Structural Deformation of a Circular Thin Plate with Combinations of Fixed and Free Edges

Transcription

1 Structural Deformation of a Circular Thin Plate with Combinations of Fixed and Free Edges Mechanical Engineering Report 2015/05 Kan Qin, Ingo Jahn School of Mechanical and Mining Engineering, The University of Queensland, Brisbane, Australia. June 2015 Abstract This report demonstrates a computational model for the structural deformation of a circular thin plate. The finite difference method is used to solve the governing equation of structural deformation. Detailed numerical procedures are described. A Python code is developed under this framework and compared with the result from the commercial software ANSYS. It is shown that the relative error is less than 3.5%. The developed Python code can be used to study the structural deformation of the circular thin plate in different applications. 1

2 Contents 1 Introduction 3 2 Computational Model Governing Equation Moments and Forces Boundary Conditions Numerical Method Discretization Numerical Treatment for Specific Points Point on Fixed Edge Point Adjacent to Fixed Edge Point Adjacent to Fixed-Free Corner Test Case 16 5 Conclusion 18 A Coefficients 20 B Discretization of Specific Points 23 B.1 Points on Fixed Edge B.2 Points Adjacent to Fixed Edge B.3 Points Adjacent to Fixed-Free Corner B.4 Points Adjacent to Fixed Corner on Free edge B.5 Points on Free Edge B.6 Points Adjacent to Free Edge B.7 Points on Free-Free Corner B.8 Points inside Free-Free Corner B.9 Points Adjacent to Free-Free Corner on Free Edge B.10 Points at Inner Domain C Source Code 43 2

3 1 Introduction The structural deformation of rectangular thin plates has been well studied [1, 2]. For the structural deformation of the circular thin plate, the governing equation in polar coordinates can be simply derived by coordinates transformation from the Cartesian system [3]. There are a few papers on introducing the numerical method for solving the governing equation for the circular plate [4, 5], but these treat cases with simplified boundary conditions (e.g. axisymmetric), and the governing equation for these specific cases can be further reduced and easily solved. In this report, the detailed numerical procedure to solve the structural deformation of the generic circular thin plate with any combination of fixed and free edges is provided. 3

4 2 Computational Model 2.1 Governing Equation The polar coordinates r and θ are used to solve the structural deformation of the circular thin plate. The governing equations for the circular thin plate is listed as [3]. ( 2 r r r + 1 w r 2 θ 2)( 2 r + 1 w 2 r r + 1 w r 2 θ 2) = p D (1) w is the local deflection, r is the local radius, θ is the local angle, p is the pressure acting on the thin plate, D is the stiffness of the thin plate, defined as: D = Eh 3 12(1 µ 3 ) (2) E is the modulus of elasticity, h is the plate thickness and µ is Poisson s ratio. The expanded form of Equation 1 can be written as. 4 w r w 4 r r 1 2 w 3 r 2 r + 1 w 2 r 3 r w r 2 r 2 θ 2 3 w 2 r 3 θ 2 r w r 4 θ w 2 r 4 θ = p 4 D (3) The above governing equation is the special case of the Kirchoff plate equation. It is simplified considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form. The basic assumptions are [6]. straight lines normal to the mid-surface remain straight after deformation; straight lines normal to the mid-surface remain normal to the mid-surface after deformation; the thickness of the plate does not change during a deformation; this equation doesn t include the in-plane tension and is only applicable for plates that carry load in bending. This governing equation is only accurate for small deflections and for thin plates subjected to forces and moments. 2.2 Moments and Forces The moments and forces used for the structural deformation of circular thin plates are shown in Figure 1. The bending or twisting moments M r, M θ, M rθ and M θr in the polar system are defined as [3]. 4

5 y PrT (, ) M T r 'r V r V T M Tr 'T M r M T x Figure 1: Schematic diagram for forces and moments. M r = D[ 2 w r +µ(1 w 2 r r w )] r 2 θ2 (4) M θ = D[ 1 w r r w w r 2 θ 2 +µ 2 r ] 2 (5) M rθ = M θr = D(1 µ)( 1 2 w r r θ w r 2 θ ) (6) The edge forces V r and V θ in radial and tangential direction are defined as [3]. V r = D[ w r ( 2 r + 1 w 2 r r w 1 µ )+ r 2 θ2 r θ (1 2 w r r θ w )] r 2 θ (7) V θ = D[ 1 w r θ r + 1 w 2 r r w r 2 θ )+(1 µ) 2 r (1 2 w r r θ w )] r 2 θ (8) 2.3 Boundary Conditions The boundary conditions for circular thin plates can be classified as free and fixed edges at radial or tangential directions, this is shown in Figure 2. The corner is also regarded as one type of boundary conditions, for example, free-free corner, this is defined as the intersection point between two free edges. The boundary conditions for circular thin plate are summarized as. 1. Fixed tangential edge 2. Fixed radial edge Zero deflection w = 0 w Zero deflection slope θ = 0 Zero deflection w = 0 w Zero deflection slope r = 0 5

6 y Corner Tangential edge Radial edge x Figure 2: Schematic diagram for different boundaries. 3. Fixed tangential edge with fixed slope 4. Fixed radial edge with fixed slope 5. Free tangential edge Zero deflection w = 0 w Constant deflection slope θ = const Zero deflection w = 0 w Constant deflection slope r = const Zero bending moment M θ = 0 Tangential edge force V θ = F θ 6. Free radial edge Zero bending moment M r = 0 Radial edge force V r = F r 7. Free-Free corner Zero bending moment M θ = M r = 0 Zero twisting moment M rθ = M θr = 0 Tangential edge force V θ = F θ Radial edge force V r = F r 6

7 To simplify the computational solution of the free-free corner boundary condition, the M θ = M r = 0 requirement can be replaced by the following deformation condition: These are obtained by combining Equation 4 and Free-Fixed (Radial-Tangential) corner 2 w r = 0 2 (9) w r w r 2 θ = 0 2 (10) Zero deflection w = 0 w Zero deflection slope θ = 0 Zero bending moment M r = 0 Radial edge force V r = F r 9. Free-Fixed (Tangential-Radial) corner Zero deflection w = 0 w Zero deflection slope r = 0 Zero bending moment M θ = 0 Tangential edge force V θ = F θ 10. Fixed-Fixed corner Zero deflection w = 0 w Zero deflection slope θ = 0 Zero deflection slope w r = 0 Where F r and F θ the force per unit length acting on the radial and tangential free edges, respectively. 7

