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CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 1 FINITE DIFFERENCE SOLUTION OF CONSOLIDATION PROBLEMS 1-D Consolidation Equation: 2 2v u uc z t = Forward difference explicit form Spatial and temporal discretization Subscript I refers to time discretization and J to spatial discretization. Writing the difference equation corresponding to the differential equation at (I,J): , 1 , 1 , 1, , 2 2I J I J I J I J I J v u u u u u c z t + ++ = ( ), 1 , 1 , 1, ,2 2v I J I J I J I J I J tc u u u u u z + + + = Let 2v tc C z α = = where C = Courant number. 1, , 1 , 1 ,(1 2 )I J I J I J I Ju u u uα α α+ += + + CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 2 To start with, at t=0, we know all the values as initial condition, so calculate step by step to get 1,I Ju + : The scheme is stable only for 2 1 2v tc z ≤ (see Fig. Below), so choose t in such a way that 2 0.5v tc z α = ≤ Comparison of explicit FD and analytical solution of Terzaghi’s consolidation equation for T=0.50. CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 3 Fully Implicit Backward Difference Write the difference equation at (I + 1, J ) 1, 1 1, 1 1, ,(1 2 )I J I J I J I Ju u u uα α α+ + + ++ + = at t=0 (known) at t= t (unknown) This is not an explicit form, so you cannot march in time step by step as before. But you have to write a matrix form of the equations and solve for each time step and proceed accordingly: Example: 1. 1 0u = 2. 1 2 3 100u u uα β α+ + = 3. 2 3 4 100u u uα β α+ + = 4. 3 4 5 100u u uα β α+ + = , (1 2 )β α= + 5. 4 5 6 100u u uα β α+ + = 6. 5 6 7 100u u uα β α+ + = 7. 7 0u = Matrix Form 1 2 3 4 5 6 7 1 0 0 0 0 0 0 0 100 100 100 100 100 0 0 0 0 0 0 1 0 u u u u u u u α β α α β α α β α α β α α β α = Tri-diagonal matrix Constant known values CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 4 The matrix eqn. is symbolically written as: [ ]{ } { }A u R= Solving for { } [ ] { }1u A R = Proceed to the next step by writing the tri-diagonal matrix at that time level and solve for {u}. This is fully implicit backward difference method is unconditionally stable (see Figure below) and any value of t can be used. Comparison of implicit FD and analytical solution of Terzaghi’s consolidation equation for T=0.50. CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 5 EXPLICIT VS. IMPLICIT TIME INTEGRATION SCHEMES 1. Explicit schemes require only information from the previous time step to advance to the next time step, while implicit methods require information for the next time step from neighboring nodes/elements to proceed to the next time step. 2. The implicit solution method requires matrix inversion of the stiffness matrix, the explicit solution does not. 3. The implicit solution scheme is unconditionally stable for large time steps, while the explicit scheme is stable only if the time step size is smaller than the critical time step size for the problem being simulated. 4. A very small time step size requirement for stability makes explicit solutions useful only for very short transient problems. But, even though the number of time steps in an explicit solution may be orders of magnitude greater than that of an implicit solution, it is significantly more efficient than an implicit solution since no matrix inversion is required. 5. Neither an implicit nor explicit solution is the clear winner in all cases. General ‘θ’ scheme Write the equation at θt time level: 2 2 t t v u uc z tθ θ = (1) Taking weighted average ( )1t t t tf f fθ θ θ + = + ( ) 2 2 2 2 2 21t t t t u u u z z zθ θ θ + = + (2) ( )1t t t t u u u t t tθ θ θ + = + (3) CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 6 One-Step Euler Integration .t t t t uu u t t θ + ≈ + .t t t t uu u t t θ + ≈ + ( )1 .t t t t t t u uu u t t t θ θ+ + ≈ + + or ( )1t t t t t t u u u u t t t θ θ+ + = + Using (2), (3), and (4) in eqn. (1) ( ) 2 2 2 21 t t t v t t t u uu uc z z t θ θ + + + = (5) Note in eqn. (4) 0θ = t t t t u u u t t + = forward difference 1θ = t t t t t u u u t t θ+ + = backward difference Writing eqn. (5) in difference form: 1, 1 1, 1, 1 , 1 , , 1 1, , 2 2 2 2 (1 )I J I J I J I J I J I J I J I Jv u u u u u u u u c z z t θ θ+ + + + + + + + + = (6) CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 7 Put 2 vc t z α = Rearrange: (
