Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Inverse Parameter Identification Technique Using PSO Algorithm Applied to Geotechnical Modeling

Inverse Parameter Identification Technique Using PSO Algorithm Applied to Geotechnical Modeling Hindawi Publishing Corporation Journal of Artificial Evolution and Applications Volume 2008, Article ID 574613, 14 pages doi:10.1155/2008/574613 Research Article Inverse Parameter Identification Technique Using PSO Algorithm Applied to Geotechnical Modeling 1 1 2 2 1 Joerg Meier, Winfried Schaedler, Lisa Borgatti, Alessandro Corsini, and Tom Schanz Labor fur ¨ Bodenmechanik, Bauhaus-Universitat ¨ Weimar, Coudraystraße 11 C, 99421 Weimar, Germany Dipartimento di Scienze della Terra, Universita ` degli Studi di Modena e Reggio Emilia, Largo Sant’ Eufemia 19, 41100 Modena, Italy Correspondence should be addressed to Winfried Schaedler, winfried.schaedler@uni-weimar.de Received 24 July 2007; Accepted 14 January 2008 Recommended by Jim Kennedy This paper presents a concept for the application of particle swarm optimization in geotechnical engineering. For the calculation of deformations in soil or rock, numerical simulations based on continuum methods are widely used. The material behavior is modeled using constitutive relations that require sets of material parameters to be specified. We present an inverse parameter identification technique, based on statistical analyses and a particle swarm optimization algorithm, to be used in the calibration process of geomechanical models. Its application is demonstrated with typical examples from the fields of soil mechanics and engineering geology. The results for two different laboratory tests and a natural slope clearly show that particle swarms are an efficient and fast tool for finding improved parameter sets to represent the measured reference data. Copyright © 2008 Joerg Meier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION tions made and the constitutive model chosen are appro- priate and provided that the performed calculation gives plausible results, parameter values are then varied by trial When compared to most other engineering tasks, geotech- nical problems are often characterized by the following and error in order to reach an improved fit of the calculation results to the measured data, the reference data. peculiarities. The materials involved are geological materials, Though this is done based on the experience of the that is, soils and rocks, which are inhomogeneous and consist geoscientist, the procedure remains to a certain extent of various phases in different states of aggregation. Initial and arbitrary or at least subjective. boundary conditions tend to be complex and heterogeneous. Furthermore, in real geotechnical field problems, the exact In recent years, due to the availability of sufficiently fast computer hardware, there has been a growing interest in the geometry is usually not known, with the available geometry application of inverse parameter identification strategies and information being limited to topographical surface data and punctual outcrops or soundings. For this reason, in geotech- optimization algorithms to geotechnical modeling in order to make this procedure automated [1–5]and thus more nics, there is always a need for a high level of simplification traceable and objective. Furthermore, this approach provides and abstraction. Frequently, continuum methods are used to calculate deformations in soil or rock, and the material statistical information, which can be used to quantify the calibration quality of the developed geotechnical model. behavior is simulated by means of constitutive models, which require a certain set of material parameters. Applications of optimization procedures in geotechnics were described by many authors, for example, in the cali- Normally, in geotechnical engineering, the values of bration process of geotechnical models [1, 2], or to identify these parameters are set based on the results of laboratory experiments, literature data, or even just experience values hydraulic parameters from field drainage tests [6]. Already in 1996, Ledesma et al. [7] and Gens et al. [8] applied gradient are used. The results of the calculation, a forward calculation, methods to a synthetic and a real example of a tunnel drift are then compared to measurement data obtained in the laboratory or in the field. Provided that the simplifica- simulation. Also during the excavation of a cavern in the 2 Journal of Artificial Evolution and Applications Calculation of Extraction of Call the Preset of relevant forward Start objective function parameter vector forward solver solver results value Optimization Stop algorithm: No criterion set of new fulfilled ? parameter vector Yes Stop Figure 1: Flowchart of the adopted iterative procedure. Spanish Pyrenees the above mentioned group of authors the unknown parameters, for example, entering estimated was applying gradient methods for the identification of values, or the preset of the parameter vector is done by a geotechnical parameters [9]. Malecot et al. [10] used inverse random generator within values margins specified by the parameter identification techniques for analyzing pressure- user. The relevant results of the forward calculation are meter tests and finite-element simulations of excavation then read out and their deviation from a set of reference problems. For the identification of soil parameters, also data, usually measured data, is determined by means of an genetic algorithms were studied [11, 12]. Feng et al. [13] used objective function. This procedure is repeated many times, an inverse technique for the determination of the parameters while at any one time, an optimization algorithm, based on of viscoelastic constitutive models for rocks, based on genetic the parameter combinations and the values of the objective programming and a particle swarm optimization algorithm. function during the previous forward calculations, identifies In the field of geoenvironmental engineering, Finsterle an improved parameter set to be used in the next forward [14] examined the potential use of standard optimization calculation. algorithms for the solution of aquifer remediation problems This sequence of cycles, illustrated in Figure 1, is inter- in three-phase and three-component flow and transport rupted when one of the following stop criteria is fulfilled: simulations of contamination plumes. As a different aspect (i) a maximum number of runs or maximum calcula- of parameter identification, Cui and Sheng [15] determined tion time is reached; the minimum parametric distance to the limit state of a strip (ii) the deviation from the reference dataset, described foundation by optimizing a reliability index. In 2006, J. Meier by the value of the objective function, falls below a and T. Schanzin [5] applied particle swarm optimization techniques to geotechnical field projects and laboratory tests, specified limit; namely, a multistage excavation and the desaturation of a (iii) the deviation could not be lowered during a certain sand column. number of cycles. All cited references agree on the fact that back-calculation Hence we use a direct approach as described by Cividini of model parameters by means of optimization routines et al. [16] to solve a back-analysis problem. In an iterative is possible in the field of geotechnics, if an appropriate procedure, the trial values of the unknown parameters are forward calculation depending on adequately realistic model corrected by minimizing an error function. It is therefore not assumptions is provided, for example, Calvello and Finno necessary to formulate the inverse problem itself, the desired [1, 2]. In this context, particle swarms represent a powerful solution is obtained by combining the results of numerous tool for finding parameter sets that best represent the forward calculations with an optimization routine. reference data, with acceptable calculation effort and time To quantify the deviation between the reference data consumption. and the modeling results, we chose the frequently used and relatively simple method of least squares. In this method, the 2. WORKING SCHEME OF THE ADOPTED PARAMETER objective function f (x) for more than one reference dataset IDENTIFICATION STRATEGY is defined as f (x) = [w f (x)] (1) The starting point of the parameter identification strategy g presented in this study is given by an ordinary geotechnical modeling-task, the so-called forward calculation. This for- with ward problem consists of a specified geometry with given initial and boundary conditions and a material model, which 1 2 calc meas f (x) = w (y (x) − y ) . (2) h, g h, g requires a set of material parameters to be determined. It h=1 is generally also possible to identify geometrical parameters [4], but this issue will not be discussed in this article. For In (1)and (2), x denotes the parameter vector to be the first run of the forward calculation, the user presets estimated and w are positive weighting factors associated g Joerg Meier et al. 3 “initialization of swarm” create particle list for each particle set initial particle position and velocity next particle “processing loop” do “get global best particle position” determine best position in past of all particles “determine current particle positions” for each particle calculate and set new velocity based on a corresponding equation calculate and set new position based on a corresponding equation next particle “parallelized calculation of objective function values” for each particle start forward calculation and calculate objective function value next particle “join all calculations” wait for all particle threads “post-processing” for each particle if current objective function is less than own best in past: save position as own best end if next particle loop until stop criterion is met Algorithm 1: Pseudocode of the used particle swarm optimizer. correspondingly with the error measure f (x). Via the of the forward problem usually has to be adapted for its weights w , the different series g can be scaled to the same use in the optimization routine. The runtime of a single value range and different precisions can be merged, for forward calculation has to be minimized in order to allow for example, a series of measuring data possessing a higher pre- a high number of calls. The number of calls needed depends cision is included with a higher weighting factor compared furthermore on the number of parameters to be identified to more uncertain data. The particular numbers for the and, of course, on the used optimization algorithm. weights have to be given manually respecting the engineer’s While the reduction of calculation time demands simpli- experience and they have to be specified depending on the fication and abstraction, the model still should be sufficiently optimization problem. The weighting factors w are used to complex to reproduce the reference data with the required provide a possibility for considering different precisions and accuracy. Furthermore, the number of required forward cal- measurement errors within one and the same data series. The culations can be reduced by applying hypersurface approxi- dimensions of the weighting factors can be taken in the way mation methods [4]. to obtain a dimensionless objective function quantity. For As a next step, it is essential to select the parameters to minimizing the objective function, we use a particle swarm be identified and to decide on the upper and lower limits optimization algorithm described by Eberhart and Kennedy of their plausible values margins. The values of some of [17, 18]. the parameters might be fixed with the aid of previous A computer program developed by the first author knowledge. These specifications must be done with care and of this paper implements this algorithm and disposes of require the experience of the geoscientist, since they influ- interfaces to several commercial finite-element packages ence the obtained results. However, due to the application of used in geotechnical engineering. A short pseudocode of a particle swarm optimizer instead of, for example, a gradient the implemented PSO used in this study is presented in method, no initial guess for the parameter set is necessary Algorithm 1. because initial positions are generated randomly within the parameter value margins. 