapping for the geometry and the logarithmic function for the physical field, are both utilized. This matches well with the BEM result that an interior trivial field yields nonzero boundary flux in case of degenerate scale. Regarding the ordinary scale, BIE results in a null field in the exterior domain owing to the Green’s third identity. It is interesting to find that ordinary and degenerate scales result in a null field in the exterior and interior domains, respectively. A parameter study for the scaling constant and the leading coefficient of the term in the Riemann conformal mapping is also done. To demonstrate this finding, different shapes were demonstrated. Theoretical derivation using the Riemann conformal mapping with the unit logarithmic capacity and the degenerate scale in the BEM/BIEM both analytically and numerically indicate the null field in the interior domain.
Sun, 01 Sep 2013 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15152013-09-01T00:00:00ZNull-field integral approach for the piezoelectricity problems with arbitrary elliptical inhomogeneitieshttp://scholars.ntou.edu.tw/handle/123456789/16750Title: Null-field integral approach for the piezoelectricity problems with arbitrary elliptical inhomogeneities
Authors: Ying-Te Lee; Jeng-Tzong Chen; Shyh-Rong Kuo
Abstract: Based on the successful experience of solving anti-plane problems containing arbitrary elliptical inclusions, we extend to deal with the piezoelectricity problems containing arbitrary elliptical inhomogeneities. In order to fully capture the elliptical geometry, the keypoint of the addition theorem in terms of the elliptical coordinates is utilized to expand the fundamental solution to the degenerate kernel and boundary densities are simulated by the eigenfunction expansion. Only boundary nodes are required instead of boundary elements. Therefore, the proposed approach belongs to one kind of meshless and semi-analytical methods. Besides, the error 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. It is worth noting that there are Jacobian terms in the degenerate kernel, boundary density and contour integral. However, they would cancel each other out in the process of the boundary contour integral. As the result, the orthogonal property of eigenfunction is preserved and the boundary integral can be easily calculated. Finally, the problem of two elliptical inhomogeneities in an infinite piezoelectric material subject to anti-plane remote shear and in-plane electric field is considered to demonstrate the validity of the present method. Besides, two circular inhomegenieties can be seen as a special case to compare with the available data by approximating the major and minor axes.
Description: 13th International Conference on Fracture, June 16–21, 2013, Beijing, China
Sun, 16 Jun 2013 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167502013-06-16T00:00:00ZAnalytical derivation and numerical experiments of degenerate scales for regular N-gon domains in two-dimensional Laplace problemshttp://scholars.ntou.edu.tw/handle/123456789/1516Title: Analytical derivation and numerical experiments of degenerate scales for regular N-gon domains in two-dimensional Laplace problems
Authors: Shyh-Rong Kuo; Jeng-Tzong Chen; Jia-Wei Lee; Yi-Wei Chen
Abstract: Degenerate scale of a regular N-gon domain is studied by using the boundary element method (BEM) and complex variables. Degenerate scale stems from either the non-uniqueness of BIE using the logarithmic kernel or the conformal radius of unit logarithmic capacity in the complex variables. Analytical formula and numerical results for the degenerate scale are obtained by using the conformal radius and boundary element program, respectively. Analytical formula of the degenerate scale contains the Gamma function for the Gamma contour which can be derived from the Schwarz–Christoffel mapping. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale (mathematics) is imbedded in the left unitary vector for the influence matrices of weakly singular (U kernel) and strongly singular (T kernel) integral operators. On the other hand, we obtain the common right unitary vector corresponding to a rigid body mode (physics) in the influence matrices of strongly singular (T kernel) and hypersingular (M kernel) operators after using the singular value decomposition. To deal with the problem of non-unique solution, the constraint of boundary flux equilibrium instead of rigid body term, CHEEF and hypersingular BIE, is added to promote the rank of influence matrices to be full rank. Null field for the exterior domain and interior nonzero field are analytically derived and numerically verified for the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in the case of the degenerate scale. It is found that the contour of nonzero exterior field for the degenerate scale using the BEM matches well with that of Schwarz–Christoffel transformation. Both analytical and numerical results agree well in the demonstrative examples of right triangle, square, regular 5-gon and regular 6-gon. It is straightforward to extend to general regular N-gon case.