)1, 1 1, 1, 1 , 1 , , 12 1 (1 ) (2 (1 ) 1) (1 )I J I J I J I J I J I Ju
u u u u uαθ αθ αθ α θ α θ α θ+ + + + + + + = + (7) In matrix form: (8) 1 2 θ ≥ unconditionally stable i.e., you can choose any t 1 2 θ < 2 1 2 (1 )v t c z θ ≤ ; 0θ = 2 1 2 vc t z ≤ For each time step, write the matrix eqn. (8) [ ]{ } { }A u R= Solve for { } [ ] { }1u A R = Types of time integration schemes: 0θ = Fully explicit (forward difference) 1θ = Fully implicit (backward difference) 1 2 θ = Crank-Nicholson 2 3 θ = Euler 1 2 θ < Conditionally stable so you choose small t but you sacrifice accuracy. unknown known ( ) 1 2 . . . . .2 1 .. . . .n u u R u αθ αθ αθ = + CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 8 Boundary conditions Both face drained FD solution for consolidation of layered soils Taylor series expansion: 2 3 ' '' '''( ) ( ) ( ) ( ) ( ) 2 6 z zf a z f a zf a f a f a + = + i.e., a function can be expanded around a neighborhood point (z = a). Expansion at point (I,J): 2 2 1 , 1 , 1 22I J I J zu uu u z z z+ = + 2 , 1 , 12 2 1 2 I J I J u uu u z z z z+ = + Layer 1 2 1 2 u uc z t = 1, ,1 , 1 , 12 1 2 I J I J I J I J u uc uu u z z z t + + + = 1, ,1 1 1 , 1 ,2 2 1 1 1 2 2 2I J I J I J I J u uc c cu u u z z t z z + + = + Impervious boundary Take a mirror Image point A’ to A Image point B’ to B CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 9 1, ,1 1 1 , 1 ,2 2 1 1 1 2 2 2 I J I J I J I J u uz c cu u u z c t z z + + = + Similarly in layer 2 1, ,2 2 2 , 1 ,2 2 2 2 2 2 2 2 I J I J I J I J u uz c cu u u z c t z z + + = + Interface B.C. (continuity of flow): 1 2 1 2 u uk k z z = 1, , 1, ,1 1 1 2 2 2 1 , 1 , 2 , 1 ,2 2 2 2 1 1 1 2 2 2 2 2 2 2 2 2 I J I J I J I J I J I J I J I J u u u uz c c z c ck u u k u u c t z z c t z z + + + + + = + 1, , 1, ,1 1 2 2 1 2 2 , 1 , , 1 ,2 2 2 2 1 1 1 1 2 2 2 2 2 2 2I J I J I J I J I J I J I J I J u u u uc c k z c c cu u u u t z z k z c t z z + + + + + = + Let 1 1 2 1 c z β = ; 2 1 1 2 1 2 2 c k z c k z β = ; 1 2 3 2 1 z k z k β = 1, , 1, , 2 2 1 , 1 1 , , 1 ,2 2 2 2 2 2 212 2I J I J I J I JI J I J I J I J u u u u c cu u u u t t z z β β β + + + + + = + 1, , 2 2 1 , 1 , 1 , 12 2 2 2 2 2 2 2 21 1 11 2 2 .I J I J I J I J I J u u c cu u u t z z β β β β β + + + + = + + Which can be written as: ( ){ }1, , 1 , 1 , 3 , 1 , 2 2 11 I J I J I J I J I J I J u u u u u u t β β β + + = + + ( ){ }11, , , 1 3 , 1 3 , 2 2 111I J I J I J I J I J tu u u u uβ β β β + + = + + CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 10 Define: 1 2 2 11 t Aβ β = + 1 3 2 2 11 t Bβ β β = + ( )1 3 2 2 1 11 t Cβ β β + = + 1, , , 1 , , 1(1 )I J I J I J I J I Ju u Au C u Bu+ + = + + Explicit forward difference For the case 1 2z z z = = , 2 1k k k= = , 2 2c c c= = 1 1 2 c z β = ; 2 1β = ; 3 1β = 2 c tA z α = = , B α= , 2C α= 1, , 1 , , 1(1 2 )I J I J I J I Ju u u uα α α+ += + + General θ scheme: 1, 1 1, 1, 1 , 1 , , 1(1 ) (1 ) ( (1 ) 1) (1 )I J I J I J I J I J I JAu C u Bu Au C u Buθ θ θ θ θ θ+ + + + + + + = + 1, , 1 , , 1(1 )I J I J I J I Ju Au C u Bu+ += + + unknown known CA4682 Advanced Geotechnical and Foundation Engineering Lecture Notes Dr. Jianfeng Jeff Wang 11 . . . . . . . . ( ) (1 ) ( ) . . . . . . . . A C B U Rθ θ θ = + The above algorithm is the most general for layered system with different thicknesses and layer properties.