3. CONCEPT FOR THE APPLICATION TO Due to the inhomogeneities of the geological materials GEOTECHNICAL PROBLEMS involved and the uncertainties related to the initial condi- If the parameter identification strategy described above is tions and the geometrical boundary conditions, geotechnical to be applied to geotechnical tasks, the geotechnical model problems tend to be underdetermined. In order to improve 4 Journal of Artificial Evolution and Applications AB ··· D and ensure the efficiency of the back-analysis, it is of significant importance to check if the set of parameters to be identified may be reduced and if for the prescribed ··· trusted zone the optimization problem is well posed. For this purpose, a statistical analysis is done based on the results of a Monte Carlo procedure including a sufficient parameter B ··· set. Figure 2 shows the principle scheme of the matrix plot used here to visualize the results of the Monte Carlo simulation. A standard mathematical tool for examining multidimensional datasets is the scatter plot matrix (see . . . . . . . . . . . . [19]), whic is included in the matrix plot presented in Figure 2, where each nondiagonal element shows the scatter plots of the respective parameters. The matrix is symmetric. Matrix element D-B, for example, may suggest that the ··· involved parameters B and D are not independent, but strongly correlated. The diagonal matrix plot elements (A- A,... , D-D) show plots where the value of the objective f (x) function is given over the parameter which is associated with the corresponding column. These plots are called hereafter objective function projections. If the problem is well posed, B D each of these plots of the objective function projections has to present one firm extreme value as it is the case in the Figure 2: Statistical analyses via matrix plot, principle scheme. diagram D-D. Otherwise, the respective parameter could not be identified reliably. By filtering out data points that have objective function values larger than a certain threshold level, Table 1: Properties of the studied soil. the distribution of the remaining points gives a rough idea of the size and shape of the extreme value (solution) space. Percentage of clay particles (diameter < 2 μm) 26–34% For further statistical analyses, the well-known linear 2D 3 Bulk density (g/cm)1.82 correlation coefficient can be calculated from the individual Dry density (g/cm)1.29 scatter plots. Referring to [20] in the analysis of our case Density of grains (g/cm)2.76 studies, we consider variables with a correlation coefficient Porosity n 53% of less than 0.5 as “noncorrelated”. Lime content 36% Loss of ignition 6.5% 4. APPLICATIONS Liquid Limit W 0.50 Plastic Limit W 0.27 4.1. Description of the studied geological material Water content 0.41 and the adopted constitutive model In the following examples of applications of the presented parameter identification technique using PSO, we are model- allows for the manufacturing of reproducible samples and ing the mechanical behavior of a natural soil. It is of geotech- test specimens. Some properties of the studied material are nical interest because it favors the development of numerous listed in Table 1. Unless otherwise expressly stated, all tests landslides, namely, rotational soil slips, earth slides, and and classifications were carried out according to the German standard DIN. earthflows. The studied material is clay that results from the weathering of structurally complex geological formations, From this description of the material, it becomes clear the San Cassiano formation (Kassianer Schichten), and the that its mechanical behavior is expected to be very complex La Valle formation (Wengener Schichten) of the Alpine and therefore it is only possible to model some important Trias. These rock formations are made up by interbedded aspects of this behavior. Like many soils with a high clay and strata of marls, tuffites, claystones, limestones, dolomites, silt content, the studied material is highly compressible and and sandstones. Like its source rocks, the soil is characterized exhibits a significant amount of creep deformations, thus by a high clay and silt content. In the field, it consists its behavior is strongly time-dependent. As a constitutive of a clayey matrix with coarser components, of diameters model, we chose the soft soil creep model, which was from centimeters up to meters, floating in it without mutual developed by Vermeer and Neher [21] to account particularly support. Therefore, it is not possible to sample the material as for these phenomena. The soft soil creep model requires the a whole representatively. As it has been completely remolded following material parameters to be specified (see Table 2). by earthflow phenomena, no preferred orientation of the A set of three parameters (c, ϕ and, ψ ) is needed to components can be observed. For this study, only the fraction model failure according to the Mohr-Coulomb criterion. smaller than 2 mm was taken, as it is considered to determine Two further parameters are used to model the amount of the relevant mechanical properties of the entire soil and it elastic and plastic strains and their stress dependency. The . Joerg Meier et al. 5 modified compression index (λ ) represents the slope of Table 2: Parameters of the soft soil creep model. the normal consolidation line during one-dimensional or isotropic logarithmic compression. In the same manner, the Parameter Description (Unit) modified swelling index (κ ) is related to the unloading or c Effective cohesion (kPa) swelling line. The modified creep index (μ )servesasa ϕ Effective friction angle ( ) measure to simulate the development of volumetric creep ◦ ψ Dilatancy angle ( ) deformations with the logarithm of time. λ Modified compression index (dimensionless) In the modeling examples of this article, we will not κ Modified swelling index (dimensionless) take into account the development and the influence of μ Modified creep index (dimensionless) water pressures, which would be also possible but imply a considerable increase in calculation effort; and in the case of the slope example in Section 4.4, more reference data would be needed. All forward calculations were carried out Table 3: Shear-test data reported by Panizza et al. 2006 [24]. applying the finite-element method using the commercial code PLAXIS (Version 8.2, professional, update-pack 8, build Effective friction Effective cohesion 1499) and considering the effect of large deformations by angle ϕ ( ) c (kPa) means of an updated Lagrangian formulation (updated mesh 18 20 analysis) [22]. 18 10 20 49 4.2. Oedometer test Direct shear 18 39 tests 16 49 A one-dimensional compression test was conducted by MFPA Weimar (Germany) [23] in a fixed oedometer ring 14 69 with an inner diameter of around 7 cm (71.45 mm) and a 18 25 height of around 2 cm (20.21 mm). Drainage was allowed on 20 20 the top and at the bottom of the soil sample. All load steps 19 7 were applied vertically, while the sample was held radially, Triaxial tests 28 14 impeding horizontal displacements. First, the sample was preloaded with 9 kPa during two days and with 13 kPa during one day. Then the load was doubled successively, with each load step lasting 24 hours, loading the sample with 25, 50, 100, 200, 400, and 800 kPa. After that, it was unloaded at We averaged these values, giving double weight to the 400, 200, 100, and 50 kPa, and finally it was reloaded again triaxial test data, which we assumed to be more precise, with 100, 200, 400, and 800 kPa (last step took 43 hours). The coming out with an average friction angle of 20 and displacements of the sample top were recorded continuously. an average cohesion of 27 kPa. The set of experimental For the numerical model of the test setting, we used parameter values is shown by Table 4 and the results of a an axisymmetric geometrical configuration with the exact forward calculation using these parameters are presented dimensions of the test specimen. In order to minimize in Figure 3 comparing them with the reference data of the calculation runtime, the discretization was done with two oedometer test. six-node triangular elements only, which is the minimum The graph shows that the deformations are underesti- possible number, as the software offers only triangular mated by the simulation. In order to test the ability of the elements. Thereby, the duration of a forward calculation PSO algorithm to find good parameter combinations, wide could be reduced to less than one minute on an ordinary search areas were chosen for the five parameters. A statistical analysis (see Section 3) comprising 2000 calls of the forward personal computer. The accuracy of the deformation results was checked by carrying out comparative analyses with calculation was then performed varying these parameters. finer meshes. Horizontal fixities were assigned to the lateral Their value margins are displayed in Table 5. boundary and to the rotation axis, simulating the stiff The scatter plot matrix of all data points with objective −7 oedometer ring and vertical fixities were attributed to the function values lower than 10 is given in Figure 4. For the ∗ ∗ ∗ basal boundary, representing the fix filter plate at the bottom. parameters λ , κ ,and μ , logarithmized margins of the After generating the initial stress state by applying the soil search intervals were used in order to avoid overrepresen- self-weight (gravity loading procedure), distributed loads tation of high parameter values. Furthermore, a parameter ∗ ∗ were applied perpendicular to the top boundary analog to constraint was prescribed, demanding for λ >κ , which has the laboratory conditions. to apply for all materials. The objective function projections ∗ ∗ ∗ ∗ ∗ Three parameters of the material model (λ , κ ,and μ ) of c, κ ,and λ indicate that the data points showing good can be determined directly from the oedometer test. As the model fits seem to concentrate in quantifiable value ranges material is known to show no dilatancy, ψ can be set to of these parameters, whereas μ and ϕ cannot be identified. ◦ ∗ 0 in all calculations. Laboratory data from shear tests on The modified compression index (λ ) and the cohesion (c) similar soil samples reported by Panizza et al. [24] is shown appear to be correlated (correlation coefficient of 0.88), to a ∗ ∗ in Table 3. lesser extent, this holds true also for λ and κ (correlation 6 Journal of Artificial Evolution and Applications Oedometer test (49.88 mm) and a height of 10 cm (98.55 mm). Drainage was −4E−03 allowed on top and at the bottom of the specimen. All load −3.5E−03 steps were performed isotropically, applying a hydrostatic −3E−03 −2.5E−03 cell pressure. The sample was preloaded at 30 kPa for three −2E−03 hours and, after that, at 50 kPa for 17 hours. Then it was −1.5E−03 gradually loaded to 800 kPa in one hour, increasing the load −1E−03 −5E−04 by steps of 100 kPa. After reaching this target load, the stress 0E+00 level was left constant for 3 weeks. Top displacements and 0 5 10 15 20 volume change of the sample were recorded during the whole Time (days) test. A reference dataset for the horizontal displacements was Reference data (experiment) calculated from the vertical displacements and the volume Simulation with parameters from laboratory tests change, assuming the shape of the specimen to remain Simulation with parameters from optimization with PSO exactly cylindrical until the end of the test. For the numerical model of the test setting, an axisym- Figure 3: Oedometer test calculation results versus reference data. metric geometrical configuration with the exact dimensions of the test specimen was used. Again, the model was dis- cretized only with two six-node triangular elements, to save Table 4: Parameters obtained from experiments and identified calculation time. The accuracy of the deformation results parameters using PSO. was checked by carrying out several comparative analyses with finer meshes. Horizontal fixities were assigned to the Experimental parameter values Identified values PSO rotation axis. Vertical fixities were attributed to the basal λ 0.064 0.082 boundary, representing the fix filter plate at the bottom. After κ 0.035 0.051 generating the initial stress state by applying the soil self- μ 1.46E–03 — weight (gravity loading procedure), two independent and ϕ ()20 — identical distributed loads were