[] [] [] Application of Finite Difference Method with Spreadsheet to Solve Seepage, Consolidation and Wave Propagation Problems Zheng-yi Feng [] Jia-Chi Liang [] Y-hsan Chang [] [] Assistant Professor, Department of Soil and Water Conservation, National Chng Hsing University, Taichng 40, Taiwan, R.O.C.(Corresponding Athor) Email: tonyfeng@dragon.nch.ed.tw [] Gradate stdent, Department of Soil and Water Conservation, National Chng Hsing University, Taichng 40, Taiwan, R.O.C.
Abstract This paper discsses how to apply finite difference method (FDM) with spreadsheet to solve partial differential eqations (PDE) of elliptic, parabolic and hyperbolic types. Examples of grondwater seepage, soil consolidation, earthqake shear stress wave propagation and sea wave propagation are illstrated. They are solved by the FDM with the proposed spreadsheet models. The nmerical reslts obtained by the spreadsheet model are verified by analytical or other nmerical soltions with satisfactions. The setp of the spreadsheet model for solving PDEs is qick and easy and the plots can be viewed immediately. The spreadsheet for PDEs in this stdy is fll of flexibility for modification to reflect varios initial and bondary conditions. The spreadsheet model developed is an effective tool and can be readily applied in solving practical engineering problems sch as seepage, consolidation and wave propagation. Keywords: finite difference method, spreadsheet, partial differential eqation, seepage, consolidation, wave propagation.
Kreyszig, Foxes Team005 999 Laplace Eqation Wave Eqation Heat A + B + C + f,,,, x t Eqation x y t x y B 4C < 0 B 4C B 4C > 0 Dirichlet Bondary Lehre 004 sccessive over- relaxation Poisson's eqation. Laplace Eqation Htchens and Gpta000 = + x y VBA Harr990 Relaxation Method Excel Laplace Relaxation Method Elliptic Eqation Parabolic Eqation x x=h Hyperbolic Eqation y y = k
(3) Laplace Eqation d r = + i, + i,,, L, n,,, L, m h k r = + r i+, ( + r ) + r h x= y r = Condction Eqation i, + i+, + i, = h 3,, 4 i + i + i, + i+, + + + = 4 4 t = k 4 a 9 φ = ( φ + φ3 + φ4 + φ5 ) 5 4 φ ( ) Dirichlet Bondary Nemann Bondary Dirichlet Bondary i + = 5Nemann Bondary Flx Flx=0 φ 5 = ( φ3 + φ6 + φ8 ) + ( φ4 + φ7 ) 8 4 8. Heat z Nemann Bondary i,,, L, n,,, L, m * α = α k / h h k () Nemann Bondary b 0 < α < / φ 6 = ( φ7 + φ8 ) + ( φ9 ) 6 4 Excel cell () Nemann Bondary Excel c iexcel φ 0 = ( φ) + ( φ ) 7 = α t, 0 < z < H, > 0 t 9 z n z = h = L / n t = z α = α t + k i, i+, ( α ) *, α i, + + α * h + i+ 0
φ 3 4 + 4 = ( φ + φ + φ φ ) φ = ( φ + φ ) + ( φ ) 5 a b Fig.a Potential of grid point not at bondary Fig.b Potential of grid point at one impervios bondary 6 4 7 8 9 φ 0 = ( φ) + ( φ ) φ 5 = ( φ3 + φ6 + φ8) + ( φ4 + φ7 ) 4 8 c d Fig.c Potential of grid point at two impervios bondary Fig.d Potential of grid point at the bottom of sheetpile
( x 0) f ( x), = t t t = k ( f x) Dirichlet Bondary Nemann Bondary Dirichlet Bondary ( H, t) ( ) 0 0, t = i, Excel 0 Nemann + + = Bondary c k z + n, n ( zn, t ) = +, h = f n n n, + i = α n, + ( α ) n, + hα f ( H, t) z n, n, f n 0 < k / h < + = n+ x i t n- 4 4 3 n, + = n, + ( α ) n, α 5 6 8 time steps Laplace Eqation Nemann - 678 Excel V i ( ) x, t = V 3. + ( xi, t0 ) t t Wave Eqation 9 + = = V i =, 0 < x < L, t > 0 6 k x c t 9 8 x α = ( U i+ + U i ) 0 n x = h = L / n + α U + kv h + i+, + = α + i,,, L, n,,, L, m ( i+, + i, ) ( α ) 7 8 α = c k / h h k 3 8 t ( i 0 ) i ( ) i i
