Chapter 3 Discretization Error in the Random Finite Element Method
3.7 Soil column results for a fixed scale of fluctuation
3.7.1 FE discretization error for a fixed RF mesh
In Figure 3.3, the effects of FE and RF discretization errors are mixed. To focus on the FE discretization error, the results for a fixed RF mesh (lRF /Lx is fixed at 1/4) but a variable FE mesh are shown in Figure 3.4. Although not shown, similar trends as in Figure 3.4 are observed for other fixed values of lRF. It is evident that a coarser FE mesh tends to make the model soil mass “stronger” than that produced by the finest FE mesh.
This is manifested by an increase in E(qu) with a coarser FE mesh. This holds true for both LA and MP discretization methods. This observation is consistent with the general understanding in the FE community that a coarse mesh tends to overestimate the response (Bathe 2014; Lepi 1998). In Section 3.7.4, the effectiveness of different types of FE elements on alleviating this issue will be discussed.
Figure 3.4 The effect of the FE discretization error for the soil column problem (lRF /Lx
fixed at 1/4)
45 3.7.2 RF discretization error for a fixed FE mesh
To focus on the RF discretization error, the results for a fixed FE mesh (finest FE mesh) but a variable RF mesh are shown in Figure 3.5. It is evident that a coarser RF mesh tends to make the model soil mass stronger or weaker than that produced by the finest RF mesh, depending on the discretization method. If the LA discretization method is used, a coarser RF mesh tends to make the model soil mass stronger because weak spots are suppressed by LA. This is manifested by an increase in E(qu) with a coarser RF mesh (see solid line in Figure 3.5). If the MP discretization method is used, a coarser RF mesh tends to make the model soil mass weaker (see dashed line in Figure 3.5), except at a very coarse RF mesh.
Figure 3.5 The effect of the RF discretization error for the soil column problem (lFE is the finest)
The different behaviors between LA and MP are explained in the following.
Figure 3.6 shows a random field realization for su. The same random field realization is simulated by LA and MP methods with two RF mesh sizes (fine and coarse), whereas
the FE mesh is fixed at the finest size. The simulated qu values for the four scenarios are shown in the plots. For this particular random field realization, there is a weak spot (white zone) indicated by the arrow in the plots. When the RF mesh is fine, this weak spot is preserved by both LA and MP (Figure 3.6a and Figure 3.6c). However, when the RF mesh is coarse, the weak spot is suppressed by LA (Figure 3.6b) while it is enlarged by MP (Figure 3.6d) by chance. As a result, a coarser RF mesh in LA tends to make the model soil mass stronger (weak spot is suppressed), whereas this is not necessarily true for MP. There is an exception: for a very coarse RF mesh in MP (e.g., lRF/Lx = 1), the weak spot can disappear (a homogeneous soil mass cannot have a “weak spot”);
therefore, even MP tends to make the model soil mass stronger.
Figure 3.6 A weak spot (the arrow) experiencing different scenarios under LA and MP methods. The finest FE mesh is adopted. The RF mesh is fine (left) and coarse (right)
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Based on the above results, the following observations are made:
1. FE discretization error: For both LA and MP, a coarser FE mesh tends to make the model soil mass stronger.
2. RF discretization error: For LA, a coarser RF mesh tends to make the model soil mass stronger. For MP, a coarser RF does not necessarily make the model soil mass stronger. (conventional RFEM) in Figure 3.3a for LA as an example. These lines are reproduced in Figure 3.7 for ease of illustration. Additionally, consider the solution produced by the finest FE mesh together with the finest RF mesh (the leftmost point in Figure 3.7). This solution will be referred to as the “reference solution” from here on. Consider three scenarios: Points A (conventional RFEM), B (finest FE mesh), and C (reference solution) in Figure 3.7. It is clear that the vertical difference between A and C is the total discretization error for the conventional RFEM. Note that A and B are with the same lRF, but B is with lFE = the finest lFE. It is then clear that the vertical difference between A and B is the FE discretization error for the conventional RFEM. Note that B and C are with the same lFE, but C is with lRF = the finest lRF. It is then clear that the vertical difference between B and C is the RF discretization error for the conventional RFEM. As a result, the FE and RF discretization errors can be separated.