applied, one perpendicular to the upper boundary (vertically) and the other one c (kPa) 27 25 perpendicular to the lateral boundary (radially). Loading was ψ ()0 — carried out the same way as in the laboratory, but instead of the stepwise application of the 800 kPa target load, this load was applied directly after the 50 kPa load step. For this reason, the 50 kPa load step in the model was prolonged in such a Table 5: Search intervals for the parameters of the oedometer test. manner that the integral of the load as a function of time equals the test conditions. Maximum Minimum ln (max.) ln (min.) In the example of the isotropic compression test, except ϕ 30 8 — — for the relatively short phases before reaching the target load, c 100 0.001 — — only one load step (800 kPa) is applied. Therefore, of the λ 1 0.002 0.00 −6.21 model parameters only μ can be determined directly from κ 0.5 0.001 −0.69 −6.91 −3 the test. This value (1.3∗10 ) is very similar to the one μ 0.75 0.00001 −0.29 −11.51 obtained from the oedometer test. Figure 5 shows the results of a forward calculation using this value together with the laboratory values of Section 4.2. Also in this example, the deformations are underesti- ∗ ∗ coefficient of 0.64). According to these findings, λ , κ ,and mated by the simulation. A statistical analysis with 1820 c were selected for the optimization procedure. The friction calls was carried out varying the parameters c (cohesion), ∗ ∗ ∗ angle (ϕ) and the modified creep index (μ )werefixedon λ (modified compression index), and μ (modified creep their experimental values. index) within the boundaries given in Table 7, logarithmic After 159 cycles (1590 calls) the particle swarm optimizer values were used for the search intervals of the latter two −9 had reduced the deviation to 5∗10 ,which wasfound to parameters. be a sufficiently low value to stop the optimization routine. As the modeled test contains no unloading phases, it The identified best parameter set and the corresponding makes no sense to identify the modified swelling index κ .Its calculation results can be seen also in Table 4 and Figure 3.It value was therefore linked to the value of λ by multiplying ∗ ∗ becomes clear that the identified parameter set represents the this parameter by 0.5, which is the typical κ /λ ratio, we measurement data much better than the available laboratory observed in our laboratory tests performed on this material parameters. and similar materials. The results of the statistical tests presented in Figure 6 suggest that good fits can be obtained for cohesion values 4.3. Isotropic compression test between 20 kPa and 90 kPa, but apart from this, the cohesion value seems to have no influence on the quality of the An isotropic compression test was performed in the triaxial apparatus on a cylindrical soil sample with a diameter of 5 cm model calibration. Whereas for the modified compression Vertical displacements (m) Joerg Meier et al. 7 ∗ ∗ ∗ cϕ ln κ ln λ ln μ 0.88 0.001 −0.69 0.64 ln κ −6.91 0.88 0.64 ln λ −6.21 −2.3 ln μ −11.51 0.001 100 8 30 −6.91 −0.69 −6.21 0 −11.51 −2.3 Figure 4: Scatter plot matrix for the oedometer test. Isotropic compression test-vertical displacements Isotropic compression test-horizontal displacements Time (days) Time (days) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0E +00 0E +00 −1E − 03 −5E − 04 −2E − 03 −3E − 03 −1E − 03 −4E − 03 −5E − 03 −1.5E − 03 −6E − 03 −7E − 03 −2E − 03 Reference data (experiment) Reference data (experiment) Simulation with parameters from laboratory tests Simulation with parameters from laboratory tests Simulation with parameters from optimization with PSO Simulation with parameters from optimization with PSO (a) (b) Figure 5: Isotropic compression test calculation results versus reference data. ∗ ∗ index (λ ) and the modified creep index (μ ), the objective Figure 6. This linear relationship seems to be valid for function projections suggest a preferred value range for the cohesion values between 20 and 90 kPa and λ values data points with low deviation values. between 0.064 and 0.165. ∗ ∗ ∗ Furthermore, it can be concluded that λ and c might Therefore, only the parameters λ and μ were selected be quite closely related to each other (correlation coefficient for the optimization procedure via PSO. We stopped this 0.96), this means, for a given λ , an appropriated cohesion procedure after 500 calls (50 cycles). The identified values value could be computed by the equation given also in are shown in Table 6.In Figure 5, the calculation results are Vertical displacements (m) Horizontal displacements (m) 8 Journal of Artificial Evolution and Applications ∗ ∗ c ln λ ln μ 0.96 0.001 0.96 ln λ −5 −0.29 ln μ −11.51 0.001 100 −50 −11.51 −0.29 ∗ ∗ ln λ = (9.48E − 03 c) − 2.7 Figure 6: Scatter plot matrix for the isotropic compression test. Table 6: Parameters from laboratory tests and identified parame- Like for the oedometer test, an improved parameter set ters using PSO. could be found also for the isotropic compression test. It can be observed that the results are much better for the vertical Experimental values Identified values (PSO) displacements, although a weighting factor of one had been λ 0.064 0.089 assigned to both datasets. κ 0.035 — This may be due to the fact that the precision of the μ 1.30E–03 2.33E–03 reference data is lower for the horizontal displacements, ϕ ()20 — since the volume change of the sample could not be c (kPa) 27 — measured with the same accuracy as the top displacements. In addition to this, small inhomogeneities of the material ψ ()0 — or a small frictional resistance at the sample top could have caused a slight distortion of the cylindrical shape of the specimen which was assumed to calculate the horizontal displacements. Table 7: Isotropic compression test search intervals for the varied parameters. 4.4. Deformations along a shear zone in Maximum Minimum ln (max.) ln (min.) a natural slope c (kPa) 100 0.001 — — 4.4.1. Analyzed section and reference data λ 1.00 6.74E–03 0.00 −5.00 μ 0.75 0.00001 −0.29 −11.51 The presented parameter identification technique was also applied to the 2D-model of a natural slope which is located in the municipality of Corvara in the Dolomites (Italy). It shows continuous creep deformations at the basis of a 20– 40 m thick soil cover consisting mainly of the above studied compared to the reference dataset and the results obtained by material and very similar materials. As slopes of this type using only laboratory data. are known to show potential acceleration phases that can Joerg Meier et al. 9 pressure at rest; this means that there is no support obtained from the soil layer further downslope. A description of the assumptions made for the foot load is shown in Figure 10. The actual main detachment zone is modeled as an open crack. No tensile forces are acting across it onto the downslope section of the sliding body, which is moving as a whole. Around section-point A, the soil body below the secondary shear zone is assumed to move at the same velocity as point B. The material properties of the secondary shear zone, which is not subject of this study, were fixed to comply with this criterion. At present, the displacement rates observed along the slope are more or less constant, 0 100 cm being superposed only by seasonal variations attributed to Displacements fluctuations of the groundwater conditions which are not sept. 2001–sept. 2004 modeled in this study. Therefore, our geotechnical model 0 200 400 600 800 1000 (meters) features displacement rates remaining constant with time. Field point 4.4.3. Numerical model Model point Figure 7: Site map with the location of the section and the GPS Figure 11 shows the characteristics of the numerical model of points. the studied slope. A plane strain geometrical configuration with the real dimensions of the slope was used. The model was discretized with 1070 triangular six-node elements. To save calculation time, the number of elements was reduced by endanger human settlements, a detailed study of the geology modeling only the uppermost 20 m of the bedrock layer. The and geomorphology as well as comprehensive monitoring upper and the lower layers were meshed with the automatic were carried out by the University of Modena and Reggio meshing procedure of the software and using a very coarse Emilia and by the National Research Council’s Group for setting. A linear elastic material model was assigned to them. Hydrogeological Catastrophes Defense (CNR-GNDCI) [25] Finally, one forward calculation took approximately three by order of the Autonomous Province of Bolzano/South- minutes on an ordinary personal computer. The interme- Tyrol [24]. diate layer was meshed manually by predefining geometry In this context, the displacements of several surface points in order to assure a sufficiently fine mesh and a points were observed regularly by means of global position- suitable orientation of the triangles in order to go against an ing system (GPS) measurements. In our study, a subarea of excessive distortion of the element shapes by the calculated the slope was modeled along a representative 2D section. A deformations. For the same reason, the updated mesh option site plan with the location of the section and the GPS points was only used in the last three years of the simulation (which is given in Figure 7,whereas Figure 8 depicts a field survey of are compared to the reference dataset). The detachment this section and shows the abstracted geometry model. zones were modeled by means of interfaces on both sides of their geometry lines. The interfaces were modeled with 4.4.2. Geotechnical model a Mohr-Coulomb material model, with negligible cohesion, the same friction angle as the basal shear zone and with The geometry was determined using all the information a constant reference stiffness in the order of magnitude available, that is, a core drilling near section-point B, the of the shear zone stiffness. The accuracy of the calculated local geomorphology, refraction seismics, and direct current deformation results was checked by carrying out several resistivity (DC-resistivity) [24]. As exposed in Figure 9, comparative analyses with finer meshes and also with a the vertical profile of the slope was divided into three horizontal basal boundary. Horizontal fixities were assigned layers interpreting various inclinometer profiles reported by to the lateral edges of the model, which extends over a total Corsini et al. [25]; the illustrated one is located near section- length of 1080 m. Vertical fixities were attributed to the basal point B. The uppermost layer, the soft soil cover, which is boundary, representing the stable bedrock. showing little internal deformations, was only considered in the form of its weight acting on the intermediate layer, the shear zone. Therefore, the displacement vectors of section- 4.4.4. Calculation phases points C and F are presumed to be equal to those of section- points B and E, respectively. The shear zone is assumed to be In the first calculation phase, all three layers are made up a thin, soft, and highly plastic layer, exhibiting a pronounced by the bedrock material. An initial stress state is generated time dependency in its mechanical behavior. The third layer by applying the self-weight of this material (gravity loading is given by the underlying weathered bedrock, which is procedure). In the second calculation phase, the two upper supposed to be stable. The earth pressure at the foot of layers are replaced by the weaker material of the soil cover. the slope was assumed to be slightly lower than the earth The third calculation phase marks the starting point of the 10 Journal of Artificial Evolution and Applications Survey 1b Survey 1a 0 50 100 m Soil cover Deeply weathered St. Cassian and La Valle beds (a) Actual main Secondary shear surface detachment zone at oversteepened front Basal shear surface not or only partially A developed At present low movement rates Detachment zones indicated by morphology No longer active Basal shear surface active for 100 s or 1000 s of years Landslide mass moving as a whole Dip angle of basal shear surface near 0 (b) Figure 8: Field survey along the section and derived geometry model. Displacements (cm) 10 0 -Soft - Little internal deformation Only considered in the form of the load imposed on the - Thin basal shear surface -Soft 40 - Highly plastic - Showing time-dependent Stable C4 behavior Inclinometer profile near model points B,C,D (Corsini et al. 2005) Figure 9: Interpretation of the inclinometer data from Corsini et al. 2005 [25]. Depth (m) Joerg Meier et al. 11 e(h) = earth pressure at depth h φ = effective friction angle of soil cover 90 m Earth pressure at rest would give For φ = 25 Assumed earth pressure distribution ∗ ∗ e(h) = γ 0.68 h ∗ ∗ e(h) = γ 0.6 h + x ◦ 3 For φ = 20 γ = 19 kN/m (specific weight of soil) ∗ ∗ e(h) = γ 0.84 h x = 20 kPa (for better technical performance) Figure 10: Earth pressure assumptions made for the foot of the slope. 