報 告 者：蕭嘉俊 指導教授：陳正宗 博士 呂學育 博士 日 期：2005/6/16 地 點：河工二館307 A semi-analytical approach for Stokes flow and plate problems with circular boundaries 報 告 者：蕭嘉俊 指導教授：陳正宗 博士 呂學育 博士 日 期：2005/6/16 地 點：河工二館307

Outlines Introduction Direct boundary integral equation method Indirect boundary integral equation method Numerical examples Conclusions

Outlines Introduction Direct boundary integral equation method Indirect boundary integral equation method Numerical examples Conclusions

Engineering problems with arbitrary boundaries Degenerate boundary (Chebyshev polynomials) Elliptic boundary (Mathieu function) Circular boundary (Fourier series) Straight boundary (Legendre polynomials) Degenerate boundary Circular boundary

Fictitious boundary method Motivation BEM/BIEM Singular Improper integral Desingular (Regular) Direct Indirect (Interior) Null-field approach Contour Fictitious boundary method Limiting process : collocation point Fictitious boundary

Fictitious boundary method Motivation BEM/BIEM Singular Improper integral Desingular (Regular) Direct Indirect (Exterior) Contour Limiting process Null-field Fictitious boundary method : collocation point Fictitious boundary Degenerate kernel Field point Present approach ill-posed CPV & HPV

Literature review Torsion bar with circular holes Laplace problems Steady state heat conduction of tube (Shen) Electromagnetic wave Engineering problems Helmholtz problems Membrane vibration Water wave and Acoustic problems (Chen) Plane elasticity：Airy stress function Biharmonic problems Solid mechanics：plate problem Fluid mechanics：Stokes flow (Hsiao)

2. Solid mechanics (Plate problem): Literature review Plane elasticity: Jeffery (1921), Howland and Knight (1939), Green (1940) and Ling (1948) 2. Solid mechanics (Plate problem): Bird and Steele (1991) 3. Viscous flow (Stokes Flow): Kamal (1966), DiPrima and Stuart (1972), Mills (1977) and Ingham and Kelmanson (1984)

Purpose A semi-analytical approach in conjunction with Fourier series, degenerate kernels and adaptive observer system is extended to biharmonic problems. Advantages： 1. Mesh free. 2. Accurate. 3. Free of CPV and HPV.

Outlines Introduction Direct boundary integral equation method Indirect boundary integral equation method Numerical examples Conclusions Further research

Problem statement Governing equation: Essential boundary condition: :lateral displacement, :slope Natural boundary condition: : moment, : shear force

Direct boundary integral equations BIEs are derived from the Rayleigh-Green identity： BIE for the domain point Null-field integral equation Interior problem

Boundary integral equation for the domain point Displacement Slope Displacement Moment Displacement Shear force : Poisson ratio

Null-field integral equation Displacement Slope Displacement Moment Displacement Shear force : Poisson ratio

Relation among the kernels Continuous (Separable form of degenerate kernel) is the fundamental solution, which satisfies

Degenerate kernels x x r s O

Fourier series The boundary densities are expanded in terms of Fourier series: M: truncating terms of Fourier series

Adaptive observer system : Collocation point : Radius of the jth circle : Origin of the jth circle : Boundary of the jth circle

Vector decomposition for normal derivative True normal direction Tangential direction Radial direction : normal derivative : tangential derivative

Linear algebraic system Null-field integral equations for and formulations H: number of circular boundaries Collocation circle index Routing circle index

Analytical Numerical Flowchart of the present method Potential Degenerate kernels Fourier series Analytical Collocation method Matching B.C. Adaptive observer system Numerical Linear algebraic system Potential Fourier coefficients BIE for domain point

Stokes flow problems (Eccentric case) Governing equation: Essential boundary condition: on on (Stationary) : stream function : normal derivative of stream function

Linear algebraic system Unknown Given Unknown constant

Constraint equation Vorticity: Constraint:

Trapezoid integral Inner circle Outer circle Vector decomposition Numerical Analytical Series sum Trapezoid integral