Tue, 15 Jan 2013 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15162013-01-15T00:00:00ZLinkage between unit logarithmic capacity in the theory of complex variables and the degenerate scale in BEM/BIEMMhttp://scholars.ntou.edu.tw/handle/123456789/16741Title: Linkage between unit logarithmic capacity in the theory of complex variables and the degenerate scale in BEM/BIEMM
Authors: Shyh-Rong Kuo; Jeng-Tzong Chen
Abstract: It is well known that BEM/BIEM result in degenerate scale for two-dimensional Laplace problem subjected to the Dirichlet boundary condition. In such a case, nontrivial boundary flux exist even the trivial boundary potential is given. It is proved that the unit logarithmic capacity in the Riemann conformal mapping with respect to the unit circle results in a null field for the interior domain. The logarithmic capacity is defined as the coefficient of the linear term in the Riemann conformal mapping. When the logarithmic capacity is equal to one, a trivial interior field can be obtained but exterior field is derived to be nonzero using ln function. Two mapping functions, Riemann conformal mapping for geometry and log function for physics, are both required. This matches well with the BEM result that an interior trivial field has nonzero boundary flux in case of degenerate scale. Regarding the ordinary scale, BIE results in a null field in the exterior domain owing to the Green's third identity. It is interesting to find that
ordinary and degenerate scales result in a null field in the exterior and interior domains, respectively. To demonstrate this finding, three cases of circle, ellipse and right triangle are demonstrated. Theoretical derivation using the Riemann conformal mapping with the unit logarithmic capacity and the degenerate scale in BEM/BIEM both indicate the null field in the interior domain analytically and numerically.
Description: ICOME2012/JASCOME2012, 12-14 December, Kyoto, Japan
Wed, 12 Dec 2012 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167412012-12-12T00:00:00ZA ROBUST TIME-INTEGRATION ALGORITHM FOR SOLVING NONLINEAR DYNAMIC PROBLEMS WITH LARGE ROTATIONS AND DISPLACEMENTShttp://scholars.ntou.edu.tw/handle/123456789/1525Title: A ROBUST TIME-INTEGRATION ALGORITHM FOR SOLVING NONLINEAR DYNAMIC PROBLEMS WITH LARGE ROTATIONS AND DISPLACEMENTS
Authors: Shyh-Rong Kuo; Yau, J. D.; Yang, Y. B.
Sat, 01 Dec 2012 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15252012-12-01T00:00:00ZBUCKLING EQUATIONS OF ORTHOTROPIC THIN PLATEShttp://scholars.ntou.edu.tw/handle/123456789/1524Title: BUCKLING EQUATIONS OF ORTHOTROPIC THIN PLATES
Authors: Shyh-Rong Kuo; Yau, J. D.
Sat, 01 Sep 2012 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15242012-09-01T00:00:00ZAnalytical derivation and numerical experiments of degenerate scale for an ellipse in BEMhttp://scholars.ntou.edu.tw/handle/123456789/1057Title: Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM
Authors: Jeng-Tzong Chen; Ying-Te Lee; Shyh-Rong Kuo; Yi-Wei Chen
Abstract: Degenerate scale of an ellipse is studied by using the dual boundary element method (BEM), degenerate kernel and unit logarithmic capacity. Degenerate scale stems from either the nonuniqueness of logarithmic kernel in the BIE or the conformal radius of unit logarithmic capacity in the complex variable. Numerical evidence of degenerate scale in BEM is given. Analytical formula for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that zero eigenvalue results in the degenerate scale. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale is imbedded in the left unitary vector for weakly singular and strongly singular integral operators. On the other hand, we obtain the common right unitary vector of a rigid body (physical) mode in the influence matrices of strongly singular and hypersingular operators after using the singular value decomposition. Null field for the exterior domain and interior nonzero fields are analytically derived and numerically verified in case of the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in case of the degenerate scale. No failure CHEEF point is confirmed in the nonzero exterior field to overcome the degenerate-scale problem. To deal with the nonuniqueness-solution problem, the constraint of boundary flux equilibrium instead of rigid body term, CHEEF and hypersingular BIE, is added to promote the rank of influence matrices to be full rank. Both analytical and numerical results agree well in the demonstrative example of an ellipse.
Sat, 01 Sep 2012 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/10572012-09-01T00:00:00ZA FAST AND ACCURATE STEP-BY-STEP SOLUTION PROCEDURE FOR DIRECT INTEGRATIONhttp://scholars.ntou.edu.tw/handle/123456789/1523Title: A FAST AND ACCURATE STEP-BY-STEP SOLUTION PROCEDURE FOR DIRECT INTEGRATION
Authors: Shyh-Rong Kuo; Yau, J. D.
