from the higher computational efforts due to the inversion of theJacobian and the larger bandwidth of the system matrices coming along with quadratic interpolations if the stable QL approximation is used.
The semi-explicit-implicit splitting scheme has been implemented in form of the predictor-corrector (P-C) algorithm. In contrast to the monolithic solution, the splitting method demands the time discretisation and the splitting of the equations to be firstly performed, which is followed by the spatial discretisation. Moreover, this scheme allows for a con-tinuous and linear equal-order FE approximation. Due to the included explicit steps, the stability of the splitting method is sensitive to the time-step size∆tbut independent of the permeability. In this regard, an optimal solution is obtained for ∆t being slightly smaller than the critical time step ∆tcr given by the CFL condition. The use of∆t≪∆tcr yields pressure oscillations and requires a stabilised formulation. The general drawback that the finest mesh patch or the highest wave speed (CFL condition) dictates the global time step might be overcome by use of multirate time-stepping strategies (Savencoet al.[152]).
Further demerits of the considered P-C algorithm are the need for finer meshes in order to obtain similar accuracy in the displacement field in comparison with the monolithic solution and the required special treatment of the volume efflux boundary condition.
6.3 Wave Propagation in an Elastic Structure-Soil
6.3 Wave Propagation in an Elastic Structure-Soil Half Space 131
The modelling of soil is restricted to the case of a materially incompressible biphasic model with linear elastic behaviour of the solid skeleton and using the balance relations and the way of treatment as explained in Section 4.4 . Such soil-structure interaction problems have been intensively discussed in the literature, cf. von Estorff [68], von Estorff
& Firuziaan [69], Kim & Yun [103] and Heider et al. [87] among others. In this regard, the overall behaviour is affected by a number of factors such as the parameters of the subsystems and the structural embedment in the foundation soil.
In the current problem, the block (4×2 m) is considered to be in a welded contact with the soil beneath and discretised with the same type of finite elements as for the soil.
The time discretisation is performed using the implicit monolithic TR-BDF2 scheme with unified time-step size ∆t = 10−3 s. The applied shear impulse force is given by
f(t) = 104[1−cos(20π t)] [1−H(t−τ)] N/m2 (6.3) with H(t−τ) being the Heaviside step function and τ = 0.1 s. As mentioned in Sec-tion 4.4, the time-dependent damping terms (4.89) enter the weak formulaSec-tion of the problem in form of nonlinear boundary integrals. Thus, an unconditional stability of the numerical solution requires that these terms are integrated over the boundary at the current time step in the sense of weakly-imposedNeumann boundary conditions. The dy-namic response of the soil and the efficiency of the considered infinite boundary treatment are investigated for two cases:
Case (1): the material parameters of the block and the soil are the same (cf. Table 6.2) with kF = 10−2m/s .
Case (2): the block is considered to be made of concrete6 with parameters given in Ta-ble 6.9, whereas the soil parameters are taken from TaTa-ble 6.2 with kF = 10−2 m/s .
Parameter Symbol Value SI unit
1st Lam´e constant µS 1.25·1010 N/m2 2nd Lam´e constant λS 8.30·109 N/m2
effective density ρSR 2800 kg/m3
initial volume fraction nS0S 0.99 –
Darcy permeability kF 10−6 m/s
Table 6.9: Material parameters of elastic concrete
A benchmark solution, referred to as Ref. FE, is generated by considering l1 =l2 = 40 m in Figure 6.29, left, where the choice of large dimensions guarantees that no reflected waves propagate back to points A, B and C during the analysis. The efficiency of the proposed boundary treatment is measured by comparing the displacements at different points in the domain (A, B and C) for two types of boundaries, i. e., for FE-IE with VDB as in Figure 6.29, right, and for FE-fix as in Figure 6.29, left, with l1 =l2 = 20 m.
6The numerical simulation of concrete using a multi-phase material model is considered by choosing nS0S = 0.99, i. e., the concrete is treated as an almost single phasic, linear elastic solid skeleton.