Figure 3.7 The decomposition of the total discretization error for the conventional RFEM into FE and RF discretization errors. The soil column problem with LA (Figure
3.3a) is used as an example.
Figure 3.8 shows how the FE and RF discretization errors vary with mesh size in the conventional RFEM (lFE = lRF = l). For LA (Figure 3.8a), the RF discretization error is larger than the FE discretization error. More importantly, the FE and RF discretization errors are both positive, so the total discretization error becomes even larger: the FE and RF discretization errors “accumulate”. For MP (Figure 3.8b), the FE discretization error is larger than the RF discretization errors (except at very large l).
More importantly, the FE and RF discretization errors have different signs (one positive and one negative, except at very large l) for the current case, so the total discretization error becomes smaller: the FE and RF discretization errors do not accumulate. This suggests that for the conventional RFEM, the MP method may outperform the LA method in the sense that the total discretization error for MP is smaller.
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(b)
Figure 3.8 The variation of FE and RF discretization errors in the conventional RFEM (lFE = lRF = l) using (a) LA and (b) MP, for the soil column problem with = 12.8 m
To demonstrate the advantage of MP over LA for the conventional RFEM, the relative error for the mean value is calculated as = (the mean of the conventional RFEM
solution – the mean of the reference solution)/(the mean of the reference solution). As shown in Figure 3.9, the relative error for MP is closer to zero than that for LA.
Figure 3.9 The relative error of the conventional RFEM (lFE = lRF = l) for the mean value of qu ( = 12.8 m)
Figure 3.9 also shows the recommended mesh size made by Ching and Phoon (2013b) and Huang and Griffiths (2015) for the conventional RFEM to achieve desirable accuracy. Ching and Phoon (2013b) recommended l/ ≤ 0.018 for LA and l/
≤ 0.012 for MP. On the other hand, Huang and Griffiths (2015) recommended l ≤ 0.5“correlation length” in general. Since they adopted the single exponential model, the correlation length = 0.5. Finally, they assigned random field values to Gauss points, that is, lRF = 0.5×lFE. Considering these factors, their recommendation is equivalent to lFE/ ≤ 0.25 or lRF/ ≤ 0.125 in our study, depending on whether random field values are assigned to elements or to Gauss points. Based on Figure 3.9, the
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(2015), while it drops to 0.001~0.004 for Ching and Phoon (2013b). Note that the recommendation made by Ching and Phoon (2013b) was based on whether the probability distribution of qu is significantly changed by a coarse mesh. The change is quantified by the p-value of the Kolmogorov-Smirnov (K-S) test between the RFEM solution and the reference solution. If the p-value is less than 0.05, the change is deemed significant. This criterion based on the p-value of the K-S test is relatively strict.
Figure 3.10 The relative error of the conventional RFEM (lFE = lRF = l) for the standard deviation of qu ( = 12.8 m)
The relative error in Figure 3.9 is for the mean value. Figure 3.10 shows the relative error for the standard deviation, calculated as = (the standard deviation of the conventional RFEM solution – the standard deviation of the reference solution)/(the standard deviation of the reference solution). Both LA and MP have comparable relative errors except at very coarse meshes, where MP has a larger relative error. Note that there is no spatial averaging for MP, so there is no variance reduction. The COV of the
su value simulated by MP is always the same as the point COV of su. In contrast, there is spatial averaging for LA, so there is significant variance reduction at a very coarse mesh.
As a result, the COV for MP is significantly larger than that for LA at a very coarse mesh. It is evident that the COV for LA is closer to that for the reference solution in general. Following Ching and Phoon (2013b) and Huang and Griffiths (2015) recommendations, the relative error for the standard deviations varies from 0.002~0.005 for the former to 0.027~0.080 for the latter.