8 3 Numerical Method The finite difference method is used to discretized the governing equation as indicated in Section 2. This section provides the detailed numerical procedure on solving the governing equation. 3.1 Discretization To assist the discretization of the governing equations, the following scheme is used at point o. ( 4 w r 4 ) o = 1 r 4(w ee 4w e +6w o 4w w +w ww ) (11) ( 3 w r 3 ) o = 1 2 r 3(w ee 2w e +2w w w ww ) (12) ( 2 w r ) 2 o = 1 r 2(w e 2w o +w w ) (13) ( w r ) o = 1 2 r (w e w w ) (14) ( 4 w 1 r 2 θ 2) o = r 2 θ 2[(w nw 2w w +w sw ) 2(w n 2w o +w s )+(w ne 2w e +w se )] (15) ( 3 w r θ 2) o = ( 3 w 1 ) = θ 2 ro 2 r θ 2[(w ne 2w e +w se ) (w nw 2w w +w sw )] (16) ( 3 w r 2 θ ) o = ( 3 w 1 θ r 2) o = 2 r 2 θ [(w ne 2w n +w nw ) (w se 2w s +w sw )] (17) ( 2 w r θ ) o = 1 4 r θ [(w ne w se ) (w nw w sw )] (18) ( w θ ) o = 1 2 θ (w n w s ) (19) ( 2 w θ 2 ) o = 1 θ 2(w n 2w o +w s ) (20) ( 3 w θ 3 ) o = 1 2 θ 3(w nn 2w n +2w s w ss ) (21) ( 4 w θ 4 ) o = 1 θ 4(w nn 4w n +6w o 4w s +w ss ) (22) The subscripts in the above equation indicate the position in the stencil. The schematic diagram of stencil used to solve the governing equation is shown in Figure 3. The discretized form of Equation 1 can be written as. a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D (23) The coefficients for these discretized equations can be found in Appendix A. The dis- 8

9 w nn w nw w n w ne w ww w w w o w e w ee w sw w s w se N, circumferential w ss E, radial Figure 3: Stencil arrangement for discretization. cretization form of M r, M θ, M rθ, V r and V θ at point o is shown as (M r ) o = b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s (24) (M θ ) o = c 1 w e +c 2 w o +c 3 w w +c 4 w n +c 5 w s (25) (M rθ ) o = (M θr ) o = k 1 w ne +k 2 w se +k 3 w nw +k 4 w sw +k 5 w n +k 6 w s (26) (V r ) o = d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se + d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o (27) (V θ ) o = e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw (28) The discretized form of Equation 9 and 10 is written as. f 1 w e +f 2 w o +f 3 w w = 0 (29) g 1 w n +g 2 w o +g 3 w s +g 4 w e +g 5 w w = 0 (30) The boundary conditions introduced in section 2.3 can be discretized using the same finite difference relations. 1. Fixed tangential edge 2. Fixed radial edge w o w ghost θ w o w ghost r 9 w o = 0 = 0 w o = 0 = 0

10 3. Fixed tangential edge with fixed slope w o w ghost θ w o = 0 = const 4. Fixed radial edge with fixed slope w o w ghost r w o = 0 = const 5. Free tangential edge c 1 w e +c 2 w o +c 3 w w +c 4 w n +c 5 w s = 0 e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ 6. Free radial edge b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se + d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r 7. Free-Free corner f 1 w e +f 2 w o +f 3 w w = 0 = 0 g 1 w n +g 2 w o +g 3 w s +g 4 w e +g 5 w w = 0 k 1 w ne +k 2 w se +k 3 w nw +k 4 w sw +k 5 w n +k 6 w s = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se + d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ 8. Free-Fixed (Radial-Tangential) corner w o = 0 w o w ghost = 0 θ b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se + d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r 10

11 9. Free-Fixed (Tangential-Radial) corner w o = 0 w o w ghost = 0 r c 1 w e +c 2 w o +c 3 w w +c 4 w n +c 5 w s = 0 e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ 10. Fixed-Fixed corner w o = 0 w o w ghost = 0 θ w o w ghost = 0 r The subscript ghost indicates the point outside the simulation domain and is aligned with point o at the same circumferential or radial direction. This point is used for discretization of governing equations and can be eliminated by implementing boundary conditions. This process can be found in section Numerical Treatment for Specific Points This section explains how to solve the governing equation in the specific points. The boundary conditions at the edge of the circular thin plate to be considered here correspond to a fixed tangential edge and three free edges. Given that this is a fourth-order governing equation, different types of points must be treated specifically, as shown in Figure Points on fixed edge; 2. Points adjacent to fixed edge; 3. Points adjacent to fixed-free corner; 4. Points adjacent to fixed-free corner on free edge; 5. Points on free edge; 6. Points adjacent to free edge; 7. Points on free corner; 8. Points inside free-free corner; 9. Points adjacent to free-free corner on free edge; 10. Points on the inner domain. 11

12 Free edge Point 1 Point 2 Point 3 Point 4 Point 5 Point 6 Point 7 Point 8 Point 9 Point 10 Free edge Free edge N, circumferential Fixed end E, radial Figure 4: Schematic diagram of different specific points Point on Fixed Edge For points on the fixed edge, the deflection is directly set as zero, the governing equation can be reduced to. w o = Point Adjacent to Fixed Edge For points adjacent to fixed edge, which are shown in Figure 5. The symbol shows this specific point, the symbol corresponds to surrounding cells used to solve the governing equation and the symbol are the ghost or zero deflection points which can be deleted by implementing boundary conditions. These definitions are also applied to the following analysis of the specific points. The governing equations and boundary conditions used to solve the points adjacent to fixed edge are. Governing equation a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D Fixed point at sw w sw = 0 12

13 N, circumferential Fixed end E, radial Figure 5: Stencil arrangement for points adjacent to fixed edge. Fixed point at se Fixed point at s w se = 0 Zero deflection slope at s w s = 0 w s w ss θ = 0 With these boundary conditions, the governing equation reduces to: a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 10 w nw +a 12 w ne = p D Point Adjacent to Fixed-Free Corner For points adjacent to fixed-free corner, these points can be clarified as west and east points, which is shown in Figure 6. The equations for points closed to the west boundary can be written as. Governing equation a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D (31) 13

14 Zero bending moment at w b 1 w o +b 2 w w +b 3 w ww +b 4 w nw +b 5 w sw = 0 (32) Fixed point at sw Fixed point at se Fixed point at s w sw = 0 (33) w se = 0 (34) w s = 0 (35) Zero deflection slope at s w s w ss θ = 0 (36) The point w sw in Equation 32 can be deleted by substituting Equation 33. b 1 w o +b 2 w w +b 3 w ww +b 4 w nw = 0 (37) The points w sw, w s, w se and w ss in Equation 31 are eliminated by substituting Equation 33, 34, 35 and 36. a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 10 w nw +a 12 w ne = p D (38) then the ghost point w ww is eliminated by combining Equation 37 and 38, which is shown as. a 1 w ee +a 2 w e +(a 3 a 5b 1 b 3 )w o +(a 4 a 5b 2 b 3 )w w +a 6 w nn +a 7 w n +(a 10 a 5b 4 b 3 )w nw +a 12 w ne = p D This is the governing equation for point adjacent to fixed-free corner. For the point at the east boundary, the governing equation and boundary condition is shown as Governing equation a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D Zero bending moment at e b 1 w ee +b 2 w e +b 3 w o +b 4 w ne +b 5 w se = 0 14

15 (a) (b) Free edge Free edge N, circumferential N, circumferential Fixed end Fixed end E, radial E, radial Figure 6: Stencil arrangement for points adjacent to fixed-free corner, (a) west point, (b) east point. Fixed point at sw Fixed point at se w sw = 0 Fixed point at s w se = 0 Zero deflection slope at s w s = 0 w s w ss θ = 0 Applying the above process to eliminate the ghost points, the governing equation becomes. a 2 w e +(a 3 a 1 b 3 b 1 )w o +(a 4 a 1 b 2 b 1 )w w +a 5 w ww +a 6 w nn +a 7 w n +a 10 w nw +(a 12 a 1b 4 b 1 )w ne = p D This section only provides the numerical treatment for the three different specific points, the numerical treatment for other points can be found in Appendix B. Please note that the treatment of points close to the edge needs to be adjusted to suit boundary conditions. The procedure follows the steps outlined above. 15

