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Null-field boundary integral formulation for boundary value problems containing circular and elliptical holes and/or inclusions
A systematic approach was proposed to deal with boundary value problems containing circular and elliptical holes and/or inclusions by using null-field boundary integral equations. The mathematical tools, degenerate kernels and Fourier series, are utilized. The kernel function was expanded to degenerate form and the boundary density was expressed in terms of Fourier series. Five advantages were achieved, (1) well-posed model, (2) singularity free, (3) elimination of boundary-layer effect, (4) exponential convergence instead of algebraic convergence in the BEM (5) free of mesh generation. This systematic approach was successfully applied to solve the Laplace, Helmhotz, biHelmholtz and biharmonic boundary value problems. Applications to anti-plane shear, torsion, anti-plane piezoelasticity, screw dislocation, acoustics, water wave, free vibration, Stokes flow, SH wave, waveguide and plate problems were published in the AOR, OE, ISOPE, IJSS, EABE, CMAME, JSV, CMES, CMC, Comp. Mech., NMPDE, IJNME, JCA, SDEE, MRC, JoM, ASME-JAM, Acta Mechanica, OE, Meccanica, APAN, AMC, Computing, BVP, EJMB/F, Composites-B, IEEE-TMTT and GJI journals. Besides, the PI won the best paper award for the JoM journal for the bending beam problem containing circular holes. The PIs former and current Ph.D. students, Y. T. Lee and J W Lee, both won the rank-one award for the student best paper competition in the Annual Mechanics Conference (2008 and 2009) for the two spherical radiators and eigenproblems by using the null-field BIEM. Another current Ph.D. student, Mr. J W Lee, also won the HIWIN award of master thesis in mechanical engineering in 2010. The PI was invited to contribute one chapter (Chapter 3) in the Springer-Verlag 2009 book Recent advances in BEM for the ceremony of Prof. Beskos 60th birthday. The PI was invited to give a plenary lecture for the ICIP 2010 conference in Hong Kong. This talk overviewed the nonuniqueness and ill-posedness when utilizing the null-field BIEM to solve the water wave problems. The PI won the National MOE Academic Award and the ICACM Fellows Award in 2011.
Mathematical education
(a) A unified point of view for solving eccentric Laplace problems
The bipolar coordinates and conformal mapping for solving eccentric-domain problems were revisited. Five models, Carrier and Pearson, Timoshenko and Goodier, Muskhelishvili, Ling and Lebedev et al. to describe the eccentric-domain Laplace problem were unified by using the conformal mapping. The results have appeared in the J. of Computer Applications in Engineering Education 2009 (SCI).
(b) Wave animation using Mathematica
One-dimensional wave phenomena were animated. Several methods, including the DAlembert solution, the diamond rule, the image method, the Laplace transform and the convolution integral, were employed in the Mathematica animation. Several examples, including an infinite string with spring, mass and damper as well as a semi-infinite string, two-media string, string and beam subject to support motions, were demonstrated to show the validity of the present formulation. Parameter study of impedance ratio and mass, spring and dashpot was also examined to see the transmission and reflection coefficient. The results were published in the J. of Computer Applications in Engineering Education 2009 (SCI).
(d) Education in engineering mathematics
The PI has published two papers in Math. Media which may be useful for students in learning mathematics.
Degenerate-boundary problem using the dual BEM
The dual BEM for the degenerate-boundary problem was successfully applied to the Helmholtz equation (membrane with stringers) and the modified Helmholtz equation (water wave with breakwaters). The mathematical structure of dual BEM was constructed by using the Calderon projector and pseudo-differential operator. In addition, the crack growth in solid propellant grain and in V-band structure of missile was solved by using the dual BEM. Based on our dual formulation, WIT (Wessex Institute of Technology) has developed a commercial code BEASY-CRACK for industry use. The company claimed that more than 50 countries used the program of dual BEM in academic universities and industry. The PI has delivered a keynote lecture on this topic in the 4th World Congress on Computational Mechanics in Buenos Aires (WCCM4). Later, the application to electrostatic problems was published by IEEE Journal of Computing in Science and Technology. Applications to electronic devices and MEMS were published in the six papers which appeared in Journals of IEEE, J. of MEMS and JMM, ... etc. In 2008, Prof. Brebbia, Editor in Chief of EABE journal invited the PI to join the board member. Besides, Prof. Atluri invited the PI to join the editorial board of CMES journal. In 2009, Prof. Brebbia, Editor in Chief of EABE journal invited the PI to be one of the Editors. In addition, the PI gave a plenary lecture on dual BEM in ICOME 2009 conference. In addition, a chapter of Dual BEM since 1986 in the Nova publisher with the title of book Advances in Computational Mechanics has appeared in 2011. Also, the PI delivered an invited lecture in ISCM III in Taipei, 2011.