Wed, 01 Jun 2011 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15232011-06-01T00:00:00ZA COMPLETE STABILITY THEORY FOR THE KIRCHHOFF THIN PLATE UNDER ALL KINDS OF ACTIONShttp://scholars.ntou.edu.tw/handle/123456789/1517Title: A COMPLETE STABILITY THEORY FOR THE KIRCHHOFF THIN PLATE UNDER ALL KINDS OF ACTIONS
Authors: Shyh-Rong Kuo; Chi, C. C.; Yang, Y. B.
Tue, 01 Sep 2009 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15172009-09-01T00:00:00ZA reliable three-node triangular plate element satisfying rigid body rule and incremental force equilibrium conditionhttp://scholars.ntou.edu.tw/handle/123456789/1518Title: A reliable three-node triangular plate element satisfying rigid body rule and incremental force equilibrium condition
Authors: Shyh-Rong Kuo; Chi, C. C.; Wei-Chung Yeih; Jiang-Ren Chang
Sat, 01 Jul 2006 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15182006-07-01T00:00:00ZAnalytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticityhttp://scholars.ntou.edu.tw/handle/123456789/1511Title: Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity
Authors: Jeng-Tzong Chen; Shyh-Rong Kuo; Lin, J. H.
Abstract: For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case.
Wed, 05 Jun 2002 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15112002-06-05T00:00:00ZAnalytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulantshttp://scholars.ntou.edu.tw/handle/123456789/1513Title: Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants
Authors: Jeng-Tzong Chen; J. H. Lin; Shyh-Rong Kuo; Y. P. Chiu
Abstract: For a potential problem, the boundary integral equation approach has been shown to yield a nonunique solution when the geometry is equal to a degenerate scale. In this paper, the degenerate scale problem in boundary element method (BEM) is analytically studied using the degenerate kernels and circulants. For the circular domain problem, the singular problem of the degenerate scale with radius one can be overcome by using the hypersingular formulation instead of the singular formulation. A simple example is shown to demonstrate the failure using the singular integral equations. To deal with the problem with a degenerate scale, a constant term is added to the fundamental solution to obtain the unique solution and another numerical example with an annular region is also considered.
Mon, 01 Oct 2001 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15132001-10-01T00:00:00ZDual Boundary Integral Equations for Helmholtz Equation at a Corner Using Contour Approach Around Singularityhttp://scholars.ntou.edu.tw/handle/123456789/16573Title: Dual Boundary Integral Equations for Helmholtz Equation at a Corner Using Contour Approach Around Singularity
Authors: I-Lin Chen; Ming-Te Liang; Shyh-Rong Kuo; Jeng-Tzong Chen
Abstract: A dual integral formulation for the Helmholtz equation problem at a corner is derived by means of the contour approach around the singularity. It is discovered that employing the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, respectively, which differs from the limiting process from an interior point to a boundary point where the jump term is descended from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials as compared to the conventional definition. The free terms of the six kernel functions in the dual integral equations for the Helmholtz equation at a corner have been examined. The kernel functions of the Helmholtz equation are quite different from those of the Laplace equation while the free terms of the Helmholtz equation are the same as those of the Laplace equation. It is worth to point out that the Laplace equation is a special case of the Helmholtz equation when the wave number approaches zero.本文探討經由推到邊界及繞道奇異點的方法導出在角點荷姆茲方程的對偶積分表示式。結果發現，利用環繞邊界法它的跳躍項是由L及M核函數經積分各貢獻一半，這與經由極限過程所得自由項完全由L核函數貢獻有所不同。在超強奇異積分方程中阿達馬主値的觀念在此從雙層勢能的法向微分推廣到切向微分以便於與傳統的定義對照。同時對於荷姆茲方程對偶邊界積分方程式中的六個核函數在角點的自由項也予以檢驗。荷姆茲方程的核函數與拉普拉斯方程的核函數完全不同，但是它們的自由項卻相同。值得一提的是拉普拉斯方程僅爲荷姆茲方程當波數k趨近於零時的一個特例而已。