3.7.4 Influence of FE element type on the conventional RFEM
In Section 3.7.1, it was observed that a coarse FE mesh tends to behave overly strong.
In the literature, the tendency of a coarse FE mesh to overestimate the response (stiffness or strength) is generally known as “overly stiff” behavior. In the context of the current study, it is more appropriate to adopt the term “overly strong” because the term
“stiff” is typically applied on the modulus, not on the strength. However, the term
“overly stiff” will still be adopted to be consistent with the literature.
As mentioned by Bathe (2014), the overly stiff behavior stems from the fact that the displacement assumptions may not accurately describe all the possible modes of deformation. This leads to “internal displacement constraints” that are implicitly imposed on the solution. These constraints can be reduced by mesh refinement or by adopting a suitable element type. This section investigates the effectiveness of different types of FE elements on alleviating the overly stiff behavior in the conventional RFEM.
In particular, the following element characteristics are considered: quadrilateral or triangular element shapes, linear or quadratic shape functions, full or reduced integration, and compatible or incompatible mode formulations. These characteristics are investigated for the soil column problem using the Abaqus software. Table 3.1 lists
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the considered element types in Abaqus. One hundred Monte Carlo simulations are performed for each of the element types in Table 3.1. This analysis is repeated for different mesh resolutions, and the results are shown in Figure 3.11.
(a)
(b)
Figure 3.11 The influence of the FE element type on the overly stiff behavior in the conventional RFEM for (a) LA and (b) MP. The soil column problem is considered with
= 12.8 m.
Table 3.1 Different types of plane strain elements for FE mesh with their names in
CPE4R 4-node bilinear, reduced integration with hourglass control
CPE6 6-node quadratic
CPE8 8-node biquadratic
CPE8R 8-node biquadratic, reduced integration
Based on the overly stiff behavior, the element types are categorized into three groups:
1. The first group consists of the 3-node triangular element (CPE3). CPE3 is extremely stiff because the strain is constant over the entire element. Its significant overly stiff behavior is evident from the fact that the convergence of E(qu)/E(qu,s) at the fine (CPE6). These element types exhibit a less overly stiff behavior than CPE3. CPE6 and CPE8 perform better than CPE4, with the cost of more computation time. The computation time for CPE4I is similar to that for CPE4, but the performance for CPE4I is comparable to that of CPE6 and CPE8 because CPE4I implements incompatible elements that can alleviate the overly stiff behavior (Bathe 2014).
3. The third group consists of 4-node and 8-node reduced integration quadrilateral elements (CPE4R and CPE8R). This is the most effective group, with the least
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overly stiff behavior. The strategy of reduced integration can alleviate the overly stiff behavior (Bathe 2014). However, the strategy of reduced integration may lead to the “hourglassing” issue (Lepi 1998), especially for a coarse mesh with linear elements. From Figure 3.11, CPE4R seems to suffer from this issue when the mesh is very coarse, while CPE8R does not.
In summary, among the FE element types investigated above, CPE4R and CPE8R are the most effective ones to alleviate the overly stiff behavior. For a very coarse mesh, CPE8R should be preferred over CPE4R because CPE4R may suffer from the hourglassing issue. Note that as mentioned in the Section 3.5, the present study employs CPE8R.
3.8 Retaining wall results for a fixed scale of fluctuation
For the retaining wall problem, = 0.2×H = 1 m is considered. Similar analyses as in the soil column problem are performed for the retaining wall problem. Figure 3.12 shows the mean value of Pa, E(Pa), normalized by the mean value of Pa,s, E(Pa,s). The qualitative observations made for the soil column problem still hold true for the retaining wall problem:
1. FE discretization error: For both LA and MP, a coarser FE mesh tends to make the model soil mass stronger (Figure 3.13).
2. RF discretization error: For LA, a coarser RF mesh tends to make the model soil mass stronger. For MP, a coarser RF mesh does not necessarily make the model soil mass stronger (Figure 3.14).