Degenerate scale problem in BIEM/BEM
The mechanism why the degenerate scale occurs in BIEM/BEM was studied analytically and numerically by using the degenerate kernels, Fourier series and circulants, respectively. The Laplace, biharmonic and Navier equations were considered. In addition, simply-connected and multiply-connected problems were solved successfully. Several regularization techniques were adopted to avoid the problem of nonunique solution. Two invited lecture were delivered in the domestic BEM conference (NCKU) and Taiwan-Japan Conference (NTU) in Numerical Analysis for the ellipse and polygon in 2011, respectively.
Large-scale problem using DBEM in conjunction with the fast multipole method
The large-scale problem for exterior acoustics was solved efficiently by employing the dual integral formulation in conjunction with the fast multipole method to accelerate the construction of influence matrix. The singular and hypersingular integrals are transformed into the summability of divergent series and regular integrals. Not only CPU time but also storage of memory were reduced in comparison with BEM without employing FMM concept. An extension to water wave was done in the IJNMF journal, 2009.
Spurious eigenvalues in simply-connected problems using BIEM/BEM
It was found that spurious eigenvalues of interior acoustics appear in the real-part, imaginary-part BEMs and MRM since information is lost. The regularization techniques including SVD updating term and updating document, domain partition, dual method, complete MRM and CHEEF method, were successfully employed to deal with the spurious-eigenvalue problem. We extended the 2-D acoustics (membrane) to plate problem and employed the SVD updating technique, Burton & Miller method, CHEEF method to filter out the spurious eigenvalues. By choosing the valid CHEEF points, we can suppress the occurrence of the spurious eigenvalues for the clamped plate in the real-part or imaginary-part BEM. We also demonstrated the existence of spurious and true eigenvalues of a rod and a circular cavity and extended it to the mixed-type boundary condition. Then, we find that the spurious eigenvalues are the same with true eigenvalues of the associated problems. It was also observed that the spurious eigenvalues are dependent on the method while the true ones are concerned with the type of boundary condition. A series of SCI papers have been published by our group. Recently, we also extended successfully to prove the true and spurious eigenvalues for an elliptical membrane through the use of the Mathieu function. The related work has been published in Meccanica 2012.
Spurious eigenvalues in multiply-connected problems using BIEM/BEM
For the 2-D multiply-connected acoustic problem, spurious eigenvalues occur even though the complex-valued BEM is employed to solve the eigenproblem. We utilized the Burton and Miller approach to suppress the appearance of spurious eigenvalues. In addition, CHIEF method was also utilized to deal with the problem. Both the continuous and discrete systems were considered. Two papers were published in the Proceeding of Royal Society London Series A. For the multiply-connected acoustic problem, we used the complex-valued BEM to derive the eigensolution in continuous and discrete systems. The occurrence of spurious eigenequation only depends on the formulation instead of the specified boundary condition, while the true eigenequation is independent of the formulation and is relevant to the specified boundary condition. In addition, we used the SVD updating technique, the Burton & Miller method and the CHIEF method to suppress the occurrence of the spurious eigenvalues successfully. The extension to annular plate problems was published in IJNME journal. The PIs current Ph.D. student, J. W. Lee, won the rank-one award for the student best paper competition in the Annual Mechanics Conference (2009) for the membrane vibration by using the null-field BIEM. The extension work of eigenproblems of the confocal elliptical membrane has appeared in IJSS, 2011.
Equivalence of the Trefftz method and MFS
The Trefftz method and MFS belong to kinds of meshless methods, their mathematical relation has not been studied to our knowledge. Based on the degenerate kernels and theory of circulants, the mathematical equivalence between the Trefftz method and MFS was derived. Later, Prof. Schabach in Gttingen University also found the equivalence and discussed with us. However, their finding was later than ours. Prof. Leitao in Portugal also noticed our finding. The extension to prove the equivalence of annular Greens functions derived by the Trefftz method and MFS (image method) has appeared in the EABE journal. Besides, the Lames problem and SCF for a hole were revisited by using the Trefttz method now and appeared in EABE, 2009. This problem was also solved by using the null-field BIE and has appeared in JOM.