Fri, 01 Jun 2001 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/165732001-06-01T00:00:00ZOn the free terms of the dual BEM for the two and three-dimensional Laplace problemshttp://scholars.ntou.edu.tw/handle/123456789/16572Title: On the free terms of the dual BEM for the two and three-dimensional Laplace problems
Authors: Jeng-Tzong Chen; Shyh-Rong Kuo; Wei-Chih Chen; Li-Wei Liu
Abstract: A dual integral formulation for the Laplace problem with a smooth boundary is derived by using the contour approach surrounding the singularity. It is found that using the contour approach the jump terms come half and half from the free terms in the L and Mkernel integrations for the two-dimensional case, which is different from the limiting process by approaching an interior point to a boundary point where the jump terms come totally from the L kernel only. The definition of the Hadamard principal value for hypersingular integral at the collocation point of a smooth boundary is extended to a generalized sense for both the tangent and normal derivatives of double-layer potentials in comparison with the conventional definition. For the three dimensional case, the jump terms come one-third and two-thirds from the free terms of L and M kernels, respectively.本文探討Laplace方程式的對偶邊界積分方程之自由項。針對圍繞在平滑邊界上奇異點周圍的半圓(二維)或半球面(三維)積分可導得自由項。在二維問題中，自由項的來源有二，一半來自Ｌ核函數，另一半來自Ｍ核函數。意即自由項分別由Ｌ核函數與Ｍ核函數各貢獻一半。這個過程雖不同于極限方法從域內點逼近到邊界點中，跳躍項完全來自Ｌ核函數所貢獻，但是最終的結果仍然是相同的。在超奇異積分方程中，阿馬主值的觀念在此從雙層勢能的法向微分推廣到切向微分。有趣的是，在三維問題中，我們發現到自由項分別來自Ｌ核函數的三分之一貢獻與Ｍ核函數的三分之二貢獻。
Thu, 01 Jun 2000 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/165722000-06-01T00:00:00Z同心圓環二維 Helmholtz 特徵方程問題真假根探討http://scholars.ntou.edu.tw/handle/123456789/16547Title: 同心圓環二維 Helmholtz 特徵方程問題真假根探討
Authors: 郭世榮; 陳正宗; 劉孟龍; 全湘偉
Abstract: 本文針對同心圓環之 Helmholtz 特徵值問題，採用實部對偶邊界元素法求得真假特徵值。以離散化系統由邊界元素法中影響係數矩陣循環的特性解析一同心圓環空間聲場真假特徵值發生的機制；在數值解方面，則發展實部對偶邊界元素法程式並配合奇異值分解法萃取真根與過濾假根。最後，比較兩者與有限元素法的結果均相當符合。In this paper, the real-part dual BEM was employed to solve the Helmholtz eigenproblems of annular domain. The circulant and degenerate kernel were used to study the true and spurious eigensolutions in discrete system for simulating the continuous system. The numerical program of real-part dual BEM was also developed to distinguish the true and spurious eigensolutions by using the singular value decomposition technique. Good agreement can be made.
Sat, 01 Jan 2000 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/165472000-01-01T00:00:00ZA nonsingular integral formulation for the Helmholtz eigenproblems of a circular domainhttp://scholars.ntou.edu.tw/handle/123456789/2391Title: A nonsingular integral formulation for the Helmholtz eigenproblems of a circular domain
Authors: Jeng-Tzong Chen; Shyh-Rong Kuo; Kwe-Hoo Chen