Dirichlet Nemann Bondary Dirichlet Bondary Excel Nemann Bondary Darcy s Law z + n, n ( zn, t ) = +, h = f n n+, h f n + n, = n f i Hydralic Gradient x fi i, i, i x = y + = 50 (Laplace ) h h + x y x h 8 + f y 8 0 36 k = k =0.03 m/sec =0 m =8 m =36 m k =k =0.03m/sec ( Harr, 990) Fig. Example of grondwater seepage of sheetpile (modified from Harr,990)
y 3 0.0000 0000 4 Excel Ie 0.6 q Excel Terzaghi q q 0.3 z α t z t α C = Ie 0.6 C v 3 Polbarinova-KochinaHarr, 990 t 5 z e tanπε = k k k = k e k 0.5 s / t T I e / h.6 0.5 Ie 0.6 = q 0.3 % 6. Excel 0 ( H, t) 5 Taylor (948) U60% 0 < z < H t > 0 3 T π v = 4 U 4 = = v U x h y 0 q / k h.5 q 0.5 B.C.0 = Nemann Bondary z
3 Fig.3 Spreadsheet setp in the case for sheetpile
4 Fig.4 Calclation reslt by Finite Difference method in the case of sheetpile
I e T/h 5 I e T/h (Harr,990) Fig.5 Analytical soltion of I e T/h(Harr,990) q/k h 6 q/k h (Harr,990) Fig.6 Analytical soltion of q/k h(harr,990)
U60% T v T = C t ( %) =.78 0.933log 00 U 5 v v / H dr 0.4 U60% H 4 4.% dr H/ T V =0.97 U=50% T V =0.848 H dr H 9 7 C v 8 0 8 m /sec 400 ( cλ z t e n ) t (, ) = sin λn z, n=,3,5... z, 0 = kpa 8 nπ ( ) 00 nπ λn =, n=,3,5... B.C. ( 0, t) l z ( H, t) I.C. ( z,0) = 0 0 kpa x y 0. 3. domainexcel 8 8 v.s. 0 kpa 88. kpa 8 T V U=90% 9 0.~0.9 9 z 0m C v = 0 < z < H t > 0 z t B.C. ( 0, t). I.C. Z ( H, t) ( z,0) 0Kpa m 7 ( m) Fig.7 Example of consolidation of soil(thickness of soil layerm)
6 = c t 0 < x < L t > 0 x 9 Fig.9 Comparison of the reslts for the consolidation of soil 6 c c = E ρ c = ( λ + G) ρ 0 c = G P h ( 0.sin πt ) H Z 400 c = gh B.C. B.C. ( 0, t).sin π t c = 998 6 I.C. 0 gh
B.C. x ( 400, t) 0 z 400m I.C ( z,0) t ( z,0) c=00 m/sec t=0 B.C. ( 0, t).sin ωt ω = πf, f = Hz ( ) x 0 Fig.0 Example of earthqake Shear wave propagation ( 400, t) 4 I.C. ( z,0) Maple003 t ( z,0) 4 Maple Excel 0.05~4.5 f n = c 4l Richart et al., 970 c = 4lf n c = 4 400 H Z =600m/sec 3 h << l << h ( h 6 I.C.B.C. 400 00m/sz z = l =400m 0 B.C. ( 0, t) x x ( 400, t) π z I.C. ( x,0).sin( ) 400 t ( x,0) c
Maple Fig. Comparison of the reslts by spreadsheet and Maple software (c=600m/sec) Fig. Realtive displacement of the srface and baserock at when resonance
x x w(x, t) l=400m h ( ) = c t x ( 0, t) = x B.C. 0 x ( 400, t) I.C. ( x,0).sin( ) t ( x,0) (c=00m/sec) πz 400 3 (,998) Fig.3 Example of the shallow water long wase 4 Fig.4 Comparison of the reslts for shallow wave long wave
Maple School of Civil Engineering, Prde University, Dover Pblications, Inc., New 5. Kreyszig, E. (999), Advanced Engineering Mathematics, 8 th edition, John Wiley & Sons, c 6. Maple (003), Maple User Manal V.9, Maplesoft, Waterloo Maple, Inc. 7. Richart, F.E., Jr., Hall, J.R., Jr. and Woods, R.D. (970), Vibrations of Soils and Fondations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey. 8. Taylor, D.W. (948), Fndamentals of Soil Mechanics, John Wiley & Sons, Inc., New York. Excel 9. (998) ISBN. Lehre, A. (004), Department of Geology, Hmboldt State University. Homepage. http://www.hmboldt.ed/~geology/corses/g eology556/556_excel_gw_models_index.html Accessed Oct 7, 005.. Foxes Team (005), Homepage. http://digilander.libero.it/foxes/diffeq/fd_spr eadsheet.htm Accessed Oct 7, 005. 3. Htchens, G.J. and Gpta, N.K. (000), A Spreadsheet Program for Steady-State Temperatre Fields, Homepage. http://sti.srs.gov/flltext/tr0003/tr0003. html Accessed Oct 7, 005. 4. Harr, M.E. (990), Grondwater and Seepage, York. I.C. B.C.Inc., New York. 957-950-63-6