20–40 m 2m 20–30 m (a) (b) (c) Figure 11: Discretization of the slope model: foot zone, vertical profile, and detachment zone. 00.51 1.52 2.53 3.54 4.55 Total displacements (mm) Figure 12: Results of a forward calculation using laboratory values of the parameters. (m) (m) 12 Journal of Artificial Evolution and Applications 010 20 30 40 50 60 Total displacements 2001–2004 (cm) Figure 13: Calibrated slope model—comparison of modeled displacement vectors (blue) with reference data derived from the GPS measurements (brown). Table 8: Laboratory values of the parameters used for the slope slope instability in the model. The shear zone material with example. time-dependent mechanical behavior is inserted and the horizontal load at the foot is set. After that, the model is Experimental parameter values left creeping with unchanged boundary conditions during λ 0.064 a period of 33 years. As the loading history of the shear κ 0.035 zone material is unknown, this time period had to be chosen μ 0.00146 arbitrarily to reach constant displacement rates as they are c (kPa) 0.01 presently observed along the natural slope. ϕ ()10 G (kPa) 5560 4.4.5. Results of initial model using parameters v 0.35 derived from experiments ∗ ∗ ∗ In a first trial forward calculation, for λ , κ ,and μ , the parameters calculated from the laboratory experiments were 4.4.6. Results of statistical analysis and used as input values. As the deformations along the shear optimization procedure zone are known to persist since hundreds or thousands of A statistical analysis was carried out (which will not be years [26], the shear strength of this zone has decreased to reported in detail here). One interesting finding of this a residual value that is characteristic for the soils originated analysis was that the friction angle and the modified creep by the weathering of the San Cassiano and La Valle beds index appeared to be closely correlated (coefficient of 0.92). outcropping in the whole slope area. Therefore, cohesion ∗ ∗ ∗ The parameters λ , κ ,and μ ; the friction angle; and the was assumed to be negligible (0.01 kPa) and a friction stiffness of the uppermost layer (represented by its shear angle of 10 was adopted, according to the average slope modulus G) were chosen for the optimization procedure inclination observed in nearby areas which were formed during which they were varied within the intervals specified since the Late Glacial by the studied processes (earth slides in Table 9. and earthflows) and covered by comparable soil covers [27]. After 82 cycles, each of them consisting of 10 forward The stiffness of the uppermost layer was set equal to the calculations, the procedure was stopped because, from then stiffness modulus observed in the oedometer test during on, the deviation could no longer be reduced significantly. unloading and reloading between the load steps 400 kPa and The resulting parameter set is also given in Table 9. Figure 13 800 kPa. For Poisson’s ratio of this layer, we used 0.35, a depicts the calculated deformations using the identified value that is considered to be characteristic of clayey soils. parameter combination, together with the displacement The experimental parameter values are shown in Table 8 vectors of the GPS measurement points. and the deformations calculated on their basis for the last It can be observed that the identified parameter set is able three years of the creep phase are presented by Figure 12. to reproduce the field measurements qualitatively. Because The latter are only in the range of millimeters, and thus not of the simplifications made in the model, no exact fit of the representing the actual situation in the field, where between displacement vectors is possible. The presented back analysis September 2001 and September 2004, displacements from procedure gives one of a number of possible approximate several centimeters to several decimeters were measured. solutions to the geotechnical problem and the result returned Joerg Meier et al. 13 Table 9: Search intervals and identified parameters for the slope example. Identified values Varied parameters Parameter Fixed parameters (PSO) Maximum Minimum v 0.35 — — — G (kPa) — 20000 200 5160 c(kPa) 0.01 ——— ψ () 0 ——— κ — 1 0.005 0.60 λ — 2 0.01 1.42 μ — 1.5 0.001 0.145 ϕ ( ) — 16 8 10.7 by the particle swarm optimizer can be seen as a parameter [2] M. Calvello and R. J. Finno, “Selecting parameters to optimize set that best represents the reference data. in model calibration by inverse analysis,” Computers and Geotechnics, vol. 31, no. 5, pp. 411–425, 2004. [3] J. Carrera, A. Alcolea, A. Medina, J. Hidalgo, and L. Slooten, 5. CONCLUSIONS “Inverse problem in hydrogeology,” Hydrogeological Journal, vol. 13, no. 1, pp. 206–222, 2005. A back analysis procedure for the identification of material [4] J.Meier,S.Rudolph, andT.Schanz, “Effektiver Algorithmus parameters of constitutive models applied to geotechnical zur Losung ¨ von inversen Aufgabenstellungen—Anwendung in problems was presented. This procedure represents a direct der Geomechanik,” Bautechnik, vol. 83, no. 7, pp. 470–481, approach based on the method of least squares, correlation analyses, and a particle swarm optimization algorithm. The [5] T. Schanz, M. M. Zimmerer, M. Datcheva, and J. Meier, “Iden- applicability and suitability of the technique was demon- tification of constitutive parameters for numerical models via strated by means of three examples from the fields of soil inverse approach,” Felsbau, vol. 24, no. 2, pp. 11–21, 2006. mechanics and engineering geology. The studied material [6] Z. F. Zhang, A. L. Ward, and G. W. Gee, “Estimating soil was a natural soil. Besides being way more objective and less hydraulic parameters of a field drainage experiment using inverse techniques,” Vadose Zone Journal, vol. 2, no. 2, pp. 201– arbitrary than the conventional trial and error procedure, 211, 2003. the outlined method provides valuable information on the [7] A. Ledesma, A. Gens, and E. E. Alonso, “Estimation of param- quality of the model calibration, the uniqueness of an eters in geotechnical backanalysis—I. Maximum likelihood obtained solution, or the determinateness of the problem. approach,” Computers and Geotechnics, vol. 18, no. 1, pp. 1– In all three examples, the particle swarm optimizer was 27, 1996. able to identify an improved parameter set after a justifiable [8] A. Gens, A. Ledesma, and E. E. Alonso, “Estimation of amount of forward calculations. Further research should parameters in geotechnical backanalysis—II. Application to also concentrate on the identification of the geometrical a tunnel excavation problem,” Computers and Geotechnics, parameters of geotechnical problems. vol. 18, no. 1, pp. 29–46, 1996. [9] A. Ledesma, A. Gens, and E. E. Alonso, “Parameter and variance estimation in geotechnical backanalysis using prior ACKNOWLEDGMENTS information,” International Journal for Numerical and Analyti- cal Methods in Geomechanics, vol. 20, no. 2, pp. 119–141, 1996. The German Academic Exchange Service (DAAD) and the [10] Y. Malecot, E. Flavigny, and M. Boulon, “Inverse analysis of Association of the Rectors of the Italian Universities (CRUI) soil parameters for finite element simulation of geotechnical are acknowledged for funding traveling expenses through a structures: pressuremeter test and excavation problem,” in VIGONI exchange project. The second author acknowledges Proceedings of the Symposium on Geotechnical Innovations,R. the support by the Konrad-Adenauer-Foundation via a B. J. Brinkgreve, H. Schad, H. f. Schweiger, and E. Willand, postgraduate scholarship. The work of the first author was Eds., pp. 659–675, Verlag Gluc ¨ kauf, Essen, Germany, 2004. funded by the German Research Foundation (DFG) via the [11] S. Levasseur, Y. Malecot, M. Boulon, and E. Flavigny, “Soil projects SCHA 675/7-2 and SCHA 675/11-2 “Geomechanical parameter identification using a genetic algorithm,” Inter- modeling of large mountainous slopes”. national Journal for Numerical and Analytical Methods in Geomechanics, vol. 32, no. 2, pp. 189–213, 2007. [12] S. Levasseur, Y. Malecot, M. Boulon, and E. Flavigny, “Soil REFERENCES parameter identification from in situ measurements using a [1] M. Calvello and R. J. Finno, “Calibration of soil models genetic algorithm and a principle component analysis,” in by inverse analysis,” in NumericalModelsinGeomechanics Proceedings of the 10th International Symposium on Numerical NUMOG VIII, G. Pande and S. Pietruszczak, Eds., pp. 107– Models in Geomechanics (NUMOG ’07), Rhodes, Greece, April 116, Balkema, Rotterdam, The Netherlands, 2002. 2007. 14 Journal of Artificial Evolution and Applications [13] X.-T. Feng, B.-R. Chen, C. Yang, H. Zhou, and X. Ding, “Identification of visco-elastic models for rocks using genetic programming coupled with the modified particle swarm opti- mization algorithm,” International Journal of Rock Mechanics and Mining Sciences, vol. 43, no. 5, pp. 789–801, 2006. [14] S. Finsterle, “Demonstration of optimization techniques for groundwater plume remediation using iTOUGH2,” Environ- mental Modelling & Software, vol. 21, no. 5, pp. 665–680, 2006. [15] L. Cui and D. Sheng, “Genetic algorithms in probabilistic finite element analysis of geotechnical problems,” Computers and Geotechnics, vol. 32, no. 8, pp. 555–563, 2005. [16] A. Cividini, L. Jurina, and G. Gioda, “Some aspects of ‘characterization’ problems in geomechanics,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 18, no. 6, pp. 487–503, 1981. [17] R. C. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the 6th International Symposium on Micro Machine and Human Science (MHS ’95), pp. 39–43, Nagoya, Japan, October 1995. [18] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, IEEE Press, Piscataway, NJ, USA, November-December 1995. [19] B. F. J. Manly, Multivariate Statistical Methods: A Primer, Chapman & Hall/CRC, Boca Raton, Fla, USA, 3rd edition, [20] J. Will,D.Roos, J. Riedel,and C. Bucher,“Robustness analysis in stochastic structural mechanics,” in NAFEMS Seminar: Use of Stochastics in FEM Analyses, Wiesbaden, Germany, May [21] P. A. Vermeer and H. P. Neher, “A soft soil model that accounts for creep,” in Proceedings of the International Symposium. Beyond 2000 in Computational Geotechnics. 10 Years of PLAXIS Inetrnational, pp. 55–58, Balkema, Amsterdam, The Nether- lands, March 1999. [22] K.-J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood-Cliffs, NJ, USA, 1982. [23] K. Lemke, H. Lorenz, A. Klimitsch, J. Koeditz, T. Schaefer, and K. J. Witt, “Results of laboratory study on earthflow materials,” Tech. Rep., Material Research and Testing Institute MFPA, Weimar, Germany, 2006. [24] M. Panizza, S. Silvano, A. Corsini, et al., “Definizione della pericolosita ` e di possibili interventi di mitigazione della frana di Corvara in Badia. Provincia Autonoma di Bolzano— Alto Adige,” Autonome Provinz Bozen—Sudtir ¨ ol, http:// www.provincia.bz.it/opere-idrauliche/attivita6 i.htm, 2006. [25] A. Corsini, A. Pasuto, M. Soldati, and A. Zannoni, “Field monitoring of the Corvara landslide (Dolomites, Italy) and its relevance for hazard assessment,” Geomorphology, vol. 66, no. 1–4, pp. 149–165, 2005. [26] M. Soldati, A. Corsini, and A. Pasuto, “Landslides and climate change in the Italian Dolomites since the Late glacial,” Catena, vol. 55, no. 2, pp. 141–161, 2004. [27] A. Corsini, “L’influenza dei fenomeni franosi sull’evoluzione geomorfologica post-glaciale dell’Alta Val Badia e della Valparola (Dolomiti),” Ph.D. thesis, University of Bologna, Bologna, Italy, 2000. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Artificial Evolution and Applications Hindawi Publishing Corporation

Inverse Parameter Identification Technique Using PSO Algorithm Applied to Geotechnical Modeling

Loading next page...
