Topology optimization for the design of geotechnical structures

K.-F. Seitz*1 1 Institute of Geotechnical Engineering and Construction Management of the Hamburg University of

Technology, Hamburg, Germany * Corresponding Author

ABSTRACT This paper presents the topology optimization for the design of two dimensional basic geotechnical structures. The SIMPmethod is applied in order to optimize the structures’ topologies concerning their deformational behaviour within the serviceability state.The finite element analysis is conducted with Abaqus standard using a hypoplastic constitutive model. Two boundary value problems – pileand retaining wall – are analysed and complement recently presented topology optimization of strip footings. This paper summarizes the re-sults in order to give an overview on the current application of topology optimization of basic geotechnical structures.

1 INTRODUCTION

Design optimization is successfully used in a wide range of engineering fields. By means of numerical methods the structural design can be adapted opti-mally to specific problems and demands. This saves expenses, economizes material use, improves the de-formational behaviour and reduces construction time. However, despite the great potential, structural opti-mization methods are rarely employed in geotechni-cal design.

Considering the design for a strip foundation (Figures 1-2) three branches of structural optimiza-tion may be employed at different stages throughout the design process. At first the basic design for the

foundation needs to be chosen. At this early stage to-pology optimization may be conducted. Using this method the most efficient basic design can be found without constraining it to preset solutions since there is no need to provide an initial design. Possible to-pologies for the strip footing are a foundation block underneath the loaded plate or a structure consisting of two or three piles (Figure 1). When the basic de-sign is chosen, e.g. the two piled construction, the form can be specified. Shape optimization can be used at this stage. Angles and the forms of the differ-ent parts are varied during optimization (Figure 2). Finally, sizing may be used to optimize construction parts, e.g. material thicknesses or profiles may be varied to specify the most efficient design.

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Figure 1. Topology optimization: three different topologies for the foundation of a strip footing.

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Within the structural optimization methods sizing is best known and often applied in geotechnical engi-neering during the detailed design. Shape optimiza-tion is less often applied. First applications of shape optimization can be found for a piled raft foundation (Kim et al. 1999; Grabe & Pucker 2011), for a pile grillage (Čiegis 2006; Kinzler 2011) and for a quay wall construction (Grabe et al. 2010).

The herein presented application of topology op-timization for basic structural geotechnical design is still a subject of research. Based only on the knowl-edge of loads and limiting boundaries it is possible to optimize the basic structural set-up, the so-called to-pology. Therefore, topology optimization is able to support the design process at early stage. The first application of topology optimization to a geotechni-cal problem is considering an underground excava-tion in linear-elastic rock material (Ren et al. 2005; Yin et al. 2000; Yin & Yang 2000; Ghabraie et al. 2010). Topology optimization of a strip footing in granular Mohr-Coulomb material is presented by Kinzler (2011). Pucker & Grabe (2011), Pucker (2013) and Seitz et al. (2015) present the application of topology optimization of a strip footing in granular hypoplastic material.

This contribution presents the numerical optimiza-tion of retaining walls and pile foundations and summarizes the application of topology optimization on basic geotechnical design in hypoplastic material, referencing the optimization of strip footings in Pucker (2013) and Seitz et al. (2015).

A commonly known numerical topology optimiza-tion algorithm, SIMP, is used in combination with the finite element (FE) method in order to determine an optimized topology. The main objective of the op-timization is to improve the deformational behaviour within the serviceability state.

The following sections present the numerical methods (2) and the numerical model (3). The opti-mization results for three different basic designs are discussed in section 4.

2 NUMERICAL METHODS

The topology optimization combines the FE-analysis with a numerical optimization algorithm in an itera-tive procedure. The density-based approach (Bendsøe 1989) to topology optimization uses a relative, virtual density of structural material within a specified de-sign domain, in which the topology may develop. Throughout the optimization process the material dis-tribution is varied inside the design domain in order to minimize the compliance; i.e. to minimize the de-formations. In each iteration step the structure’s compliance is evaluated with FE-analysis. Subse-quently, the numerical optimization algorithm up-dates the material distribution based on the current design sensitivity.

The topology optimization code from Sigmund (2001) is applied to the geotechnical design optimiza-tions. The code is based on a power-law approach called SIMP (Solid Isotropic Material with penaliza-tion). The relative material density varies between 0 (no structural material – only soil) and 1 (only struc-tural material – no soil). A distinct design, however, can only consist of relative material densities 0 and 1. Therefore, an iteratively increasing penalization fac-tor p, in accordance with Groenwold & Etman (2010), is applied to the relative density in order to obtain a design without intermediate densities. The penalization factor p will be chosen depending on the iteration step k (1).

10)021.1 ,5min(

10110

k

kp k

(1)

Two materials exist in the design domain, which is contrary to Sigmund’s formulation. Therefore, the optimization problem is adapted by adding a stiffness difference between the structure’s stiffness Estruc and the soil material’s stiffness Esoil. It can be written as in equation (2), where c is the compliance, N is the number of elements e, k0 is the element stiffness ma-trix, ρe is the relative density in element e and vf is

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Figure 2. Shape optimization: three different shapes for the foundation of a strip footing.

