Analysis of circular torsion bar with circular holes

Analysis of circular torsion bar with circular holes using null-field approach 研究生：沈文成指導教授：陳正宗教授時間： 15: 10 ~ 15: 25 地點：Room 47450 -3 九十四年電子計算機於土木水利程應用研討會 1

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 2

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 3

Motivation and literature review BEM/BIEM Singular and hypersingular Bump contour Improper integral Limit process Regular Fictitious BEM Null-field approach CPV and HPV Fictitiou s boundar y Collocation point Ill-posed 4

Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV Advantages of degenerate kernel value 1. No principal 2. Wellposed 5

Engineering problem with arbitrary geometries Straight boundary (Legendre polynomial) (Fourier series) Degenerate boundary (Chebyshev polynomial) Circular boundary Elliptic boundary (Mathieu function) 6

Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics Bipolar coordinate Special solution Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics Limited to doubly connected domain 7

Fourier series approximation n n Ling (1943) - torsion of a circular tube Caulk et al. (1983) - steady heat conduction with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Mogilevskaya et al. (2002) - elasticity problems with circular boundaries 8

Contribution and goal However, they didn’t employ the nullfield integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. n To develop a systematic approach for solving Laplace problems with multiple holes is our goal. n 9

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 10

Boundary integral equation and null-field integral equation Interior case Exterior case Null-field integral equation 11

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 12

Expansions of fundamental solution and boundary density n Degenerate kernel - fundamental solution n Fourier series expansions - boundary density 13

Separable form of fundamental solution (1 D) Separable property continuous discontinuo us 14

Separable form of fundamental solution (2 D) 15

Boundary density discretization Fourier series Ex. constant element Present method Conventional BEM 16

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 17

Adaptive observer system collocation point 18

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 19

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 20

Linear algebraic equation wher e Index of collocation circle Index of routing circle Column vector of Fourier coefficients (Nth routing circle) 21

Explicit form of each submatrix [Upk] and vector {tk} Truncated terms of Fourier series Number of collocation points Fourier coefficients 22

Flowchart of present method Potential gradient Degenerate kernel Analytical Fourier series Adaptive observer system Vector decompositio n Potential of domain point Collocation point and matching B. C. Linear algebraic Fourier coefficients equation Numerical 23

Comparisons of conventional BEM and present method Boundary density Auxiliary Formulatio Observ n er discretizatio system n system Constant, Boundary Fixed Convention Linear, Fundament integral observe al al r Qurdrature equation BEM … solution system Fourier Null-field Adaptiv e Present series Degenerate integral method expansion kernel equation observe r Singularit y CPV, RPV and HPV No principal value 24

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 25

Torsion bar with circular holes removed Torque The warping function Boundary condition on where 26

Torsion problems Torque Eccentric case Two holes (N=2) 27

Torsion problems Torque Three holes (N=3) Four holes (N=4) 28

Torsional rigidity (Eccentric case) 0. 20 0. 40 0. 60 0. 80 0. 98 Present method (M=10) 0. 97872 0. 95137 0. 90312 0. 82473 0. 76168 0. 66705 Exact solution BIE formulation 0. 97872 0. 95137 0. 90312 0. 82473 0. 76168 0. 66555 0. 97872 0. 95137 0. 90316 0. 82497 0. 76252 0. 66732 29

Axial displacement with two circular holes Dashed line: exact solution Solid line: first-order solution Caulk’s data (1983) ASME Journal of Applied Mechanics Present method (M=10) 30

Axial displacement with three circular holes Dashed line: exact solution Solid line: first-order solution Caulk’s data (1983) ASME Journal of Applied Mechanics Present method (M=10) 31

Axial displacement with four circular holes Dashed line: exact solution Solid line: first-order solution Caulk’s data (1983) ASME Journal of Applied Mechanics Present method (M=10) 32

Torsional rigidity (N=2, 3, 4) Number of holes BIE formulation First-order solution 2 Present method (M=10) 0. 8657 0. 8661 3 0. 8214 0. 8224 4 0. 7893 0. 7934 33

Outlines Motivation and literature review n Mathematical formulation n Expansions of fundamental solution and boundary density Adaptive observer system Linear algebraic equation Numerical examples n Conclusions n 34

Conclusions n n A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve torsion problems with circular boundaries. Numerical results agree well with available exact solutions and Caulk’s data for only few terms of Fourier series. 35

Conclusions n Engineering problems with circular boundaries which satisfy the Laplace equation can be solved by using the proposed approach in a more efficient and accurate manner. 36