 
/lp/hindawi-publishing-corporation/inverse-parameter-identification-technique-using-pso-algorithm-applied-G3DFKoWEWg

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Hindawi Publishing Corporation
Copyright
Copyright © 2008 Joerg Meier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN
1687-6229
DOI
10.1155/2008/574613
Publisher site
See Article on Publisher Site

Abstract

Hindawi Publishing Corporation Journal of Artificial Evolution and Applications Volume 2008, Article ID 574613, 14 pages doi:10.1155/2008/574613 Research Article Inverse Parameter Identification Technique Using PSO Algorithm Applied to Geotechnical Modeling 1 1 2 2 1 Joerg Meier, Winfried Schaedler, Lisa Borgatti, Alessandro Corsini, and Tom Schanz Labor fur ¨ Bodenmechanik, Bauhaus-Universitat ¨ Weimar, Coudraystraße 11 C, 99421 Weimar, Germany Dipartimento di Scienze della Terra, Universita ` degli Studi di Modena e Reggio Emilia, Largo Sant’ Eufemia 19, 41100 Modena, Italy Correspondence should be addressed to Winfried Schaedler, winfried.schaedler@uni-weimar.de Received 24 July 2007; Accepted 14 January 2008 Recommended by Jim Kennedy This paper presents a concept for the application of particle swarm optimization in geotechnical engineering. For the calculation of deformations in soil or rock, numerical simulations based on continuum methods are widely used. The material behavior is modeled using constitutive relations that require sets of material parameters to be specified. We present an inverse parameter identification technique, based on statistical analyses and a particle swarm optimization algorithm, to be used in the calibration process of geomechanical models. Its application is demonstrated with typical examples from the fields of soil mechanics and engineering geology. The results for two different laboratory tests and a natural slope clearly show that particle swarms are an efficient and fast tool for finding improved parameter sets to represent the measured reference data. Copyright © 2008 Joerg Meier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION tions made and the constitutive model chosen are appro- priate and provided that the performed calculation gives plausible results, parameter values are then varied by trial When compared to most other engineering tasks, geotech- nical problems are often characterized by the following and error in order to reach an improved fit of the calculation results to the measured data, the reference data. peculiarities. The materials involved are geological materials, Though this is done based on the experience of the that is, soils and rocks, which are inhomogeneous and consist geoscientist, the procedure remains to a certain extent of various phases in different states of aggregation. Initial and arbitrary or at least subjective. boundary conditions tend to be complex and heterogeneous. Furthermore, in real geotechnical field problems, the exact In recent years, due to the availability of sufficiently fast computer hardware, there has been a growing interest in the geometry is usually not known, with the available geometry application of inverse parameter identification strategies and information being limited to topographical surface data and punctual outcrops or soundings. For this reason, in geotech- optimization algorithms to geotechnical modeling in order to make this procedure automated [1–5]and thus more nics, there is always a need for a high level of simplification traceable and objective. Furthermore, this approach provides and abstraction. Frequently, continuum methods are used to calculate deformations in soil or rock, and the material statistical information, which can be used to quantify the calibration quality of the developed geotechnical model. behavior is simulated by means of constitutive models, which require a certain set of material parameters. Applications of optimization procedures in geotechnics were described by many authors, for example, in the cali- Normally, in geotechnical engineering, the values of bration process of geotechnical models [1, 2], or to identify these parameters are set based on the results of laboratory experiments, literature data, or even just experience values hydraulic parameters from field drainage tests [6]. Already in 1996, Ledesma et al. [7] and Gens et al. [8] applied gradient are used. The results of the calculation, a forward calculation, methods to a synthetic and a real example of a tunnel drift are then compared to measurement data obtained in the laboratory or in the field. Provided that the simplifica- simulation. Also during the excavation of a cavern in the 2 Journal of Artificial Evolution and Applications Calculation of Extraction of Call the Preset of relevant forward Start objective function parameter vector forward solver solver results value Optimization Stop algorithm: No criterion set of new fulfilled ? parameter vector Yes Stop Figure 1: Flowchart of the adopted iterative procedure. Spanish Pyrenees the above mentioned group of authors the unknown parameters, for example, entering estimated was applying gradient methods for the identification of values, or the preset of the parameter vector is done by a geotechnical parameters [9]. Malecot et al. [10] used inverse random generator within values margins specified by the parameter identification techniques for analyzing pressure- user. The relevant results of the forward calculation are meter tests and finite-element simulations of excavation then read out and their deviation from a set of reference problems. For the identification of soil parameters, also data, usually measured data, is determined by means of an genetic algorithms were studied [11, 12]. Feng et al. [13] used objective function. This procedure is repeated many times, an inverse technique for the determination of the parameters while at any one time, an optimization algorithm, based on of viscoelastic constitutive models for rocks, based on genetic the parameter combinations and the values of the objective programming and a particle swarm optimization algorithm. function during the previous forward calculations, identifies In the field of geoenvironmental engineering, Finsterle an improved parameter set to be used in the next forward [14] examined the potential use of standard optimization calculation. algorithms for the solution of aquifer remediation problems This sequence of cycles, illustrated in Figure 1, is inter- in three-phase and three-component flow and transport rupted when one of the following stop criteria is fulfilled: simulations of contamination plumes. As a different aspect (i) a maximum number of runs or maximum calcula- of parameter identification, Cui and Sheng [15] determined tion time is reached; the minimum parametric distance to the limit state of a strip (ii) the deviation from the reference dataset, described foundation by optimizing a reliability index. In 2006, J. Meier by the value of the objective function, falls below a and T. Schanzin [5] applied particle swarm optimization techniques to geotechnical field projects and laboratory tests, specified limit; namely, a multistage excavation and the desaturation of a (iii) the deviation could not be lowered during a certain sand column. number of cycles. All cited references agree on the fact that back-calculation Hence we use a direct approach as described by Cividini of model parameters by means of optimization routines et al. [16] to solve a back-analysis problem. In an iterative is possible in the field of geotechnics, if an appropriate procedure, the trial values of the unknown parameters are forward calculation depending on adequately realistic model corrected by minimizing an error function. It is therefore not assumptions is provided, for example, Calvello and Finno necessary to formulate the inverse problem itself, the desired [1, 2]. In this context, particle swarms represent a powerful solution is obtained by combining the results of numerous tool for finding parameter sets that best represent the forward calculations with an optimization routine. reference data, with acceptable calculation effort and time To quantify the deviation between the reference data consumption. and the modeling results, we chose the frequently used and relatively simple method of least squares. In this method, the 2. WORKING SCHEME OF THE ADOPTED PARAMETER objective function f (x) for more than one reference dataset IDENTIFICATION STRATEGY is defined as f (x) = [w f (x)] (1) The starting point of the parameter identification strategy g presented in this study is given by an ordinary geotechnical modeling-task, the so-called forward calculation. This for- with ward problem consists of a specified geometry with given initial and boundary conditions and a material model, which 1 2 calc meas f (x) = w (y (x) − y ) . (2) h, g h, g requires a set of material parameters to be determined. It h=1 is generally also possible to identify geometrical parameters [4], but this issue will not be discussed in this article. For In (1)and (2), x denotes the parameter vector to be the first run of the forward calculation, the user presets estimated and w are positive weighting factors associated g Joerg Meier et al. 3 “initialization of swarm” create particle list for each particle set initial particle position and velocity next particle “processing loop” do “get global best particle position” determine best position in past of all particles “determine current particle positions” for each particle calculate and set new velocity based on a corresponding equation calculate and set new position based on a corresponding equation next particle “parallelized calculation of objective function values” for each particle start forward calculation and calculate objective function value next particle “join all calculations” wait for all particle threads “post-processing” for each particle if current objective function is less than own best in past: save position as own best end if next particle loop until stop criterion is met Algorithm 1: Pseudocode of the used particle swarm optimizer. correspondingly with the error measure f (x). Via the of the forward problem usually has to be adapted for its weights w , the different series g can be scaled to the same use in the optimization routine. The runtime of a single value range and different precisions can be merged, for forward calculation has to be minimized in order to allow for example, a series of measuring data possessing a higher pre- a high number of calls. The number of calls needed depends cision is included with a higher weighting factor compared furthermore on the number of parameters to be identified to more uncertain data. The particular numbers for the and, of course, on the used optimization algorithm. weights have to be given manually respecting the engineer’s While the reduction of calculation time demands simpli- experience and they have to be specified depending on the fication and abstraction, the model still should be sufficiently optimization problem. The weighting factors w are used to complex to reproduce the reference data with the required provide a possibility for considering different precisions and accuracy. Furthermore, the number of required forward cal- measurement errors within one and the same data series. The culations can be reduced by applying hypersurface approxi- dimensions of the weighting factors can be taken in the way mation methods [4]. to obtain a dimensionless objective function quantity. For As a next step, it is essential to select the parameters to minimizing the objective function, we use a particle swarm be identified and to decide on the upper and lower limits optimization algorithm described by Eberhart and Kennedy of their plausible values margins. The values of some of [17, 18]. the parameters might be fixed with the aid of previous A computer program developed by the first author knowledge. These specifications must be done with care and of this paper implements this algorithm and disposes of require the experience of the geoscientist, since they influ- interfaces to several commercial finite-element packages ence the obtained results. However, due to the application of used in geotechnical engineering. A short pseudocode of a particle swarm optimizer instead of, for example, a gradient the implemented PSO used in this study is presented in method, no initial guess for the parameter set is necessary Algorithm 1. because initial positions are generated randomly within the parameter value margins. 