3. The conventional RFEM: The FE and RF discretization errors tend to accumulate for LA, but they may not accumulate for MP. For LA, the RF discretization error is larger, whereas for MP, the FE discretization error is larger (Figure 3.15).
4. The conventional RFEM: The relative error for the mean value of Pa for MP is closer to zero than that for LA (Figure 3.16). On the other hand, both LA and MP have comparable relative errors for the standard deviation of Pa, except at very coarse meshes where MP has a larger relative error (Figure 3.17). The MP method may outperform the LA method in the sense that the total discretization error is smaller.
(a)
(b)
Figure 3.12 The normalized mean Pa for δ=1 m using (a) LA and (b) MP
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Figure 3.13 The effect of the FE discretization error for the retaining wall problem (lRF/H fixed at 1/2.5)
Figure 3.14 The effect of the RF discretization error for the retaining wall problem (lFE
is the finest)
(a)
(b)
Figure 3.15 The variation of FE and RF discretization errors in the conventional RFEM (lFE = lRF = l) using (a) LA and (b) MP for the retaining wall problem with = 1 m
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Figure 3.16 The relative error of the conventional RFEM (lFE = lRF = l) for the mean value of Pa ( = 1 m)
Figure 3.17 The relative error of the conventional RFEM (lFE = lRF = l) for the standard deviation of Pa ( = 1 m)
3.9 Monte Carlo results for various scales of fluctuation
This section studies the discretization error for various SOFs. For the soil column problem, a dimensionless ratio of SOF/(width of column), /Lx, is considered. For the retaining wall problem, a dimensionless ratio of SOF/(height of wall), /H, is considered. Both /Lx and /H are then chosen from {0.02, 0.1, 0.2, 0.5, 1, 2, 10, 50, 300}. Although not shown, the qualitative observations made in the previous sections still hold true for various SOFs. In the following, the discretization error for the conventional RFEM for various SOFs is investigated.
3.9.1 Discretization error for conventional RFEM for various SOFs
Figure 3.18 and Figure 3.19 show, respectively, the normalized mean of qu for the soil column and that of Pa for the retaining wall using the conventional RFEM for various SOFs. The results are based on two thousand Monte Carlo simulations for each combination of SOF and mesh resolution. It is interesting to note that the “worst case”
SOF depends on the mesh size l. The worst case SOF refers to the SOF where the discrepancy between the mean response [E(qu) or E(Pa)] and the nominal response [E(qu,s) or E(Pa,s)] is the largest. The worst case SOF has been of particular interest in recent studies (e.g., Allahverdizadeh et al. 2016; Ching et al. 2017a; Fenton et al. 2005), as it can be used as a basis for design when it is difficult to obtain a site-specific SOF from limited soil data. From Figure 3.18 and Figure 3.19, it can be seen that, as the mesh size becomes coarser, not only the extremum point of E(qu)/E(qu,s) or E(Pa)/E(Pa,s) becomes closer to unity but also the corresponding worst case SOF may change as well.
Moreover, this behavior of changing is also affected by whether LA or MP is adopted.
In principle, the worst case SOF obtained from the reference solution (i.e., the solid lines in Figure 3.18 and Figure 3.19) with the finest FE and RF mesh should be close to
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the “actual” worst case SOF. The worst case SOF obtained from a coarse mesh can be far away from the actual.
(a)
(b)
Figure 3.18 The normalized mean qu for various SOFs for the soil column problem. The conventional RFEM (lFE = lRF = l) is employed with (a) LA and (b) MP
(a)
(b)
Figure 3.19 The normalized mean Pa for various SOFs for the retaining wall problem.