16 Free edge Free edge Free edge y x Fixed end 4 Test Case Figure 7: Schematic diagram of the circular thin plate. This finite difference method for the circular thin plate was implemented in the Python programming language. The source code is shown in Appendix C. In order to verify this developed Python code for the deformation of a circular thin plate, a test case of a 45 segment of a circular thin plate was selected. The thickness of this plate is chosen as 150 µm, the inner and outer radius is 25.4 mm and 50.8 mm, respectively. A uniform pressure of 70 Pa is prescribed on the top. The schematic diagram as well the boundary conditions are shown in Figure 7. The material selected is stainless steel with the elasticity module of 200 GPa and the Poisson s ration of 0.3. The commercial software ANSYS [7] was used to generate the verification case. The comparison of results of the structural deformation contour are shown in Figure 8. Note the legend in these two cases are not in the same scale, and the deformation result from ANSYS are negative. The comparison of the contour is reasonable. The deflection at the inner, middle and outer radii from these two cases are also extracted, and shown in Figure 9. The maximum relative error between the developed Python code and ANSYS is less than 3.5%. This proves that the developed Python code for the structural deformation is suitable for studying the structural deformation of circular thin plates. 16

17 (a) (b) Figure 8: Structural deformation contour (a): ANSYS, (b) the Developed Python code. 17

18 Defomration, m outer radii middle radii inner radii FDM-inner radii Ansys-inner radii FDM-middle radii Ansys-middle radii FDM-outer radii Ansys-outer radii 5e Angle, deg Figure 9: Deflection comparison between ANSYS and the developed Python code at different radii. 5 Conclusion This report provides the detailed numerical procedures to solve the structural deformation of the circular thin plate. Comparison of the finite difference code to ANSYS, shows that for the standard geometry (45, r inner =25.4 mm and r outer =50.8 mm), the difference is less than 3.5%. This demonstrates that the code is suitable for the analysis of small deflections of thin circular plates. 18

19 References [1] M.V. Barton. Finite difference equation for the analysis of thin rectangular plates with combinations of fixed and free edges. Technical report, The University of Texas, August [2] C.B. Dolicanin, V.B. Nikolic, and D.C. Dolicanin. Application of finite difference method to study of the phenonmenon in the theory of thin plate. Scientific Publications of the State University of Novi Pazar, 2(1):29 43, [3] S. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw-Hill, New York, [4] H.G. Sabiniak. The method of finite differences in solutions concerning circular plate. Key Engineering Materials, 490: , [5] A.K. Chakravorty. Finite difference solution for circular plates on elastic foundations. International Journal For Numerical Methods in Engineering, 9:73 84, [6] J.N. Reddy. Theory and analysis of elastic plates and shells. Taylor and Francis, Philadelphia, PA, [7] ANSYS, Inc. ANSYS Mechanical User s Guide, 15.0 edition,

20 A Coefficients a 1 = 1 r r 3 r a 2 = 1 2 rr 4 3 r 1 4 r 2 r 2 2 r 3 r + 2 r θ 2 r 4 3 r 2 θ 2 r 2 a 3 = 6 r r 2 r 8 2 θ 2 r θ 4 r r 2 θ 2 r 2 a 4 = 2 r 3 r 1 2 rr 1 3 r 2 r 4 2 r 2 4 r θ 2 r 4 3 r 2 θ 2 r 2 a 5 = 1 r 1 4 r 3 r a 6 = 1 θ 4 r 4 a 7 = 4 θ 2 r 4 4 θ 4 r 4 4 r 2 θ 2 r 2 a 8 = 4 θ 2 r 4 4 θ 4 r 4 4 r 2 θ 2 r 2 a 9 = 1 θ 4 r 4 1 a 10 = r θ 2 r r 2 θ 2 r 2 1 a 11 = r θ 2 r r 2 θ 2 r 2 2 a 12 = r 2 θ 2 r 1 2 r θ 2 r 3 2 a 13 = r 2 θ 2 r 1 2 r θ 2 r 3 b 1 = 1 r + µ 2 2 rr b 2 = 2 r 2µ 2 θ 2 r 2 b 3 = 1 r µ 2 2 rr b 4 = µ θ 2 r 2 b 5 = µ θ 2 r 2 20

21 c 1 = µ r rr c 2 = 2µ r 2 2 θ 2 r 2 c 3 = µ r rr c 4 = 1 θ 2 r 2 c 5 = 1 θ 2 r 2 d 1 = 1 2 r 3 d 2 = 1 r 2 r 1 2 rr 1 2 r 1 3 r θ 2 r + µ 1 2 r θ 2 r 2 d 3 = 1 r rr r 2 r + 1 r θ 2 r µ 1 2 r θ 2 r 2 d 4 = 1 2 r 3 d 5 = 2 µ 2 r θ 2 r 2 d 6 = 2 µ 2 r θ 2 r 2 d 7 = µ 2 2 r θ 2 r 2 d 8 = µ 2 2 r θ 2 r 2 d 9 = µ 3 θ 2 r 3 d 10 = µ 3 θ 2 r 3 d 11 = 6 2µ θ 2 r 2 3 r 2 r 21

22 1 e 1 = 2 θ 3 r 3 e 2 = µ 1 r 2 θr 1 r 2 θr µ 1 θr 1 3 θ 3 r 3 1 e 3 = θ 3 r r 2 θr + µ 1 θr µ 1 3 r 2 θr e 4 = 1 2 θ 3 r 3 1 e 5 = 4 r θr r 2 θr + µ 1 2 r θr µ r 2 θr 1 e 6 = 2 r 2 θr 1 4 r θr µ r θr µ r 2 θr e 7 = µ 1 2 r 2 θr 1 2 r 2 θr µ 1 2 r θr r θr 2 1 e 8 = 4 r θr r 2 θr + µ 1 2 r θr + µ r 2 θr f 1 = 1 r 2 f 2 = 2 r 2 f 3 = 1 r 2 1 g 1 = θ 2 r 2 g 2 = 2 θ 2 r 2 1 g 3 = θ 2 r 2 g 4 = 1 2 rr g 5 = 1 2 rr 1 k 1 = 4 r θr 1 k 2 = 4 r θr 1 k 3 = 4 r θr 1 k 4 = 4 r θr k 5 = 1 2 θr 2 k 5 = 1 2 θr 2 22

23 N, circumferential Fixed end E, radial Figure 10: Stencil arrangement for points adjacent to fixed edge. B Discretization of Specific Points This appendix will address the governing equation for specific points and the MATLAB script for extracting the coefficients of the reduced governing equation at the specific point. B.1 Points on Fixed Edge w o = 0 B.2 Points Adjacent to Fixed Edge a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D w sw = 0 w s = 0 w se = 0 w s w ss = 0 θ 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 23