3. CONCEPT FOR THE APPLICATION TO Due to the inhomogeneities of the geological materials GEOTECHNICAL PROBLEMS involved and the uncertainties related to the initial condi- If the parameter identification strategy described above is tions and the geometrical boundary conditions, geotechnical to be applied to geotechnical tasks, the geotechnical model problems tend to be underdetermined. In order to improve 4 Journal of Artificial Evolution and Applications AB ··· D and ensure the efficiency of the back-analysis, it is of significant importance to check if the set of parameters to be identified may be reduced and if for the prescribed ··· trusted zone the optimization problem is well posed. For this purpose, a statistical analysis is done based on the results of a Monte Carlo procedure including a sufficient parameter B ··· set. Figure 2 shows the principle scheme of the matrix plot used here to visualize the results of the Monte Carlo simulation. A standard mathematical tool for examining multidimensional datasets is the scatter plot matrix (see . . . . . . . . . . . . [19]), whic is included in the matrix plot presented in Figure 2, where each nondiagonal element shows the scatter plots of the respective parameters. The matrix is symmetric. Matrix element D-B, for example, may suggest that the ··· involved parameters B and D are not independent, but strongly correlated. The diagonal matrix plot elements (A- A,... , D-D) show plots where the value of the objective f (x) function is given over the parameter which is associated with the corresponding column. These plots are called hereafter objective function projections. If the problem is well posed, B D each of these plots of the objective function projections has to present one firm extreme value as it is the case in the Figure 2: Statistical analyses via matrix plot, principle scheme. diagram D-D. Otherwise, the respective parameter could not be identified reliably. By filtering out data points that have objective function values larger than a certain threshold level, Table 1: Properties of the studied soil. the distribution of the remaining points gives a rough idea of the size and shape of the extreme value (solution) space. Percentage of clay particles (diameter < 2 μm) 26–34% For further statistical analyses, the well-known linear 2D 3 Bulk density (g/cm)1.82 correlation coefficient can be calculated from the individual Dry density (g/cm)1.29 scatter plots. Referring to [20] in the analysis of our case Density of grains (g/cm)2.76 studies, we consider variables with a correlation coefficient Porosity n 53% of less than 0.5 as “noncorrelated”. Lime content 36% Loss of ignition 6.5% 4. APPLICATIONS Liquid Limit W 0.50 Plastic Limit W 0.27 4.1. Description of the studied geological material Water content 0.41 and the adopted constitutive model In the following examples of applications of the presented parameter identification technique using PSO, we are model- allows for the manufacturing of reproducible samples and ing the mechanical behavior of a natural soil. It is of geotech- test specimens. Some properties of the studied material are nical interest because it favors the development of numerous listed in Table 1. Unless otherwise expressly stated, all tests landslides, namely, rotational soil slips, earth slides, and and classifications were carried out according to the German standard DIN. earthflows. The studied material is clay that results from the weathering of structurally complex geological formations, From this description of the material, it becomes clear the San Cassiano formation (Kassianer Schichten), and the that its mechanical behavior is expected to be very complex La Valle formation (Wengener Schichten) of the Alpine and therefore it is only possible to model some important Trias. These rock formations are made up by interbedded aspects of this behavior. Like many soils with a high clay and strata of marls, tuffites, claystones, limestones, dolomites, silt content, the studied material is highly compressible and and sandstones. Like its source rocks, the soil is characterized exhibits a significant amount of creep deformations, thus by a high clay and silt content. In the field, it consists its behavior is strongly time-dependent. As a constitutive of a clayey matrix with coarser components, of diameters model, we chose the soft soil creep model, which was from centimeters up to meters, floating in it without mutual developed by Vermeer and Neher [21] to account particularly support. Therefore, it is not possible to sample the material as for these phenomena. The soft soil creep model requires the a whole representatively. As it has been completely remolded following material parameters to be specified (see Table 2). by earthflow phenomena, no preferred orientation of the A set of three parameters (c, ϕ and, ψ ) is needed to components can be observed. For this study, only the fraction model failure according to the Mohr-Coulomb criterion. smaller than 2 mm was taken, as it is considered to determine Two further parameters are used to model the amount of the relevant mechanical properties of the entire soil and it elastic and plastic strains and their stress dependency. The . Joerg Meier et al. 5 modified compression index (λ ) represents the slope of Table 2: Parameters of the soft soil creep model. the normal consolidation line during one-dimensional or isotropic logarithmic compression. In the same manner, the Parameter Description (Unit) modified swelling index (κ ) is related to the unloading or c Effective cohesion (kPa) swelling line. The modified creep index (μ )servesasa ϕ Effective friction angle ( ) measure to simulate the development of volumetric creep ◦ ψ Dilatancy angle ( ) deformations with the logarithm of time. λ Modified compression index (dimensionless) In the modeling examples of this article, we will not κ Modified swelling index (dimensionless) take into account the development and the influence of μ Modified creep index (dimensionless) water pressures, which would be also possible but imply a considerable increase in calculation effort; and in the case of the slope example in Section 4.4, more reference data would be needed. All forward calculations were carried out Table 3: Shear-test data reported by Panizza et al. 2006 [24]. applying the finite-element method using the commercial code PLAXIS (Version 8.2, professional, update-pack 8, build Effective friction Effective cohesion 1499) and considering the effect of large deformations by angle ϕ ( ) c (kPa) means of an updated Lagrangian formulation (updated mesh 18 20 analysis) [22]. 18 10 20 49 4.2. Oedometer test Direct shear 18 39 tests 16 49 A one-dimensional compression test was conducted by MFPA Weimar (Germany) [23] in a fixed oedometer ring 14 69 with an inner diameter of around 7 cm (71.45 mm) and a 18 25 height of around 2 cm (20.21 mm). Drainage was allowed on 20 20 the top and at the bottom of the soil sample. All load steps 19 7 were applied vertically, while the sample was held radially, Triaxial tests 28 14 impeding horizontal displacements. First, the sample was preloaded with 9 kPa during two days and with 13 kPa during one day. Then the load was doubled successively, with each load step lasting 24 hours, loading the sample with 25, 50, 100, 200, 400, and 800 kPa. After that, it was unloaded at We averaged these values, giving double weight to the 400, 200, 100, and 50 kPa, and finally it was reloaded again triaxial test data, which we assumed to be more precise, with 100, 200, 400, and 800 kPa (last step took 43 hours). The coming out with an average friction angle of 20 and displacements of the sample top were recorded continuously. an average cohesion of 27 kPa. The set of experimental For the numerical model of the test setting, we used parameter values is shown by Table 4 and the results of a an axisymmetric geometrical configuration with the exact forward calculation using these parameters are presented dimensions of the test specimen. In order to minimize in Figure 3 comparing them with the reference data of the calculation runtime, the discretization was done with two oedometer test. six-node triangular elements only, which is the minimum The graph shows that the deformations are underesti- possible number, as the software offers only triangular mated by the simulation. In order to test the ability of the elements. Thereby, the duration of a forward calculation PSO algorithm to find good parameter combinations, wide could be reduced to less than one minute on an ordinary search areas were chosen for the five parameters. A statistical analysis (see Section 3) comprising 2000 calls of the forward personal computer. The accuracy of the deformation results was checked by carrying out comparative analyses with calculation was then performed varying these parameters. finer meshes. Horizontal fixities were assigned to the lateral Their value margins are displayed in Table 5. boundary and to the rotation axis, simulating the stiff The scatter plot matrix of all data points with objective −7 oedometer ring and vertical fixities were attributed to the function values lower than 10 is given in Figure 4. For the ∗ ∗ ∗ basal boundary, representing the fix filter plate at the bottom. parameters λ , κ ,and μ , logarithmized margins of the After generating the initial stress state by applying the soil search intervals were used in order to avoid overrepresen- self-weight (gravity loading procedure), distributed loads tation of high parameter values. Furthermore, a parameter ∗ ∗ were applied perpendicular to the top boundary analog to constraint was prescribed, demanding for λ >κ , which has the laboratory conditions. to apply for all materials. The objective function projections ∗ ∗ ∗ ∗ ∗ Three parameters of the material model (λ , κ ,and μ ) of c, κ ,and λ indicate that the data points showing good can be determined directly from the oedometer test. As the model fits seem to concentrate in quantifiable value ranges material is known to show no dilatancy, ψ can be set to of these parameters, whereas μ and ϕ cannot be identified. ◦ ∗ 0 in all calculations. Laboratory data from shear tests on The modified compression index (λ ) and the cohesion (c) similar soil samples reported by Panizza et al. [24] is shown appear to be correlated (correlation coefficient of 0.88), to a ∗ ∗ in Table 3. lesser extent, this holds true also for λ and κ (correlation 6 Journal of Artificial Evolution and Applications Oedometer test (49.88 mm) and a height of 10 cm (98.55 mm). Drainage was −4E−03 allowed on top and at the bottom of the specimen. All load −3.5E−03 steps were performed isotropically, applying a hydrostatic −3E−03 −2.5E−03 cell pressure. The sample was preloaded at 30 kPa for three −2E−03 hours and, after that, at 50 kPa for 17 hours. Then it was −1.5E−03 gradually loaded to 800 kPa in one hour, increasing the load −1E−03 −5E−04 by steps of 100 kPa. After reaching this target load, the stress 0E+00 level was left constant for 3 weeks. Top displacements and 0 5 10 15 20 volume change of the sample were recorded during the whole Time (days) test. A reference dataset for the horizontal displacements was Reference data (experiment) calculated from the vertical displacements and the volume Simulation with parameters from laboratory tests change, assuming the shape of the specimen to remain Simulation with parameters from optimization with PSO exactly cylindrical until the end of the test. For the numerical model of the test setting, an axisym- Figure 3: Oedometer test calculation results versus reference data. metric geometrical configuration with the exact dimensions of the test specimen was used. Again, the model was dis- cretized only with two six-node triangular elements, to save Table 4: Parameters obtained from experiments and identified calculation time. The accuracy of the deformation results parameters using PSO. was checked by carrying out several comparative analyses with finer meshes. Horizontal fixities were assigned to the Experimental parameter values Identified values PSO rotation axis. Vertical fixities were attributed to the basal λ 0.064 0.082 boundary, representing the fix filter plate at the bottom. After κ 0.035 0.051 generating the initial stress state by applying the