Linear algebraic augmented system Unknown

Outlines Introduction Direct boundary integral equation method Indirect boundary integral equation method Numerical examples Conclusions Further research

Indirect boundary integral equation Indirect boundary integral equation is originated from the physical concept of superposition : Vorticity : single layer fictitious densities : double layer fictitious densities

Stokes flow problems (Eccentric case) Governing equation: Essential boundary condition: on on (Stationary) : stream function : normal derivative of stream function

Linear algebraic system Unknown Given Unknown constant :Collocation point

Constraint equation Vorticity: Constraint:

Trapezoid integral Inner circle Outer circle Vector decomposition Numerical Analytical Series sum Trapezoid integral

Linear algebraic augmented system Unknown

Outlines Introduction Direct boundary integral equation method Indirect boundary integral equation method Numerical examples Conclusions

Numerical examples Direct BIEM Indirect BIEM Plate problems 1. Annular plate 2. Circular plate with three circular holes Stokes flow problems 1. Doubly-connected case 1. Simply-connected case 2. Doubly-connected case

Numerical examples Direct BIEM Indirect BIEM Plate problems 1. Annular plate 2. Circular plate with three circular holes Stokes flow problems 1. Doubly-connected case 1. Simply-connected case 2. Doubly-connected case

Plate problems (Case 1) Exact solution： Geometric data: and Essential boundary conditions: and on and on Exact solution：

Contour plot of displacement (No. of nodes=1,920, No. of elements=3,600) Exact solution FEM mesh Present method (M=10) FEM (ABAQUS)

Boundary densities for outer circle Exact solution: Exact solution:

Boundary densities for inner circle Exact solution: Exact solution:

Plate problems (Case 2) Geometric data: Essential boundary conditions: and on and on and on and on (Bird & Steele, 1991)

Contour plot of displacement Present method (N=21) Present method (N=41) Present method (N=61) Present method (N=81)

Contour plot of displacement Present method (N=101) Bird and Steele (1991) (No. of nodes=3,462, No. of elements=6,606) FEM mesh FEM (ABAQUS)

Parseval sum for convergence

Parseval sum for convergence

Stokes flow problems (Case 1) Boundary conditions: Exact solution: (Mills, 1977) where

Contour plot of Streamline Exact solution (Mills, 1977) Analytical solution, P=100 (Wu, 2004) Present method (N=161) (Null-field + collocation) Exact solution: (Mills, 1977) (Closed-form solution) Analytical solution: (Wu, 2004) (Null-field equation solution) (Trefftz solution)

Stokes flow problems (Case 2) Governing equation: Angular velocity: Boundary conditions: and on and on (Stationary) Eccentricity:

Comparison of stream function Kelmanson & Ingham (BIE) Analytical solution Present method (Direct BIEM) (Indirect BIEM) n=80 n=160 n=320 Limit n→∞ 0.0 0.1066 0.1062 0.1061 0.1060 0.1060 (N=5) 0.1 0.1052 0.1048 0.1047 0.1046 0.1046 (N=7) 0.2 0.1011 0.1006 0.1005 0.1005 (N=7) 0.3 0.0944 0.0939 0.0938 0.0938 (N=7) 0.4 0.0854 0.0850 0.0848 0.0846 0.0848 (N=9) 0.5 0.0748 0.0740 0.0739 0.0738 0.0738 (N=11) 0.6 0.0622 0.0615 0.0613 0.0612 0.0611 0.0611 (N=17) 0.7 0.0484 0.0477 0.0474 0.0472 0.0472 (N=17) 0.8 0.0347 0.0332 0.0326 0.0322 0.0322 (N=21) 0.9 0.0191 0.0175 0.0168 0.0163 0.0164 0.0164 (N=31) n: number of boundary nodes N: number of collocation points

Comparison for (160) BIE (Kelmanson) Present method Analytical solution (28) u1 (320) (640) (36) (∞) (44) DOF of BIE (Kelmanson) DOF of present method

Contour plot of Streamline for -Q/90 Q/20 Q/5 -Q/30 Q/2 Q Present method (N=81) -Q/90 Q/20 Q/5 -Q/30 Q/2 Kelmanson (Q=0.0740, n=160) Q e Kamal (Q=0.0738)