Abstract: A nonsingular integral formulation for the Helmholtz eigenproblem is developed in this paper. This novel method contains only imaginary‐part kernels instead of complex‐part kernels in the complex‐valued BEM. Based on the imaginary‐part formulation without singular source, no singular or hypersingular integrals are present. Although this formulation avoids the computation of singular and hypersingular integrals, this approach results in spurious eigensolutions. After comparing the results from the dual formulation, the true and spurious solutions can be separated. An analytical example for the eigensolutions of a two‐dimensional circular domain is studied. The continuous system can be transformed to a discrete system with circulants. Based on the spectral properties of circulants, the true and spurious solutions for the eigenvalues, boundary modes, interior modes and multiplicities are all examined. The possible failure of Hutchinson's sorting technique of looking at modal shapes is also discussed.本文針對Helmholtz特徵值問題發展一非奇異積分推導解法。此法僅採用負數核函數中的虛部。基於此不含奇異元的輔助系統，奇異與超奇異積分將不會出現。然而，卻會導致假的特徵解。配合對偶架構後，真假特徵解可被分辨出來。利用循環對稱矩陣可解析的特性，一個圓形範例，將可解析推導並說明真假特徵解的發生機制。同時，包括真假特徵值、真假特徵邊界狀態、真假內域模態與真假重根數均有所探討。另Hutchinson由模態鑑別真假解的技巧可行性，在本文亦加以討論。
Mon, 01 Nov 1999 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/23911999-11-01T00:00:00Z同心圓環二維Helmholtz方程特徵值問題 真假根探討http://scholars.ntou.edu.tw/handle/123456789/16778Title: 同心圓環二維Helmholtz方程特徵值問題 真假根探討
Authors: 郭世榮; 陳正宗; 劉孟龍
Abstract: 本文針對同心圓環之Helmholtz特徵值問題利用實部對偶邊界元素法來求得真假特徵值。在解析解方面，以離散系統模擬連續系統，並利用邊界元素法中影響係數矩陣循環的特性來求得一同心圓環形聲場空間真假特徵值發生的機制；在數值解方面，則發展實部對偶邊界元素法程式並配合奇異值分解法分辨真根與過濾假根。最後，比較兩者的結果均相當符合。In this paper, the real-part dual BEM was employed to solve the Helmholtz eigenproblems of annular domain. The circulant was used to study the true and spurious eigensolutions in discrete system for simulating the continuous system. The numerical program of real-part dual BEM was also developed to distinguish the true and spurious eigensolutions by using the singular value decomposition technique. Good agreement can be made.
Fri, 01 Jan 1999 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167781999-01-01T00:00:00ZHouseholder正交矩陣的新觀點http://scholars.ntou.edu.tw/handle/123456789/16777Title: Householder正交矩陣的新觀點
Authors: 陳正宗; 郭世榮; 李慶鋒
Abstract: 文獻中有許多的方法可建立正交矩陣，Householder利用鏡射導得正交對稱矩陣。本文提出奇數階與偶數階Householder矩陣，均可分別由eAt與eiBt導得，其中A為反對稱實矩陣，B為對稱實矩陣，t為某特定時間。本文並分別舉一階到五階的例子，說明Householder矩陣均可由本文新觀點導得。It is well known that many approaches can obtain the orthogonal matrix. Householder employed the mirror mapping to derive the symmetric orthogonal matrix. The Householder matrices of odd and even orders are found to have the matrix forms of eAt and eiBt, respectively, where A is an anti-symmetric matrix, B is a symmetric matrix and t is a specified time. One by one to five by five Householder matrices are constructed by the proposed formulation.
Fri, 01 Jan 1999 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167771999-01-01T00:00:00Z解析及數值實驗探討積分方程外域問題CHIEF方法http://scholars.ntou.edu.tw/handle/123456789/16776Title: 解析及數值實驗探討積分方程外域問題CHIEF方法
Authors: 陳義麟; 朱皖山; 陳正宗; 梁明德; 郭世榮
Abstract: 積分方程用於解外域聲場Helmholtz問題由來已久。如果僅使用奇異積分方程（ 式）會產生虛擬頻率的非物理現象。CHIEF法因易於使用而受到歡迎，且可以克服外域虛擬頻率的問題；但是如果所選取內域點配置不當則會導致失效。本文主要目的係以CHIEF方法為基礎，以圓形循環矩陣（circulant）的特性及可分離核函數（degenerate kernel function）配合SVD的技巧，用解析的方法詳細探討虛擬頻率發生的機制及過濾虛擬頻率時可能的失敗的位置，並且以圓柱外域問題為例，數值實驗結果與解析的結果亦非常吻合。
Fri, 01 Jan 1999 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/167761999-01-01T00:00:00ZFrequency-independent infinite elements for analysing semi-infinite problemshttp://scholars.ntou.edu.tw/handle/123456789/1528Title: Frequency-independent infinite elements for analysing semi-infinite problems
Authors: Yang, Y. B.; Shyh-Rong Kuo; Hung, H. H.
Tue, 01 Oct 1996 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15281996-10-01T00:00:00ZTRACING POSTBUCKLING PATHS OF STRUCTURES CONTAINING MULTI-LOOPShttp://scholars.ntou.edu.tw/handle/123456789/1522Title: TRACING POSTBUCKLING PATHS OF STRUCTURES CONTAINING MULTI-LOOPS
Authors: Shyh-Rong Kuo; Yang, Y. B.
Fri, 01 Dec 1995 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/15221995-12-01T00:00:00Z