24 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 10syms k1 k2 k3 k4 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 12 13% points adjacent to fixed edge eq = a1 w ee + a2 w e + a3 w o + a4 w w + a5 w ww + a6 w nn + a7 w n + a10 w nw + a12 w ne; 16 simplify(eq) B.3 Points Adjacent to Fixed-Free Corner (a): west point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + (b): east point a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w o +b 2 w w +b 3 w ww +b 4 w nw +b 5 w sw = 0 w sw = 0 w s = 0 w se = 0 w s w ss = 0 θ a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w ee +b 2 w w +b 3 w o +b 4 w ne +b 5 w se = 0 w sw = 0 w s = 0 w se = 0 w s w ss = 0 θ 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 24

25 (a) (b) Free edge Free edge N, circumferential N, circumferential Fixed end Fixed end E, radial E, radial Figure 11: Stencil arrangement for points adjacent to fixed-free corner, (a) west point, (b) east point. 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 12 13% points adjacent to fixed corner eq1 = b1 w o + b2 w w + b3 w ww + b4 w nw + b5 w sw; 16 sol ww = solve (eq1 == 0, w ww) ; 17 simplifyfraction (sol ww) clear all 20 21syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 22syms b1 b2 b3 b4 b5 23syms c1 c2 c3 c4 c5 24syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 25syms e1 e2 e3 e4 e5 e6 e7 e8 26syms f1 f2 f3 g1 g2 g3 g4 g5 27syms k1 k2 k3 k4 k5 k6 28syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se eq1 = b1 w ee + b2 w e + b3 w o + b4 w ne + b5 w se; 31 sol ee = solve (eq1 == 0, w ee) ; 32 simplifyfraction ( sol ee ) 25

26 B.4 Points Adjacent to Fixed Corner on Free edge (a): west point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 b 1 w ne +b 2 w n +b 3 w nw +b 4 w nn +b 5 w o = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r (b): east point w sw = 0 w s = 0 w se = 0 w s w ss = 0 θ a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 b 1 w ne +b 2 w n +b 3 w nw +b 4 w nn +b 5 w o = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r w sw = 0 w s = 0 w se = 0 w s w ss = 0 θ 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 12syms Vr Vt V 13 14% potins adjacent to fixed corner on free edge 15 26

27 (a) (b) Free edge Free edge N, circumferential N, circumferential Fixed end Fixed end E, radial E, radial Figure 12: Stencil arrangement for points adjacent to fixed corner on free edge, (a) west point, (b) east point. 16 eq1 = b1 w e + b2 w o + b3 w w + b4 w n; 17 sol w = solve(eq1 == 0, w w) ; 18 simplifyfraction (sol w) 19 eq2 = b1 w ne + b2 w n + b3 w nw + b4 w nn + b5 w o; 20 sol nw = solve (eq2 == 0, w nw) ; 21 simplifyfraction (sol nw) 22 eq3 = d1 w ee + d2 w e + d3 sol w + d4 w ww + d5 w ne + d7 sol nw + d9 w n + d11 w o; 23 sol ww = solve (eq3 == Vr, w ww) ; 24 simplifyfraction (sol ww) clear all 27 28syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 29syms b1 b2 b3 b4 b5 30syms c1 c2 c3 c4 c5 31syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 32syms e1 e2 e3 e4 e5 e6 e7 e8 33syms f1 f2 f3 g1 g2 g3 g4 g5 34syms k1 k2 k3 k4 k5 k6 35syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 36syms Vr Vt V eq1 = b1 w e + b2 w o + b3 w w + b4 w n; 40 sol e = solve(eq1 == 0, w e) ; 41 simplifyfraction ( sol e ) 42 eq2 = b1 w ne + b2 w n + b3 w nw + b4 w nn + b5 w o; 43 sol ne = solve (eq2 == 0, w ne) ; 44 simplifyfraction ( sol ne ) 45 eq3 = d1 w ee + d2 sol e + d3 w w + d4 w ww + d5 sol ne + d7 w nw + d9 w n + d11 w o; 46 sol ee = solve (eq3 == Vr, w ee) ; 47 simplifyfraction ( sol ee ) 27

28 B.5 Points on Free Edge (a): west and east points a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 b 1 w ne +b 2 w n +b 3 w nw +b 4 w nn +b 5 w o = 0 b 1 w se +b 2 w s +b 3 w sw +b 4 w o +b 5 w ss = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r (b): north point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D c 1 w e +c 2 w o +c 3 w w +c 4 w n +c 5 w s = 0 c 1 w ee +c 2 w e +c 3 w o +c 4 w ne +c 5 w se = 0 c 1 w o +c 2 w w +c 3 w ww +c 4 w nw +c 5 w sw = 0 e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 12syms Vr Vt V eq1 = b1 w e + b2 w o + b3 w w + b4 w n + b5 w s; 15 sol w = solve(eq1 == 0, w w) ; 16 simplifyfraction (sol w) 17 eq2 = b1 w ne + b2 w n + b3 w nw + b4 w nn + b5 w o; 18 sol nw = solve (eq2 == 0, w nw) ; 19 simplifyfraction (sol nw) 20 eq3 = b1 w se + b2 w s + b3 w sw + b4 w o + b5 w ss ; 21 sol sw = solve (eq3 == 0, w sw) ; 28

29 (a) (b) Free edge Free edge N, circumferential N, circumferential E, radial E, radial (c) Free edge N, circumferential E, radial Figure 13: Stencil arrangement for points on free edge, (a) west point, (b) east point, (c) north point. 29

30 22 simplifyfraction (sol sw) 23 eq4 = d1 w ee + d2 w e + d3 sol w + d4 w ww + d5 w ne + d6 w se + d7 sol nw + d8 sol sw + d9 w n + d10 w s + d11 w o; 24 sol ww = solve (eq4 == Vr, w ww) ; 25 simplifyfraction (sol ww) clear all 28 29syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 30syms b1 b2 b3 b4 b5 31syms c1 c2 c3 c4 c5 32syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 33syms e1 e2 e3 e4 e5 e6 e7 e8 34syms f1 f2 f3 g1 g2 g3 g4 g5 35syms k1 k2 k3 k4 k5 k6 36syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 37syms Vr Vt V eq1 = b1 w e + b2 w o + b3 w w + b4 w n + b5 w s; 40 sol e = solve(eq1 == 0, w e) ; 41 simplifyfraction ( sol e ) 42 eq2 = b1 w ne + b2 w n + b3 w nw + b4 w nn + b5 w o; 43 sol ne = solve (eq2 == 0, w ne) ; 44 simplifyfraction ( sol ne ) 45 eq3 = b1 w se + b2 w s + b3 w sw + b4 w o + b5 w ss ; 46 sol se = solve (eq3 == 0, w se) ; 47 simplifyfraction ( sol se ) 48 eq4 = d1 w ee + d2 sol e + d3 w w + d4 w ww + d5 sol ne + d6 sol se + d7 w nw + d8 w sw + d9 w n + d10 w s + d11 w o; 49 sol ee = solve (eq4 == Vr, w ee) ; 50 simplifyfraction ( sol ee ) clear all 53 54syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 55syms b1 b2 b3 b4 b5 56syms c1 c2 c3 c4 c5 57syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 58syms e1 e2 e3 e4 e5 e6 e7 e8 59syms f1 f2 f3 g1 g2 g3 g4 g5 60syms k1 k2 k3 k4 k5 k6 61syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 62syms Vr Vt V eq1 = c1 w e + c2 w o + c3 w w + c4 w n + c5 w s; 65 sol n = solve(eq1 == 0, w n) ; 66 simplifyfraction ( sol n ) 67 eq2 = c1 w ee + c2 w e + c3 w o + c4 w ne + c5 w se; 68 sol ne = solve (eq2 == 0, w ne) ; 69 simplifyfraction ( sol ne ) 70 eq3 = c1 w o + c2 w w + c3 w ww + c4 w nw + c5 w sw; 30