The conventional RFEM (lFE = lRF = l) is employed with (a) LA and (b) MP
3.9.2 Suggestions for allowable mesh size
This section provides suggestions for the allowable normalized mesh size, (l/)allow, to achieve a certain error tolerance. The suggestions are based on the relative error with
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respect to the reference solution. Recall that the relative error in Figure 3.9 and Figure 3.10 (soil column) and Figure 3.16 and Figure 3.17 (retaining wall) is defined as (the mean of the conventional RFEM solution – the mean of the reference solution)/(the mean of the reference solution) or (the standard deviation of the conventional RFEM solution – the standard deviation of the reference solution)/(the standard deviation of the reference solution). (l/)allow is selected such that the absolute value of the relative error does not exceed a certain “error tolerance” for both the relative errors in the mean value and standard deviation. Two error tolerances are considered: 0.01 and 0.05. Taking the LA discretization method for the soil column problem as an example, Figure 3.9 and Figure 3.10 indicate that (l/)allow = 0.034 if the 0.01-tolerance is adopted, and (l/)allow = 0.126 if the 0.05-tolerance is adopted. However, Figure 3.9, Figure 3.10, Figure 3.16, and Figure 3.17 are for fixed SOFs ( = 12.8 m for the soil column and = 1 m for the retaining wall). In general, (l/)allow may depend on SOF. The most conservative (l/)allow value over various SOFs should be adopted for conservatism. Table 3.2 lists the values of (l/)allow for the soil column and retaining wall problems. According to Table 3.2, the following suggestions can be obtained:
1. If the error tolerance is 0.01 (|relative error| 0.01), (l/)allow is approximately 0.03 for LA and is approximately 0.04 for MP.
2. If the error tolerance is 0.05 (|relative error| 0.05), (l/)allow is approximately 0.1 for LA and is approximately 0.2 for MP.
The (l/)allow values for MP are larger than those for LA. This is consistent with the previous observation: for the conventional RFEM, the MP method outperforms the LA method in the sense that the total discretization error is smaller. Table 3.2 also lists the recommendations made by Ching and Phoon (2013b) and Huang and Griffiths
(2015). The recommendation made by Ching and Phoon (2013b) is even stricter than the 0.01-tolerance, whereas the recommendation made by Huang and Griffiths (2015) is comparable to the 0.05-tolerance.
Table 3.2 Allowable l/, (l/)allow, for 0.01-tolerance and 0.05-tolerance
Case undrained shear strength related to the soil column and retaining wall examples with a separable single exponential model. These suggestions may be restricted to the current numerical problems. They may no longer be valid if, for example, a different autocorrelation model or a different response quantity of interest is adopted. Moreover, the suggestions should not be applied to problems with a spatially variable Young’s modulus, as Ching and Hu (2016) pointed out that the allowable mesh size for a spatially variable Young’s modulus is quite different from that for spatially variable undrained shear strength.
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Chapter 4 Weakest-Path Model for Mobilized Shear Strength
4.1 Introduction
This chapter focuses on the weakest-path model (WPM). The WPM is a simplified model that receives statistics of “point” shear strength, and outputs statistics of
“mobilized” shear strength. This is of practical value because, on one hand (input), engineers may be able to obtain statistics of point shear strength from site investigation data. On the other hand (output), the mobilized shear strength is a desirable strength parameter for engineers because it is the shear strength that affects the occurrence of the limit state. Moreover, compared to the RFEM or RLEM, the WPM is cheap to calculate.
The WPM, however, requires calibration. Hence, this chapter also presents six geotechnical problems, together with three scenarios of ground properties to be used for the calibration of the WPM. The calibration results will be presented in the next chapter.
4.2 Concepts of WPM
The WPM is originally developed by Ching and Phoon (2013a). This model has been applied to a retaining wall (Hu and Ching 2015), a 3D soil column (Ching et al. 2016b), and a basal heave problem (Ching et al. 2017a). The WPM is based on the observation that the mobilized shear strength (Xmob) is approximately equal to the spatial average along the critical slip curve (Ching and Phoon 2013a). Note that the critical slip curve is
The WPM is originally developed by Ching and Phoon (2013a). This model has been applied to a retaining wall (Hu and Ching 2015), a 3D soil column (Ching et al. 2016b), and a basal heave problem (Ching et al. 2017a). The WPM is based on the observation that the mobilized shear strength (Xmob) is approximately equal to the spatial average along the critical slip curve (Ching and Phoon 2013a). Note that the critical slip curve is