soil self- μ 1.46E–03 — weight (gravity loading procedure), two independent and ϕ ()20 — identical distributed loads were applied, one perpendicular to the upper boundary (vertically) and the other one c (kPa) 27 25 perpendicular to the lateral boundary (radially). Loading was ψ ()0 — carried out the same way as in the laboratory, but instead of the stepwise application of the 800 kPa target load, this load was applied directly after the 50 kPa load step. For this reason, the 50 kPa load step in the model was prolonged in such a Table 5: Search intervals for the parameters of the oedometer test. manner that the integral of the load as a function of time equals the test conditions. Maximum Minimum ln (max.) ln (min.) In the example of the isotropic compression test, except ϕ 30 8 — — for the relatively short phases before reaching the target load, c 100 0.001 — — only one load step (800 kPa) is applied. Therefore, of the λ 1 0.002 0.00 −6.21 model parameters only μ can be determined directly from κ 0.5 0.001 −0.69 −6.91 −3 the test. This value (1.3∗10 ) is very similar to the one μ 0.75 0.00001 −0.29 −11.51 obtained from the oedometer test. Figure 5 shows the results of a forward calculation using this value together with the laboratory values of Section 4.2. Also in this example, the deformations are underesti- ∗ ∗ coefficient of 0.64). According to these findings, λ , κ ,and mated by the simulation. A statistical analysis with 1820 c were selected for the optimization procedure. The friction calls was carried out varying the parameters c (cohesion), ∗ ∗ ∗ angle (ϕ) and the modified creep index (μ )werefixedon λ (modified compression index), and μ (modified creep their experimental values. index) within the boundaries given in Table 7, logarithmic After 159 cycles (1590 calls) the particle swarm optimizer values were used for the search intervals of the latter two −9 had reduced the deviation to 5∗10 ,which wasfound to parameters. be a sufficiently low value to stop the optimization routine. As the modeled test contains no unloading phases, it The identified best parameter set and the corresponding makes no sense to identify the modified swelling index κ .Its calculation results can be seen also in Table 4 and Figure 3.It value was therefore linked to the value of λ by multiplying ∗ ∗ becomes clear that the identified parameter set represents the this parameter by 0.5, which is the typical κ /λ ratio, we measurement data much better than the available laboratory observed in our laboratory tests performed on this material parameters. and similar materials. The results of the statistical tests presented in Figure 6 suggest that good fits can be obtained for cohesion values 4.3. Isotropic compression test between 20 kPa and 90 kPa, but apart from this, the cohesion value seems to have no influence on the quality of the An isotropic compression test was performed in the triaxial apparatus on a cylindrical soil sample with a diameter of 5 cm model calibration. Whereas for the modified compression Vertical displacements (m) Joerg Meier et al. 7 ∗ ∗ ∗ cϕ ln κ ln λ ln μ 0.88 0.001 −0.69 0.64 ln κ −6.91 0.88 0.64 ln λ −6.21 −2.3 ln μ −11.51 0.001 100 8 30 −6.91 −0.69 −6.21 0 −11.51 −2.3 Figure 4: Scatter plot matrix for the oedometer test. Isotropic compression test-vertical displacements Isotropic compression test-horizontal displacements Time (days) Time (days) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0E +00 0E +00 −1E − 03 −5E − 04 −2E − 03 −3E − 03 −1E − 03 −4E − 03 −5E − 03 −1.5E − 03 −6E − 03 −7E − 03 −2E − 03 Reference data (experiment) Reference data (experiment) Simulation with parameters from laboratory tests Simulation with parameters from laboratory tests Simulation with parameters from optimization with PSO Simulation with parameters from optimization with PSO (a) (b) Figure 5: Isotropic compression test calculation results versus reference data. ∗ ∗ index (λ ) and the modified creep index (μ ), the objective Figure 6. This linear relationship seems to be valid for function projections suggest a preferred value range for the cohesion values between 20 and 90 kPa and λ values data points with low deviation values. between 0.064 and 0.165. ∗ ∗ ∗ Furthermore, it can be concluded that λ and c might Therefore, only the parameters λ and μ were selected be quite closely related to each other (correlation coefficient for the optimization procedure via PSO. We stopped this 0.96), this means, for a given λ , an appropriated cohesion procedure after 500 calls (50 cycles). The identified values value could be computed by the equation given also in are shown in Table 6.In Figure 5, the calculation results are Vertical displacements (m) Horizontal displacements (m) 8 Journal of Artificial Evolution and Applications ∗ ∗ c ln λ ln μ 0.96 0.001 0.96 ln λ −5 −0.29 ln μ −11.51 0.001 100 −50 −11.51 −0.29 ∗ ∗ ln λ = (9.48E − 03 c) − 2.7 Figure 6: Scatter plot matrix for the isotropic compression test. Table 6: Parameters from laboratory tests and identified parame- Like for the oedometer test, an improved parameter set ters using PSO. could be found also for the isotropic compression test. It can be observed that the results are much better for the vertical Experimental values Identified values (PSO) displacements, although a weighting factor of one had been λ 0.064 0.089 assigned to both datasets. κ 0.035 — This may be due to the fact that the precision of the μ 1.30E–03 2.33E–03 reference data is lower for the horizontal displacements, ϕ ()20 — since the volume change of the sample could not be c (kPa) 27 — measured with the same accuracy as the top displacements. In addition to this, small inhomogeneities of the material ψ ()0 — or a small frictional resistance at the sample top could have caused a slight distortion of the cylindrical shape of the specimen which was assumed to calculate the horizontal displacements. Table 7: Isotropic compression test search intervals for the varied parameters. 4.4. Deformations along a shear zone in Maximum Minimum ln (max.) ln (min.) a natural slope c (kPa) 100 0.001 — — 4.4.1. Analyzed section and reference data λ 1.00 6.74E–03 0.00 −5.00 μ 0.75 0.00001 −0.29 −11.51 The presented parameter identification technique was also applied to the 2D-model of a natural slope which is located in the municipality of Corvara in the Dolomites (Italy). It shows continuous creep deformations at the basis of a 20– 40 m thick soil cover consisting mainly of the above studied compared to the reference dataset and the results obtained by material and very similar materials. As slopes of this type using only laboratory data. are known to show potential acceleration phases that can Joerg Meier et al. 9 pressure at rest; this means that there is no support obtained from the soil layer further downslope. A description of the assumptions made for the foot load is shown in Figure 10. The actual main detachment zone is modeled as an open crack. No tensile forces are acting across it onto the downslope section of the sliding body, which is moving as a whole. Around section-point A, the soil body below the secondary shear zone is assumed to move at the same velocity as point B. The material properties of the secondary shear zone, which is not subject of this study, were fixed to comply with this criterion. At present, the displacement rates observed along the slope are more or less constant, 0 100 cm being superposed only by seasonal variations attributed to Displacements fluctuations of the groundwater conditions which are not sept. 2001–sept. 2004 modeled in this study. Therefore, our geotechnical model 0 200 400 600 800 1000 (meters) features displacement rates remaining constant with time. Field point 4.4.3. Numerical model Model point Figure 7: Site map with the location of the section and the GPS Figure 11 shows the characteristics of the numerical model of points. the studied slope. A plane strain geometrical configuration with the real dimensions of the slope was used. The model was discretized with 1070 triangular six-node elements. To save calculation time, the number of elements was reduced by endanger human settlements, a detailed study of the geology modeling only the uppermost 20 m of the bedrock layer. The and geomorphology as well as comprehensive monitoring upper and the lower layers were meshed with the automatic were carried out by the University of Modena and Reggio meshing procedure of the software and using a very coarse Emilia and by the National Research Council’s Group for setting. A linear elastic material model was assigned to them. Hydrogeological Catastrophes Defense (CNR-GNDCI) [25] Finally, one forward calculation took approximately three by order of the Autonomous Province of Bolzano/South- minutes on an ordinary personal computer. The interme- Tyrol [24]. diate layer was meshed manually by predefining geometry In this context, the displacements of several surface points in order to assure a sufficiently fine mesh and a points were observed regularly by means of global position- suitable orientation of the triangles in order to go against an ing system (GPS) measurements. In our study, a subarea of excessive distortion of the element shapes by the calculated the slope was modeled along a representative 2D section. A deformations. For the same reason, the updated mesh option site plan with the location of the section and the GPS points was only used in the last three years of the simulation (which is given in Figure 7,whereas Figure 8 depicts a field survey of are compared to the reference dataset). The detachment this section and shows the abstracted geometry model. zones were modeled by means of interfaces on both sides of their geometry lines. The interfaces were modeled with 4.4.2. Geotechnical model a Mohr-Coulomb material model, with negligible cohesion, the same friction angle as the basal shear zone and with The geometry was determined using all the information a constant reference stiffness in the order of magnitude available, that is, a core drilling near section-point B, the of the shear zone stiffness. The accuracy of the calculated local geomorphology, refraction seismics, and direct current deformation results was checked by carrying out several resistivity (DC-resistivity) [24]. As exposed in Figure 9, comparative analyses with finer meshes and also with a the vertical profile of the slope was divided into three horizontal basal boundary. Horizontal fixities were assigned layers interpreting various inclinometer profiles reported by to the lateral edges of the model, which extends over a total Corsini et al. [25]; the illustrated one is located near section- length of 1080 m. Vertical fixities were attributed to the basal point B. The uppermost layer, the soft soil cover, which is boundary, representing the stable bedrock. showing little internal deformations, was only considered in the form of its weight acting on the intermediate layer, the shear zone. Therefore, the displacement vectors of section- 4.4.4. Calculation phases points C and F are presumed to be equal to those of section- points B and E, respectively. The shear zone is assumed to be In the first calculation phase, all three layers are made up a thin, soft, and highly plastic layer, exhibiting a pronounced by the bedrock material. An initial stress state is generated time dependency in its mechanical behavior. The third layer by applying the self-weight of this material (gravity loading is given by the underlying weathered bedrock, which is procedure). In the second calculation phase, the two upper supposed to be stable. The earth pressure at the foot of layers are replaced by the weaker material of the soil cover. the slope was assumed to be slightly lower than the earth The third calculation phase marks the starting point of the 10 Journal of Artificial Evolution and Applications Survey 1b Survey 1a 0 50 100 m Soil cover Deeply weathered St. Cassian and La Valle beds (a) Actual main Secondary shear surface detachment zone at oversteepened front Basal shear surface not or only partially A developed At present low movement rates Detachment zones indicated by morphology No longer active Basal shear surface active for 100 s or 1000 s of years Landslide mass moving as a whole Dip angle of basal shear surface near 0 (b) Figure 8: Field survey along the section and derived geometry model. Displacements (cm) 10 0 -Soft - Little internal deformation Only considered in the form of the load imposed on the - Thin basal shear surface -Soft 40 - Highly plastic - Showing time-dependent Stable C4 behavior Inclinometer profile near model points B,C,D (Corsini et al. 2005) Figure 9: Interpretation of the inclinometer data from Corsini et al. 2005 [25]. Depth (m) Joerg Meier et al. 11 e(h) = earth pressure at depth h φ = effective friction angle of soil cover 90 m Earth pressure at rest would give For φ = 25 Assumed earth pressure distribution ∗ ∗ e(h) = γ 0.68 h ∗ ∗ e(h) = γ 0.6 h + x ◦ 3 For φ = 20 γ = 19 kN/m (specific weight of soil) ∗ ∗ e(h) = γ 0.84 h x = 20 kPa (for better technical performance) Figure 10: Earth pressure assumptions made for the foot of the slope. 