Contour plot of Streamline for Present method (N=21) Present method (N=41) Present method (N=81) Kelmanson (Q=0.0740, n=160)

Contour plot of vorticity for Present method (N=21) Present method (N=41) Kelmanson (n=160)

Contour plot of vorticity for Present method (N=21) Present method (N=41) Kelmanson (n=160)

Numerical examples Direct BIEM Indirect BIEM Plate problems 1. Annular plate 2. Circular plate with three circular holes Stokes flow problems 1. Doubly-connected case 1. Simply-connected case 2. Doubly-connected case

Stokes flow problems Governing equation: Angular velocity: Boundary conditions: and on and on (Stationary) Eccentricity:

Comparison of n: number of boundary nodes Kelmanson & Ingham Analytical solution Present method (Direct BIEM) (Indirect BIEM) n=80 n=160 n=320 Limit n→∞ 0.0 0.1066 0.1062 0.1061 0.1060 0.1060 (N=5) 0.1 0.1052 0.1048 0.1047 0.1046 0.1046 (N=7) 0.2 0.1011 0.1006 0.1005 0.1005 (N=7) 0.3 0.0944 0.0939 0.0938 0.0938 (N=7) 0.4 0.0854 0.0850 0.0848 0.0846 0.0848 (N=9) 0.5 0.0748 0.0740 0.0739 0.0738 0.0738 (N=11) 0.6 0.0622 0.0615 0.0613 0.0612 0.0611 0.0611 (N=17) 0.7 0.0484 0.0477 0.0474 0.0472 0.0472 (N=17) 0.8 0.0347 0.0332 0.0326 0.0322 0.0322 (N=21) 0.9 0.0191 0.0175 0.0168 0.0163 0.0164 0.0164 (N=31) n: number of boundary nodes N: number of collocation points

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,222 No. of elements=2,244) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,230 No. of elements=2,260) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,245 No. of elements=2,290) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,257 No. of elements=2,314) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,397 No. of elements=3,228) FEM result

Contour plot of streamline for Separation point Reattachment point Present method (N=161) FEM mesh (No. of nodes=1,734 No. of elements=3,218) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,735 No. of elements=3,220) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,635 No. of elements=3,010) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,196 No. of elements=2,162) FEM result

Contour plot of streamline for Present method (N=161) FEM mesh (No. of nodes=1,254 No. of elements=2,268) FEM result

Degree of separation and reattachment point versus eccentricity Separation point Reattachment point

Outlines Introduction Direct boundary integral equation method Indirect boundary integral equation method Numerical examples Conclusions

Conclusions Successful extension to biharmonic problems with circular boundaries. Both direct and indirect BIEMs were presented to solve biharmonic problems. Good agreement was obtained after compared with previous results, exact solution and ABAQUS data. Stream function and vorticity were found to be independent of Poisson ratio as we predicted.

Conclusions Boundary layer effect in the present method was also examined. A general-purpose program for solving the biharmonic problems with circular boundaries was developed successfully. Once engineering problems satisfy the biharmonic equation with circular boundaries, our present method can be used.

Conclusions Direct BIEM Indirect BIEM Null-field integral equation available? Direct BIEM Indirect BIEM Null-field ! No !

Thank you for your kind attention! The end Thank you for your kind attention!

Further research Viewpoint Finished Further research Direct BIEM & formulation Indirect BIEM Single & double layer potentials Triple & Quadruple layer potentials Post processing Lateral displacement Stress or moment diagram Boundary condition Essential boundary condition Natural boundary condition...etc.

Further research Viewpoint Finished Further research Degenerate scale Simply-connected problem had finished by Wu (2004) Doubly-connected & multiply-connected problems Shape of domain Circular domain Arbitrary domain

: deflection of the circular plate General form : flexure rigidity uniform pressure a B w=constant Governing equation: Boundary condition: Splitting method Governing equation: Boundary condition: : deflection of the circular plate General form : flexure rigidity Governing equation: : uniform distributed load Boundary condition: : domain of interest