31 71 sol nw = solve (eq3 == 0, w nw) ; 72 simplifyfraction (sol nw) 73 eq4 = e1 w nn + e2 sol n + e3 w s + e4 w ss + e5 sol ne + e6 sol nw + e7 w se + e8 w sw; 74 sol nn = solve (eq4 == Vt, w nn) ; 75 simplifyfraction (sol nn) B.6 Points Adjacent to Free Edge (a): west point (b): east point (c): north point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w o +b 2 w w +b 3 w ww +b 4 w nw +b 5 w sw = 0 a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w ee +b 2 w e +b 3 w o +b 4 w ne +b 5 w se = 0 a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D c 1 w ne +c 2 w n +c 3 w nw +c 4 w nn +c 5 w o = 0 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se eq1 = b1 w o + b2 w w + b3 w ww + b4 w nw + b5 w sw; 14 sol ww = solve (eq1 == 0, w ww) ; 15 simplifyfraction (sol ww) 16 31

32 (a) (b) Free edge Free edge N, circumferential N, circumferential E, radial E, radial (c) Free edge N, circumferential E, radial Figure 14: Stencil arrangement for points adjacent to free edge, (a) west point, (b) east point, (c) north point. 32

33 17 clear all 18 19syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 20syms b1 b2 b3 b4 b5 21syms c1 c2 c3 c4 c5 22syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 23syms e1 e2 e3 e4 e5 e6 e7 e8 24syms f1 f2 f3 g1 g2 g3 g4 g5 25syms k1 k2 k3 k4 k5 k6 26syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se eq1 = b1 w ee + b2 w e + b3 w o + b4 w ne + b5 w se; 29 sol ee = solve (eq1 == 0, w ee) ; 30 simplifyfraction ( sol ee ) clear all 33 34syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 35syms b1 b2 b3 b4 b5 36syms c1 c2 c3 c4 c5 37syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 38syms e1 e2 e3 e4 e5 e6 e7 e8 39syms f1 f2 f3 g1 g2 g3 g4 g5 40syms k1 k2 k3 k4 k5 k6 41syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se eq1 = c1 w ne + c2 w n + c3 w nw + c4 w nn + c5 w o; 44 sol nn = solve (eq1 == 0, w nn) ; 45 simplifyfraction (sol nn) B.7 Points on Free-Free Corner (a): northwest point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w se +b 2 w s +b 3 w sw +b 4 w o +b 5 w ss = 0 c 1 w ee +c 2 w e +c 3 w o +c 4 w ne +c 5 w se = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ f 1 w e +f 2 w o +f 3 w w = 0 g 1 w n +g 2 w o +g 3 w s +g 4 w e +g 5 w w = 0 k 1 w ne +k 2 w se +k 3 w nw +k 4 w sw +k 5 w n +k 6 w s = 0 33

34 (a) (b) Free edge N, circumferential N, circumferential Free edge Free edge E, radial E, radial Figure 15: Stencil arrangement for points adjacent to free corner, (a) northwest point, (b) northeast point. (b): northeast point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w se +b 2 w s +b 3 w sw +b 4 w o +b 5 w ss = 0 c 1 w o +c 2 w w +c 3 w ww +c 4 w nw +c 5 w sw = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ f 1 w e +f 2 w o +f 3 w w = 0 g 1 w n +g 2 w o +g 3 w s +g 4 w e +g 5 w w = 0 k 1 w ne +k 2 w se +k 3 w nw +k 4 w sw +k 5 w n +k 6 w s = 0 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 12syms Vr Vt V eq1 = f1 w e + f2 w o + f3 w w; 15 sol w = solve(eq1 == 0, w w) ; 16 simplifyfraction (sol w) 34

35 17 eq2 = g1 w n + g2 w o + g3 w s + g4 w e + g5 sol w ; 18 sol n = solve(eq2 == 0, w n) ; 19 simplifyfraction ( sol n ) 20 eq3 = b1 w se + b2 w s + b3 w sw + b4 w o + b5 w ss ; 21 sol sw = solve (eq3 == 0, w sw) ; 22 simplifyfraction (sol sw) 23 eq4 = c1 w ee + c2 w e + c3 w o + c4 w ne + c5 w se; 24 sol ne = solve (eq4 == 0, w ne) ; 25 simplifyfraction ( sol ne ) 26 eq5 = k1 sol ne + k2 w se + k3 w nw + k4 sol sw + k5 sol n + k6 w s; 27 sol nw = solve (eq5 == V, w nw) ; 28 simplifyfraction (sol nw) 29 eq6 = e1 w nn + e2 sol n + e3 w s + e4 w ss + e5 sol ne + e6 sol nw + e7 w se + e8 sol sw ; 30 sol nn = solve (eq6 == Vt, w nn) ; 31 simplifyfraction (sol nn) 32 eq7 = d1 w ee + d2 w e + d3 sol w + d4 w ww + d5 sol ne + d6 w se + d7 sol nw + d8 sol sw + d9 sol n + d10 w s + d11 w o; 33 sol ww = solve (eq7 == Vr, w ww) ; 34 simplifyfraction (sol ww) clear all 37 38syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 39syms b1 b2 b3 b4 b5 40syms c1 c2 c3 c4 c5 41syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 42syms e1 e2 e3 e4 e5 e6 e7 e8 43syms f1 f2 f3 g1 g2 g3 g4 g5 44syms k1 k2 k3 k4 k5 k6 45syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 46syms Vr Vt V eq1 = f1 w e + f2 w o + f3 w w; 49 sol e = solve(eq1 == 0, w e) ; 50 simplifyfraction ( sol e ) 51 eq2 = g1 w n + g2 w o + g3 w s + g4 sol e + g5 w w; 52 sol n = solve(eq2 == 0, w n) ; 53 simplifyfraction ( sol n ) 54 eq3 = b1 w se + b2 w s + b3 w sw + b4 w o + b5 w ss ; 55 sol se = solve (eq3 == 0, w se) ; 56 simplifyfraction ( sol se ) 57 eq4 = c1 w o + c2 w w + c3 w ww + c4 w nw + c5 w sw; 58 sol nw = solve (eq4 == 0, w nw) ; 59 simplifyfraction (sol nw) 60 eq5 = k1 w ne + k2 sol se + k3 sol nw + k4 w sw + k5 sol n + k6 w s; 61 sol ne = solve (eq5 == V, w ne) ; 62 simplifyfraction ( sol ne ) 63 eq6 = e1 w nn + e2 sol n + e3 w s + e4 w ss + e5 sol ne + e6 sol nw + e7 sol se + e8 w sw; 64 sol nn = solve (eq6 == Vt, w nn) ; 35