Fictitious frequency in exterior acoustics
The occurring mechanism why fictitious frequency appears in the BEM for exterior Helmholtz problems was understood analytically and numerically by using the degenerate kernel, the Fourier series and circulants. Both the radiation and scattering problems were solved free of irregular frequencies by using regularization techniques. We have proposed a new concept of modal participation factor for the numerical instability and applied it to the acoustic problem in the MRC journal. In addition, the fictitious frequency embedded in the singular or hypersingular integral equations has been discussed in MRC. We also studied the Helmholtz equation with the mixed-type boundary condition for a semi-infinite rod and a circular radiator. Good agreement was made. Although the irregular values occur at the same positions for BEM as they appear in the problems with the Dirichlet, Neumann and mixed-type boundary conditions in BEM, they do not appear obviously by using our semi-analytical approach. Fictitious frequency is studied in a unified manner by using the SVD and a EABE paper was published in 2009.
Meshless method
Based on the imaginary-part kernel in the dual formulation, we have developed a new meshless method for determining 2-D and 3-D acoustic modes. The NDIF (Nondimensional Dynamic Influence Function) method developed by Kang et al. is only our special case. The extension to vibration of elliptical membrane vibration was aaccepted by ASME JVA Journal and Meccanica in 2011. The PIs master student, W. C. Lee, has won the student competition award on this topic in the 34th National Conference of Theoretical and Applied Mechanics.
Modified MFS
Instead of putting the sources outside the domain in the conventional MFS, the modified MFS can successfully put the singularities on the real boundary. Successful applications to the Laplace, Helmholtz and Cauchy problems were achieved and published in the CMES, EABE, IJNME and Comp. Mech. journals.
A unified point of view for rank deficiency in BIEM/BEM
Degenerate boundary, degenerate scale, corner, spurious eigenvalue and fictitious frequency all stem from the rank-deficiency problem in deriving the influence matrix. A unified formulation for treating the nonuniqueness problem was proposed by using regularization techniques. Rank deficiency problems can be understood in the SVD structure for the four influence matrices in the dual BEM. In our works, we have examined the null field and nonzero field in depth. Spurious data are embedded in the left unitary vector, while true information can be extracted from the right unitary vector. The PI delivered a plenary lecture in CFD 15, 2008. The PIs master student, Y Fan, has won the student competition award on this topic in the 35th National Conference of Theoretical and Applied Mechanics (2011). An invited lecture by ISCM III in Taipei was delivered in December, 2011. The related works were published on EABE 2012. In 2012, the PI was invited to give a keynote lecture and a plenary lecture for ACMFMS 2012 and ICOME 2012, respectively.
Numerical study on degenerate scale and logarithmic capacity
The relation of degenerate scale and unit logarithmic capacity is study numerically by using the boundary element method and complex variables. 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 and analytical formula for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity. 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. The related works were published on EABE 2012 for elliptical geometry and AMC 2013 for regular N-gon domains. Besides, in this project, we will theoretically construct the relation of degenerate scale and unit logarithmic capacity by using the complex-variable boundary integral equation.
Inverse problems
Two dimensional Cauchy problems were solved by using desingularized MFS and appeared in Comp. Mech., 2008.
Free vibration of plates with holes and/or inclusions and flexural wave of the infinite plate containing the holes and/or inclusions
In the recent years, the PI and Dr. W M Lee have successfully applied the multipole Trefftz method, BEM, null-field BIEM to study the vibration of plates and flexural waves of an infinite plate containing holes and/or inclusions. Several papers in CMES, JSV, IJSS, JoM, EABE, CM and ASME-JAM were published.
Trapped modes
We have applied the null-filed BIEM/BEM to deal with water-wave problems containing multiple cylinders subject to incident wave. It is interesting that the near-trapped mode (physically realizable) and fictitious wave number (mathematically realizable) were both observed. The results were published in AOR, 2009. Later, we extended to water-wave problems containing multiple porous cylinders. The results were published in OE. Related works on water wave problems were published in ISOPE 2011, AOR 2009, OE 2011, EJMB/F 2012, EABE 2012 and APAN 2012.