20–40 m 2m 20–30 m (a) (b) (c) Figure 11: Discretization of the slope model: foot zone, vertical profile, and detachment zone. 00.51 1.52 2.53 3.54 4.55 Total displacements (mm) Figure 12: Results of a forward calculation using laboratory values of the parameters. (m) (m) 12 Journal of Artificial Evolution and Applications 010 20 30 40 50 60 Total displacements 2001–2004 (cm) Figure 13: Calibrated slope model—comparison of modeled displacement vectors (blue) with reference data derived from the GPS measurements (brown). Table 8: Laboratory values of the parameters used for the slope slope instability in the model. The shear zone material with example. time-dependent mechanical behavior is inserted and the horizontal load at the foot is set. After that, the model is Experimental parameter values left creeping with unchanged boundary conditions during λ 0.064 a period of 33 years. As the loading history of the shear κ 0.035 zone material is unknown, this time period had to be chosen μ 0.00146 arbitrarily to reach constant displacement rates as they are c (kPa) 0.01 presently observed along the natural slope. ϕ ()10 G (kPa) 5560 4.4.5. Results of initial model using parameters v 0.35 derived from experiments ∗ ∗ ∗ In a first trial forward calculation, for λ , κ ,and μ , the parameters calculated from the laboratory experiments were 4.4.6. Results of statistical analysis and used as input values. As the deformations along the shear optimization procedure zone are known to persist since hundreds or thousands of A statistical analysis was carried out (which will not be years [26], the shear strength of this zone has decreased to reported in detail here). One interesting finding of this a residual value that is characteristic for the soils originated analysis was that the friction angle and the modified creep by the weathering of the San Cassiano and La Valle beds index appeared to be closely correlated (coefficient of 0.92). outcropping in the whole slope area. Therefore, cohesion ∗ ∗ ∗ The parameters λ , κ ,and μ ; the friction angle; and the was assumed to be negligible (0.01 kPa) and a friction stiffness of the uppermost layer (represented by its shear angle of 10 was adopted, according to the average slope modulus G) were chosen for the optimization procedure inclination observed in nearby areas which were formed during which they were varied within the intervals specified since the Late Glacial by the studied processes (earth slides in Table 9. and earthflows) and covered by comparable soil covers [27]. After 82 cycles, each of them consisting of 10 forward The stiffness of the uppermost layer was set equal to the calculations, the procedure was stopped because, from then stiffness modulus observed in the oedometer test during on, the deviation could no longer be reduced significantly. unloading and reloading between the load steps 400 kPa and The resulting parameter set is also given in Table 9. Figure 13 800 kPa. For Poisson’s ratio of this layer, we used 0.35, a depicts the calculated deformations using the identified value that is considered to be characteristic of clayey soils. parameter combination, together with the displacement The experimental parameter values are shown in Table 8 vectors of the GPS measurement points. and the deformations calculated on their basis for the last It can be observed that the identified parameter set is able three years of the creep phase are presented by Figure 12. to reproduce the field measurements qualitatively. Because The latter are only in the range of millimeters, and thus not of the simplifications made in the model, no exact fit of the representing the actual situation in the field, where between displacement vectors is possible. The presented back analysis September 2001 and September 2004, displacements from procedure gives one of a number of possible approximate several centimeters to several decimeters were measured. solutions to the geotechnical problem and the result returned Joerg Meier et al. 13 Table 9: Search intervals and identified parameters for the slope example. Identified values Varied parameters Parameter Fixed parameters (PSO) Maximum Minimum v 0.35 — — — G (kPa) — 20000 200 5160 c(kPa) 0.01 ——— ψ () 0 ——— κ — 1 0.005 0.60 λ — 2 0.01 1.42 μ — 1.5 0.001 0.145 ϕ ( ) — 16 8 10.7 by the particle swarm optimizer can be seen as a parameter [2] M. Calvello and R. J. Finno, “Selecting parameters to optimize set that best represents the reference data. in model calibration by inverse analysis,” Computers and Geotechnics, vol. 31, no. 5, pp. 411–425, 2004. [3] J. Carrera, A. Alcolea, A. Medina, J. Hidalgo, and L. Slooten, 5. CONCLUSIONS “Inverse problem in hydrogeology,” Hydrogeological Journal, vol. 13, no. 1, pp. 206–222, 2005. A back analysis procedure for the identification of material [4] J.Meier,S.Rudolph, andT.Schanz, “Effektiver Algorithmus parameters of constitutive models applied to geotechnical zur Losung ¨ von inversen Aufgabenstellungen—Anwendung in problems was presented. This procedure represents a direct der Geomechanik,” Bautechnik, vol. 83, no. 7, pp. 470–481, approach based on the method of least squares, correlation analyses, and a particle swarm optimization algorithm. The [5] T. Schanz, M. M. Zimmerer, M. Datcheva, and J. Meier, “Iden- applicability and suitability of the technique was demon- tification of constitutive parameters for numerical models via strated by means of three examples from the fields of soil inverse approach,” Felsbau, vol. 24, no. 2, pp. 11–21, 2006. mechanics and engineering geology. The studied material [6] Z. F. Zhang, A. L. Ward, and G. W. Gee, “Estimating soil was a natural soil. Besides being way more objective and less hydraulic parameters of a field drainage experiment using inverse techniques,” Vadose Zone Journal, vol. 2, no. 2, pp. 201– arbitrary than the conventional trial and error procedure, 211, 2003. the outlined method provides valuable information on the [7] A. Ledesma, A. Gens, and E. E. Alonso, “Estimation of param- quality of the model calibration, the uniqueness of an eters in geotechnical backanalysis—I. Maximum likelihood obtained solution, or the determinateness of the problem. approach,” Computers and Geotechnics, vol. 18, no. 1, pp. 1– In all three examples, the particle swarm optimizer was 27, 1996. able to identify an improved parameter set after a justifiable [8] A. Gens, A. Ledesma, and E. E. Alonso, “Estimation of amount of forward calculations. Further research should parameters in geotechnical backanalysis—II. Application to also concentrate on the identification of the geometrical a tunnel excavation problem,” Computers and Geotechnics, parameters of geotechnical problems. vol. 18, no. 1, pp. 29–46, 1996. [9] A. Ledesma, A. Gens, and E. E. Alonso, “Parameter and variance estimation in geotechnical backanalysis using prior ACKNOWLEDGMENTS information,” International Journal for Numerical and Analyti- cal Methods in Geomechanics, vol. 20, no. 2, pp. 119–141, 1996. The German Academic Exchange Service (DAAD) and the [10] Y. Malecot, E. Flavigny, and M. Boulon, “Inverse analysis of Association of the Rectors of the Italian Universities (CRUI) soil parameters for finite element simulation of geotechnical are acknowledged for funding traveling expenses through a structures: pressuremeter test and excavation problem,” in VIGONI exchange project. The second author acknowledges Proceedings of the Symposium on Geotechnical Innovations,R. the support by the Konrad-Adenauer-Foundation via a B. J. Brinkgreve, H. Schad, H. f. Schweiger, and E. Willand, postgraduate scholarship. The work of the first author was Eds., pp. 659–675, Verlag Gluc ¨ kauf, Essen, Germany, 2004. funded by the German Research Foundation (DFG) via the [11] S. Levasseur, Y. Malecot, M. Boulon, and E. Flavigny, “Soil projects SCHA 675/7-2 and SCHA 675/11-2 “Geomechanical parameter identification using a genetic algorithm,” Inter- modeling of large mountainous slopes”. national Journal for Numerical and Analytical Methods in Geomechanics, vol. 32, no. 2, pp. 189–213, 2007. [12] S. Levasseur, Y. Malecot, M. Boulon, and E. Flavigny, “Soil REFERENCES parameter identification from in situ measurements using a [1] M. Calvello and R. J. Finno, “Calibration of soil models genetic algorithm and a principle component analysis,” in by inverse analysis,” in NumericalModelsinGeomechanics Proceedings of the 10th International Symposium on Numerical NUMOG VIII, G. Pande and S. Pietruszczak, Eds., pp. 107– Models in Geomechanics (NUMOG ’07), Rhodes, Greece, April 116, Balkema, Rotterdam, The Netherlands, 2002. 2007. 14 Journal of Artificial Evolution and Applications [13] X.-T. Feng, B.-R. Chen, C. Yang, H. Zhou, and X. Ding, “Identification of visco-elastic models for rocks using genetic programming coupled with the modified particle swarm opti- mization algorithm,” International Journal of Rock Mechanics and Mining Sciences, vol. 43, no. 5, pp. 789–801, 2006. [14] S. Finsterle, “Demonstration of optimization techniques for groundwater plume remediation using iTOUGH2,” Environ- mental Modelling & Software, vol. 21, no. 5, pp. 665–680, 2006. [15] L. Cui and D. Sheng, “Genetic algorithms in probabilistic finite element analysis of geotechnical problems,” Computers and Geotechnics, vol. 32, no. 8, pp. 555–563, 2005. [16] A. Cividini, L. Jurina, and G. Gioda, “Some aspects of ‘characterization’ problems in geomechanics,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 18, no. 6, pp. 487–503, 1981. [17] R. C. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the 6th International Symposium on Micro Machine and Human Science (MHS ’95), pp. 39–43, Nagoya, Japan, October 1995. [18] J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, IEEE Press, Piscataway, NJ, USA, November-December 1995. [19] B. F. J. Manly, Multivariate Statistical Methods: A Primer, Chapman & Hall/CRC, Boca Raton, Fla, USA, 3rd edition, [20] J. Will,D.Roos, J. Riedel,and C. Bucher,“Robustness analysis in stochastic structural mechanics,” in NAFEMS Seminar: Use of Stochastics in FEM Analyses, Wiesbaden, Germany, May [21] P. A. Vermeer and H. P. Neher, “A soft soil model that accounts for creep,” in Proceedings of the International Symposium. Beyond 2000 in Computational Geotechnics. 10 Years of PLAXIS Inetrnational, pp. 55–58, Balkema, Amsterdam, The Nether- lands, March 1999. [22] K.-J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood-Cliffs, NJ, USA, 1982. [23] K. Lemke, H. Lorenz, A. Klimitsch, J. Koeditz, T. Schaefer, and K. J. Witt, “Results of laboratory study on earthflow materials,” Tech. Rep., Material Research and Testing Institute MFPA, Weimar, Germany, 2006. [24] M. Panizza, S. Silvano, A. Corsini, et al., “Definizione della pericolosita ` e di possibili interventi di mitigazione della frana di Corvara in Badia. Provincia Autonoma di Bolzano— Alto Adige,” Autonome Provinz Bozen—Sudtir ¨ ol, http:// www.provincia.bz.it/opere-idrauliche/attivita6 i.htm, 2006. [25] A. Corsini, A. Pasuto, M. Soldati, and A. Zannoni, “Field monitoring of the Corvara landslide (Dolomites, Italy) and its relevance for hazard assessment,” Geomorphology, vol. 66, no. 1–4, pp. 149–165, 2005. [26] M. Soldati, A. Corsini, and A. Pasuto, “Landslides and climate change in the Italian Dolomites since the Late glacial,” Catena, vol. 55, no. 2, pp. 141–161, 2004. [27] A. Corsini, “L’influenza dei fenomeni franosi sull’evoluzione geomorfologica post-glaciale dell’Alta Val Badia e della Valparola (Dolomiti),” Ph.D. thesis, University of Bologna, Bologna, Italy, 2000.

Journal

Journal of Artificial Evolution and ApplicationsHindawi Publishing Corporation

Published: May 4, 2008

References