36 (a) (b) Free edge N, circumferential N, circumferential Free edge Free edge E, radial E, radial Figure 16: Stencil arrangement for points inside free-free corner, (a) northwest point, (b) northeast point. 65 simplifyfraction (sol nn) 66 eq7 = d1 w ee + d2 sol e + d3 w w + d4 w ww + d5 sol ne + d6 sol se + d7 sol nw + d8 w sw + d9 sol n + d10 w s + d11 w o; 67 sol ee = solve (eq7 == Vr, w ee) ; 68 simplifyfraction ( sol ee ) B.8 Points inside Free-Free Corner (a): northwest point (b): northeast point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w o +b 2 w w +b 3 w ww +b 4 w nw +b 5 w sw = 0 c 1 w ne +c 2 w n +c 3 w nw +c 4 w nn +c 5 w o = 0 a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w ee +b 2 w e +b 3 w o +b 4 w ne +b 5 w se = 0 c 1 w ne +c 2 w n +c 3 w nw +c 4 w nn +c 5 w o = 0 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 36

37 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se eq1 = b1 w o + b2 w w + b3 w ww + b4 w nw + b5 w sw; 15 sol ww = solve (eq1 == 0, w ww) ; 16 simplifyfraction (sol ww) 17 eq2 = c1 w ne + c2 w n + c3 w nw + c4 w nn + c5 w o; 18 sol nn = solve (eq2 == 0, w nn) ; 19 simplifyfraction (sol nn) clear all 22 23syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 24syms b1 b2 b3 b4 b5 25syms c1 c2 c3 c4 c5 26syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 27syms e1 e2 e3 e4 e5 e6 e7 e8 28syms f1 f2 f3 g1 g2 g3 g4 g5 29syms k1 k2 k3 k4 k5 k6 30syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se eq1 = b1 w ee + b2 w e + b3 w o + b4 w ne + b5 w se; 34 sol ee = solve (eq1 == 0, w ee) ; 35 simplifyfraction ( sol ee ) 36 eq2 = c1 w ne + c2 w n + c3 w nw + c4 w nn + c5 w o; 37 sol nn = solve (eq2 == 0, w nn) ; 38 simplifyfraction (sol nn) B.9 Points Adjacent to Free-Free Corner on Free Edge (a): northwest point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D c 1 w e +c 2 w o +c 3 w w +c 4 w n +c 5 w s = 0 c 1 w ee +c 2 w e +c 3 w o +c 4 w ne +c 5 w se = 0 e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ f 1 w o +f 2 w w +f 3 w ww = 0 g 1 w nw +g 2 w w +g 3 w sw +g 4 w o +g 5 w ww = 0 37

38 (b): southwest point a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 b 1 w se +b 2 w s +b 3 w sw +b 4 w o +b 5 w ss = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r (c): southeast point f 1 w ne +f 2 w n +f 3 w nw = 0 g 1 w nn +g 2 w n +g 3 w o +g 4 w ne +g 5 w nw = 0 a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D b 1 w e +b 2 w o +b 3 w w +b 4 w n +b 5 w s = 0 b 1 w se +b 2 w s +b 3 w sw +b 4 w o +b 5 w ss = 0 d 1 w ee +d 2 w e +d 3 w w +d 4 w ww +d 5 w ne +d 6 w se +d 7 w nw +d 8 w sw +d 9 w n +d 10 w s +d 11 w o = F r (d): northeast point f 1 w ne +f 2 w n +f 3 w nw = 0 g 1 w nn +g 2 w n +g 3 w o +g 4 w ne +g 5 w nw = 0 a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D c 1 w e +c 2 w o +c 3 w w +c 4 w n +c 5 w s = 0 c 1 w o +c 2 w w +c 3 w ww +c 4 w nw +c 5 w sw = 0 e 1 w nn +e 2 w n +e 3 w s +e 4 w ss +e 5 w ne +e 6 w nw +e 7 w se +e 8 w sw = F θ f 1 w ee +f 2 w e +f 3 w o = 0 g 1 w ne +g 2 w e +g 3 w se +g 4 w ee +g 5 w o = 0 1 clear all 2 clc 3 4syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 5syms b1 b2 b3 b4 b5 6syms c1 c2 c3 c4 c5 7syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 8syms e1 e2 e3 e4 e5 e6 e7 e8 9syms f1 f2 f3 g1 g2 g3 g4 g5 38

39 (a) (b) N, circumferential N, circumferential Free edge Free edge E, radial E, radial (c) (d) Free edge Free edge Free edge Free edge N, circumferential N, circumferential E, radial E, radial Figure 17: Stencil arrangement for points adjacent to free-free corner on free edge, (a) northwest point, (b) southwest point, (c) southeast point, (d) northeast point. 39

40 10syms k1 k2 k3 k4 k5 k6 11syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 12syms Vr Vt V eq1 = f1 w o + f2 w w + f3 w ww; 15 sol ww = solve (eq1 == 0, w ww) ; 16 simplifyfraction (sol ww) 17 eq2 = g1 w nw + g2 w w + g3 w sw + g4 w o + g5 sol ww; 18 sol nw = solve (eq2 == 0, w nw) ; 19 simplifyfraction (sol nw) 20 eq3 = c1 w e + c2 w o + c3 w w + c4 w n + c5 w s; 21 sol n = solve(eq3 == 0, w n) ; 22 simplifyfraction ( sol n ) 23 eq4 = c1 w ee + c2 w e + c3 w o + c4 w ne + c5 w se; 24 sol ne = solve (eq4 == 0, w ne) ; 25 simplifyfraction ( sol ne ) 26 eq5 = e1 w nn + e2 sol n + e3 w s + e4 w ss + e5 sol ne + e6 sol nw + e7 w se + e8 w sw; 27 sol nn = solve (eq5 == Vt, w nn) ; 28 simplifyfraction (sol nn) clear all 31 32syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 33syms b1 b2 b3 b4 b5 34syms c1 c2 c3 c4 c5 35syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 36syms e1 e2 e3 e4 e5 e6 e7 e8 37syms f1 f2 f3 g1 g2 g3 g4 g5 38syms k1 k2 k3 k4 k5 k6 39syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 40syms Vr Vt V eq1 = f1 w ne + f2 w n + f3 w nw; 43 sol nw = solve (eq1 == 0, w nw) ; 44 simplifyfraction (sol nw) 45 eq2 = g1 w nn + g2 w n + g3 w o + g4 w ne + g5 sol nw ; 46 sol nn = solve (eq2 == 0, w nn) ; 47 simplifyfraction (sol nn) 48 eq3 = b1 w e + b2 w o + b3 w w + b4 w n + b5 w s; 49 sol w = solve(eq3 == 0, w w) ; 50 simplifyfraction (sol w) 51 eq4 = b1 w se + b2 w s + b3 w sw + b4 w o + b5 w ss ; 52 sol sw = solve (eq4 == 0, w sw) ; 53 simplifyfraction (sol sw) 54 eq5 = d1 w ee + d2 w e + d3 sol w + d4 w ww + d5 w ne + d6 w se + d7 sol nw + d8 sol sw + d9 w n + d10 w s + d11 w o; 55 sol ww = solve (eq5 == Vr, w ww) ; 56 simplifyfraction (sol ww) clear all 40