Antiplane problem
We extended the successful experience of solving an infinite medium containing elliptical holes and/or inclusions subject to remote shears. Since there is Mobius transform in complex-variable analysis for the problems containing two circular holes/inclusions, many literatures were found on studying this problem. However, due to lack relating conformal mapping function, very a few literatures were discussed on this issue for two elliptical inclusions. To the PIs best knowledge, only Noda & Matsuo (IJF 2000) and Lee & Kim (Composites-B 2012) have investigated this problem by using the singular integral equation and the volume integral equation, respectively. We have used null-field integral equation to study the problem and our results were published in Composites-B, 2013.
SH-wave scattering problem by a semi-elliptical hill
By taking free body for the half-plane with a half-elliptical arc, it is designed to be imbedded in an infinite domain with an elliptical boundary. A linear algebraic equation was constructed easily by six constraint equations through two subdomains and four boundary conditions. The focusing effect was also found in a semi-elliptical hill impinging by SH-wave. Besides, results of the hybrid method (FEM) were also provided for comparisons. For the scattering problems of SH wave by successive semi-elliptical (circular) hills or canyons, our proposed approach can be straightforward applied to solve by the using proposed approach.
An elliptical waveguide with a nonconfocal suspended strip
A nonconfocal suspended strip in an elliptical waveguide was analyzed by using the proposed approach. A closed-form fundamental solution was expressed in terms of the degenerate kernel containing the Mathieu and modified Mathieu functions in the elliptical coordinates. The hypersingular integral formulation was also used for overcoming the problem of rank deficiency due to the degenerate boundary. After comparing with the numerical data using FEM, we found wrong numerical results in the literature. Besides, we also found that there are some roots with multiplicity 2. The work has been published in IEEE TMTT 2013.
Following the success of the SVD mathematical structure for the rank-deficient problems, we will extend to link the relation between the degenerate scale and logarithmic capacity. The main focus of this project is shown in Table A.
During the past years, more than 1061 papers and books have cited our research articles as shown in the Table B. Two papers, ASME and ASCE, were cited more than 161 and 123 times in WOS, respectively. More than 178 SCI papers were published during 1986-2012 by Prof. J.T. Chen (PI) as shown in Tables C and D (growth curve). The principal investigator is now the member of editorial board for eight international journals and four domestic journals as shown in Table E. Table F shows the ranking of impact factor for each journal. Table G shows the number of citing of our top 20 papers. During the ten years, several lectures including plenary lectures, keynote lectures and invited lectures were delivered was shown in Table H. Also, the PI has reviewed ASME books and SCI papers for more than 83 different journals as shown in Table I.
For convenience of reviewers or readers owing to page limit, our website http://ind.ntou.edu.tw/~msvlab was constructed for your reference and more than 160000 entries were recorded since 1999. It contains all the PDF files of our 178 SCI papers.
Reviewers and readers can find the following tables from the home page of our web site HYPERLINK "http://ind.ntou.edu.tw/~msvlab" http://ind.ntou.edu.tw/~msvlab. Your comments on the web site are welcome.
Table A. The focus of the project
EMBED Equation.DSMT4
Table A: The focus of the project
Table B: Papers (1059) citing our NTOU/MSV articles
Table C: List of SCI papers (178) by NTOU/MSV group since 1988
Table D: Growth of number of SCI papers by NTOU/MSV group
Table E: Editorship of PI (13 board members of Journals)
Table F: Ranking of impact factor for our SCI papers
Table G: Top citing 20 papers by NTOU/MSV group
Table H: Plenary lectures, keynote lectures and invited lectures by PI.
Table I: Reviewed journals (76) and books (2) by PI.
(filename: NSC2011-five year.doc revise by Lee J W Dec.26, 2011.)
Research of NTOU/MSV Group since 1994
Degenerate scale
(2013)
The present project will focus on the degenerate scale
Annular case
Circle
Ellipse
Confocal ellipse
(2012-2015)
(NSC 101-2221-E-019-050-MY3)
SVD mathematical structure, addition theorem and degenerate kernel for rank-deficient problems
Past successful experiences of solving rank-deficient problems on the BEM/BIEM
(2003~2012)
Conformal mapping
BEM
Unit logarithmic capacity
Extension
Two-dimensional Laplace and Navier equations
Relation linking
Regular N-gon
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