In case (1) with unified material parameters, the stiffness ratio of the concrete block to that of soil is EB/ES = 1, with EB, ES being the elasticity moduli7 of the concrete and the soil solid skeleton, respectively8. In this case, the vibration of the block damps out in a weak manner and results in a successive wave transition into the supporting soil. In this connection, Figure 6.30 shows exemplary contour plots of the computed solid displacement field, which makes the wave propagation and the weak damping behaviour apparent.
t= 0.15 s t= 0.30 s
t= 0.45 s t= 0.60 s
8 4
1
x1
x2
uS
uS
[10−3m ]
Figure 6.30: Time sequence of solid displacement uS = q
(u2S1+u2S2) contour plots for case (1) with EB/ES = 1
In Figure 6.31, the time history of the horizontal displacement uS1 of point C at the top of the block with weakly damped motion is depicted. Therein, a good agreement among the different solution strategies is obtained as far as the reflected waves do not propagate back to point C. However, the solution with FE-fix deteriorates after a certain time due to the interfere of the reflected waves.
For the two points A and B in the soil domain, the time history of the horizontal dis-placement uS1 is plotted in Figure 6.32. The efficiency of the FE-IE scheme in impeding the reflecting waves is obvious by comparison with the reference solution Ref. FE. How-ever, the FE-fix solution violates this agreement due to the overlapping of progressing
7The Young’s modulus is computed asES=µS(2µS+ 3λS)/(µS+λS) .
8EB/ES ≪1 represents the case of a soft block founded on a rigid base [68], which is not a case of study in this contribution.
6.3 Wave Propagation in an Elastic Structure-Soil Half Space 133
-80 -40 0 40 80 120
0 0.2 0.4 0.6 0.8 1
Ref. FE FE-fix FE-IE
uS1[10−4 m]
timet [s]
Figure 6.31: Horizontal displacement time history at point C for case (1) withEB/ES = 1 and reflected waves.
-12 -8 -4 0 4 8 12
0 0.2 0.4 0.6 0.8 1
Ref. FE FE-fix FE-IE
uS1[10−4m]
time t[s]
-12 -8 -4 0 4 8 12
0 0.2 0.4 0.6 0.8 1
Ref. FE FE-fix FE-IE
uS1[10−4 m]
time t[s ]
Figure 6.32: Horizontal displacement time history for case (1) with EB/ES = 1 at point A(0, 15) (left) and at point B(-5, 10) (right)
In case (2), the stiffness of the concrete block is higher than that of soil (EB/ES = 2.1×103), which leads to a strong damping of the block motion. For this case, Figure 6.33 shows exemplary contour plots of the computed horizontal solid displacement for the discussed three different boundary cases, and clearly reveals the influence of the boundary conditions on the reflected waves.
The maximum response of the block decreases in comparison with case (1) and the motion damps out very strongly. Accordingly, only one wave corresponding to the impulse loading appears and radiates towards the infinity. Moreover, Figure 6.34, left, shows that due to the high stiffness difference between the block and the soil beneath, the reflected waves do not propagate into the block as the displacement at point C(2,-2) is not disturbed.
For a point in the domain (here: B(-5,10)), Figure 6.34, right, depicts the time history of the horizontal displacement uS1 considering the different boundary treatments. Here, the role of the damping boundary in reducing the effects of the reflecting waves is further
t= 0.30 s, wave progression from the near field
t= 0.60 s, waves approach the far field
7 0
−4 uS1 [10−7m]
Figure 6.33: Time sequence of displacement uS1 contour plots for case (2): Ref. FE (left), FE-IE with VDB (middle) and FE-fix (right)
enhanced by comparing the FE-fix and the FE-IE results with the reference Ref. FE solution.
In conclusion, it is apparent that the proposed VDB method can significantly but not perfectly prevent the wave reflection back to the near field. This is due to the fact that absorption of the approaching waves cannot be made perfect over the whole range of the
-20 0 20 40 60 80
0 0.2 0.4 0.6 0.8 1
Ref. FE FE-fix FE-IE
uS1[10−7 m]
timet [s ]
-4 0 4 8
0 0.2 0.4 0.6 0.8 1
Ref. FE FE-fix FE-IE
uS1[10−7 m]
timet [s]
Figure 6.34: Horizontal solid displacement time history for case (2) with EB/ES ≫ 1 at point C(2,-2) (left), and at point B(-5,10) (right)