41 59 60syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 61syms b1 b2 b3 b4 b5 62syms c1 c2 c3 c4 c5 63syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 64syms e1 e2 e3 e4 e5 e6 e7 e8 65syms f1 f2 f3 g1 g2 g3 g4 g5 66syms k1 k2 k3 k4 k5 k6 67syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 68syms Vr Vt V eq1 = f1 w ne + f2 w n + f3 w nw; 71 sol ne = solve (eq1 == 0, w ne) ; 72 simplifyfraction ( sol ne ) 73 eq2 = g1 w nn + g2 w n + g3 w o + g4 sol ne + g5 w nw; 74 sol nn = solve (eq2 == 0, w nn) ; 75 simplifyfraction (sol nn) 76 eq3 = b1 w e + b2 w o + b3 w w + b4 w n + b5 w s; 77 sol e = solve(eq3 == 0, w e) ; 78 simplifyfraction ( sol e ) 79 eq4 = b1 w se + b2 w s + b3 w sw + b4 w o + b5 w ss ; 80 sol se = solve (eq4 == 0, w se) ; 81 simplifyfraction ( sol se ) 82 eq5 = d1 w ee + d2 sol e + d3 w w + d4 w ww + d5 sol ne + d6 sol se + d7 w nw + d8 w sw + d9 w n + d10 w s + d11 w o; 83 sol ee = solve (eq5 == Vr, w ee) ; 84 simplifyfraction ( sol ee ) clear all 87 88syms a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 89syms b1 b2 b3 b4 b5 90syms c1 c2 c3 c4 c5 91syms d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 92syms e1 e2 e3 e4 e5 e6 e7 e8 93syms f1 f2 f3 g1 g2 g3 g4 g5 94syms k1 k2 k3 k4 k5 k6 95syms w ee w e w o w w w ww w nn w n w s w ss w nw w sw w ne w se 96syms Vr Vt V eq1 = f1 w ee + f2 w e + f3 w o; 99 sol ee = solve (eq1 == 0, w ee) ; 100 simplifyfraction ( sol ee ) 101 eq2 = g1 w ne + g2 w e + g3 w se + g4 sol ee + g5 w o; 102 sol ne = solve (eq2 == 0, w ne) ; 103 simplifyfraction ( sol ne ) 104 eq3 = c1 w e + c2 w o + c3 w w + c4 w n + c5 w s; 105 sol n = solve(eq3 == 0, w n) ; 106 simplifyfraction ( sol n ) 107 eq4 = c1 w o + c2 w w + c3 w ww + c4 w nw + c5 w sw; 108 sol nw = solve (eq4 == 0, w nw) ; 41

42 N, circumferential E, radail Figure 18: Stencil arrangement for points at the inner domain. 109 simplifyfraction (sol nw) 110 eq5 = e1 w nn + e2 sol n + e3 w s + e4 w ss + e5 sol ne + e6 sol nw + e7 w se + e8 w sw; 111 sol nn = solve (eq5 == Vt, w nn) ; 112 simplifyfraction (sol nn) B.10 Points at Inner Domain a 1 w ee +a 2 w e +a 3 w o +a 4 w w +a 5 w ww +a 6 w nn +a 7 w n +a 8 w s + a 9 w ss +a 10 w nw +a 11 w sw +a 12 w ne +a 13 w se = p D 42

43 C Source Code 1#!/usr/bin/env python 2# Validation case for developed 3# structural model for thin circular plate 4# Author: Jason Qin, Nov import sys, os, gzip 7 sys.path.append(os.path. expandvars( $HOME/e3bin )) 8 from math import 9 import numpy 10 import matplotlib. pyplot as plt 11 from scipy. sparse import 12 from scipy import 13 from scipy. sparse. linalg import spsolve 14 15# Geometry for top foil 16 inch = 25.4e 3 17 r0 = 1.0 inch 18 r1 = 2.0 inch 19 mean r = (r0+r1)/ theta0 = theta1 = 45.0 pi/ A sector = 45.0/360.0 pi (r1 2.0 r0 2.0) 23 24# Gird 25 ni = nj = grd posr = numpy. zeros ((ni, nj)) 28 grd post = numpy. zeros ((ni, nj)) 29 grd posx = numpy. zeros ((ni, nj)) 30 grd posy = numpy. zeros ((ni, nj)) 31 dr = (r1 r0)/(ni 1) 32 dt = (theta1 theta0)/(nj 1) 33 for j in range(nj) : 34 for i in range(ni) : 35 grd posr [ i, j ] = dr i + r0 36 grd post [ i, j ] = dt j + theta0 37 grd posx [ i, j ] = grd posr [ i, j ] cos(grd post [ i, j ]) 38 grd posy [ i, j ] = grd posr [ i, j ] sin (grd post [ i, j ]) 39 grd imin = 0; grd imax = ni 1; 40 grd jmin = 0; grd jmax = nj 1; 41 42# Properties for top foil 43E = 2.0e+11 44h = 150.0e 6 45mu = D = E h 3.0/(12.0 (1.0 mu 2.0) ) 47F N = 70.0 # Modulus of elasticity # The thickness of top foil # Poisson s ratio # plate stiffness # Forces acting on the node 43

44 48 p o = numpy. zeros ((ni, nj)) 49Vr = numpy. zeros ((ni, nj)) 50Vt = numpy. zeros ((ni, nj)) 51V = numpy. zeros ((ni, nj)) 52 for j in range(nj) : 53 for i in range(ni) : 54 if ( i > grd imin and i < grd imax ) and ( j > grd jmin and j < grd jmax ) : 55 p o[ i, j ] = F N 56 57# Coefficient functions 58 def get local a coefficient (ra) : calculate the load equation coefficient l a1 = 1.0/dr /(dr 3.0 ra) 65 l a2 = 1.0/(2.0 dr ra 3.0) 4.0/dr /(dr 2.0 ra 2.0) 2.0/(dr 3.0 ra) + 2.0/(dr dt 2.0 ra 3.0) 4.0/(dr 2.0 dt 2.0 ra 2.0) 66 l a3 = 6.0/dr /(dr 2.0 ra 2.0) 8.0/(dt 2.0 ra 4.0) + 6.0/(dt 4.0 ra 4.0) + 8.0/(dr 2.0 dt 2.0 ra 2.0) 67 l a4 = 2.0/(dr 3.0 ra) 1.0/(2.0 dr ra 3.0) 1.0/(dr 2.0 ra 2.0) 4.0/dr /(dr dt 2.0 ra 3.0) 4.0/(dr 2.0 dt 2.0 ra 2.0) 68 l a5 = 1.0/dr /(dr 3.0 ra) 69 l a6 = 1.0/(dt 4.0 ra 4.0) 70 l a7 = 4.0/(dt 2.0 ra 4.0) 4.0/(dt 4.0 ra 4.0) 4.0/(dr 2.0 dt 2.0 ra 2.0) 71 l a8 = 4.0/(dt 2.0 ra 4.0) 4.0/(dt 4.0 ra 4.0) 4.0/(dr 2.0 dt 2.0 ra 2.0) 72 l a9 = 1.0/(dt 4.0 ra 4.0) 73 l a10 = 1.0/(dr dt 2.0 ra 3.0) + 2.0/(dr 2.0 dt 2.0 ra 2.0) 74 l a11 = 1.0/(dr dt 2.0 ra 3.0) + 2.0/(dr 2.0 dt 2.0 ra 2.0) 75 l a12 = 2.0/(dr 2.0 dt 2.0 ra 2.0) 1.0/(dr dt 2.0 ra 3.0) 76 l a13 = 2.0/(dr 2.0 dt 2.0 ra 2.0) 1.0/(dr dt 2.0 ra 3.0) return l a1, l a2, l a3, l a4, l a5, l a6, l a7, \ 79 l a8, l a9, l a10, l a11, l a12, l a def get local b coefficient (rb) : calculate moment r coefficient l b1 = 1.0/dr mu/(2.0 dr rb) 88 l b2 = 2.0/dr 2.0 (2.0 mu)/(dt 2.0 rb 2.0) 89 l b3 = 1.0/dr 2.0 mu/(2.0 dr rb) 90 l b4 = mu/(dt 2.0 rb 2.0) 44

45 91 l b5 = mu/(dt 2.0 rb 2.0) return l b1, l b2, l b3, l b4, l b def get local c coefficient (rc) : calculate moment t coefficient l c1 = mu/dr /(2.0 dr rc) 102 l c2 = (2.0 mu)/dr /(dt 2.0 rc 2.0) 103 l c3 = mu/dr /(2.0 dr rc) 104 l c4 = 1.0/(dt 2.0 rc 2.0) 105 l c5 = 1.0/(dt 2 rc 2.0) return l c1, l c2, l c3, l c4, l c def get local d coefficient (rd) : calculate force x coefficient l d1 = 1.0/(2.0 dr 3.0) 116 l d2 = 1.0/(dr 2.0 rd) 1.0/(2.0 dr rd 2.0) 1.0/dr /(dr dt 2.0 rd 2.0) + (mu 1.0) /(dr dt 2.0 rd 2.0) 117 l d3 = 1.0/dr /(2.0 dr rd 2.0) + 1.0/(dr 2.0 rd) + 1.0/(dr dt 2.0 rd 2.0) (mu 1.0) /(dr dt 2.0 rd 2.0) 118 l d4 = 1.0/(2.0 dr 3.0) 119 l d5 = 1.0/(2.0 dr dt 2.0 rd 2.0) (mu 1.0) /(2.0 dr dt 2.0 rd 2.0) 120 l d6 = 1.0/(2.0 dr dt 2.0 rd 2.0) (mu 1.0) /(2.0 dr dt 2.0 rd 2.0) 121 l d7 = (mu 1.0) /(2.0 dr dt 2.0 rd 2.0) 1.0/(2.0 dr dt 2.0 rd 2.0) 122 l d8 = (mu 1.0) /(2.0 dr dt 2.0 rd 2.0) 1.0/(2.0 dr dt 2.0 rd 2.0) 123 l d9 = (mu 1.0) /(dt 2.0 rd 3.0) 2.0/(dt 2.0 rd 3.0) 124 l d10 = (mu 1.0) /(dt 2.0 rd 3.0) 2.0/(dt 2.0 rd 3.0) 125 l d11 = 4.0/(dt 2.0 rd 3.0) 2.0/(dr 2.0 rd) (2.0 (mu 1.0) )/(dt 2.0 rd 3.0) return l d1, l d2, l d3, l d4, l d5, l d6, l d7, l d8, l d9, l d10, l d def get local e coefficient (re) : calculate force y coefficient

46 l e1 = 1.0/(2.0 dt 3.0 re 3.0) 136 l e2 = (mu 1.0) /(dr 2.0 dt re) 1.0/(dr 2.0 dt re) (mu 1.0) /(dt re 3.0) 1.0/(dt 3.0 re 3.0) 137 l e3 = 1.0/(dt 3.0 re 3.0) + 1.0/(dr 2.0 dt re) + (mu 1.0) /( dt re 3.0) (mu 1.0) /(dr 2.0 dt re) 138 l e4 = 1.0/(2.0 dt 3.0 re 3.0) 139 l e5 = 1.0/(4.0 dr dt re 2.0) + 1.0/(2.0 dr 2.0 dt re) + (mu 1.0) /(2.0 dr dt re 2.0) (mu 1.0) /(2.0 dr 2.0 dt re) 140 l e6 = 1.0/(2.0 dr 2.0 dt re) 1.0/(4.0 dr dt re 2.0) (mu 1.0) /(2.0 dr dt re 2.0) (mu 1.0) /(2.0 dr 2.0 dt re) 141 l e7 = (mu 1.0) /(2.0 dr 2.0 dt re) 1.0/(2.0 dr 2.0 dt re) (mu 1.0) /(2.0 dr dt re 2.0) 1.0/(4.0 dr dt re 2.0) 142 l e8 = 1.0/(4.0 dr dt re 2.0) 1.0/(2.0 dr 2.0 dt re) + (mu 1.0) /(2.0 dr dt re 2.0) + (mu 1.0) /(2.0 dr 2.0 dt re) return l e1, l e2, l e3, l e4, l e5, l e6, l e7, l e def get local f coefficient ( rf ) : calculate moment x and y coefficient l f1 = 1.0/dr l f2 = 2.0/dr l f3 = 1.0/dr return l f1, l f2, l f def get local g coefficient (rg) : calculate moment x and y coefficient l g1 = 1.0/(dt 2.0 rg 2.0) 165 l g2 = 2.0/(dt 2.0 rg 2.0) 166 l g3 = 1.0/(dt 2.0 rg 2.0) 167 l g4 = 1.0/(2.0 dr rg) 168 l g5 = 1.0/(2.0 dr rg) return l g1, l g2, l g3, l g4, l g def get local k coefficient (rk) : calculate corner force coefficient l k1 = 1.0/(4.0 dr dt rk) 46

47 179 l k2 = 1.0/(4.0 dr dt rk) 180 l k3 = 1.0/(4.0 dr dt rk) 181 l k4 = 1.0/(4.0 dr dt rk) 182 l k5 = 1.0/(2.0 dt rk 2.0) 183 l k6 = 1.0/(2.0 dt rk 2.0) return l k1, l k2, l k3, l k4, l k5, l k # coefficient matrix for deformation 188row = [] 189 col = [] 190 data = [] 191 f matrix = numpy. zeros (( ni nj)) # calculate coefficient matrix now 194 mi indx = mj indx = for j in range(nj) : 197 for i in range(ni) : ################################################################ 200 # for points at fixed edge, y=0 201 if j == grd jmin : 202 # deflection zero : w=0 203 mj indx = i + j ni 204 row.append(mi indx) ; col.append(mj indx) ; data.append (1.0) 205 f matrix [ mi indx ] = p o [ i, j ]/D 206 mi indx = mi indx ################################################################ 209 # for points adjacent to fixed edge 210 elif j == (grd jmin+1) and ( i >= (grd imin+2) and i<= ( grd imax 2) ) : radius = grd posr [ i, j ] 213 a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13 = get local a coefficient (radius) mj indx = ( i+2) + j ni 216 row.append(mi indx) ; col.append(mj indx) ; data.append(a1) 217 mj indx = ( i+1) + j ni 218 row.append(mi indx) ; col.append(mj indx) ; data.append(a2) 219 mj indx = i + j ni 220 row.append(mi indx) ; col.append(mj indx) ; data.append(a3) 221 mj indx = (i 1) + j ni 222 row.append(mi indx) ; col.append(mj indx) ; data.append(a4) 223 mj indx = (i 2) + j ni 224 row.append(mi indx) ; col.append(mj indx) ; data.append(a5) 47