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the prescribed volume fraction of the structural mate-rial )(V inside the design domain V0.

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EE

KUUxc

eTe

N

e

pe

T

uku

(2)

The optimization problem (2) is solved using the optimality criteria method. The material distribution is determined using a heuristic updating scheme (3), see Bendsøe (1995), where m is a positive move-limit and η (= 0.5) a numerical damping coefficient.

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e

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Be is defined as

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(4)

where λ is a Lagrangian multiplier. The sensitivities of the objective function c are given by

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)()(

.

(5)

For further information on the optimality criteria method, the updating scheme and the derivation of sensitivities, see Bendsøe (1995).

3 NUMERICAL MODELS

The FE-models are chosen to be planar with granular soil and solved using the FE-code Abaqus

standard. Different boundary value problems are ana-lysed:

Pile – a pile modelled through coupled nodes at the ground surface which is loaded with a hori-zontal force H = 5 kN, a vertical force V = 1 kN and a momentum M = 5 kN m (Figure 3).

loaded coupled nodes(width = 0,8 m)

design domain

granular soil

10 m

10 m

20 m

20 m

20 m

grid0.1 x 0.1

grid1.0 x 1.0

Figure 3. 2d pile model.

Retaining wall – modelled as symmetrical con-

struction excavation (Figure 4). The steel wall is 10 cm thick.

steel wall

design domain

granular soil

5 m

5 m4 m

2 m

2.5 m

10 m

10 m

grid0.1 x 0.1

grid1.0 x 1.0

Figure 4. 2d retaining wall model.

Vertically loaded plate (Pucker 2013) – in a

symmetrical model a 1 m wide rigid plate is loaded centrically. The design domain measures 2 x 3 m. The model boundaries are at 6 m in horizontal direction and 7 m in vertical direc-tion. The discretization is in accordance with the presented models in Figures 3 & 4.

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Vertically loaded plate with eccentricity (Seitz et al. 2015) – in a 2d model a 1 m wide rigid plate is loaded with a concentrated vertical force with 0.3 m eccentricity. The design do-main measures 5 x 5 m and is in the centre of the 45 x 25 m model. The discretization is in accordance with the presented models in Fig-ures 3 & 4.

The soil is modelled with a hypoplastic constitu-tive model which is not only able to reproduce hy-poplastic material behaviour but also capable of modelling material transition between soil and con-struction material such as concrete and reproducing linear elastic, ideal plastic material behaviour. An in-terpolation factor is used in accordance with the rela-tive material densities in order to reproduce either the hypoplastic soil material (ρe = 0) or the linear elastic, ideal plastic structural material (ρe = 1). The soil con-sists of a granular material and the structural material is modelled as cement-sand mixture, that may arise e.g. from jet grouting. For detailed information on the constitutive model see Pucker (2013).

4 OPTIMIZATION RESULTS

One hundred iteration steps are carried out for the topology optimization. The optimization results for the retaining wall model and the pile model are shown in Figure 5 and Figure 6. Reference to the to-pology optimization for a strip footing is made with Figure 7 and Figure 8. Depicted are the topologies for the volume fractions (vf) 1 %, 5 % and 10 % of the design domain volume. It is clearly visible that the optimal topology is dependent on the structural system:

The optimal topology for a system loaded with a horizontal force, a vertical force and a mo-mentum (pile) is located directly underneath the loading. For smaller material volumes (vf = 1 %, Figure 5, left) the material concentrates underneath the loading transfer point. At 5 % vf (Figure 5, middle) the structure expands in horizontal direction but keeps the core under-neath the loading point. At 10 % vf (Figure 5, right) the optimal topology cannot expand any further in horizontal direction and therefore ex-pands downwards; the core underneath the loading point persists; a structure with a similar

shape as an anchor develops additionally on the left side.

The optimal topology to support the retaining wall is located at the level of surface of the ex-cavation. The material concentrates at the lower end of the earth wedge which develops when the soil is excavated. Therefore, the load on the retaining wall is reduced as well as the defor-mation.

For a vertically loaded rigid plate the optimized topology is oriented vertically downwards (Figures 7 & 8). The topologies for the centri-cally loaded plate develop at the plate edges, where the maximal stresses are located (Figure 7). The topologies for the load with eccentricity develop directly underneath the load transfer point, such that a rotation of the rigid plate is hindered and therefore the deformations are re-duced (Figure 8).

The improvement of the deformational behaviour is evaluated by loading an alternative foundation which has the same material volume. For the pile model the alternative foundation is depicted in Figure 5. The alternative pile is as wide as the loaded sur-face (0.8 m) and its length depends on the available material volume. The settlement of loaded centre as well as the horizontal displacement and the rotation are compared to the deformations of the optimized topologies. The displacements can be reduced up to 84.4 % and the rotation up to 88.6 %.

For the support of the retaining wall the alternative topology is a structure with a similar shape as an an-chor at 1m depth and 3 m length. The width is ad-justed to the available material volume. The dis-placement of the retaining wall in horizontal direction at ground surface is compared to the dis-placement of the optimized topology. The horizontal displacement can be reduced up to 20.6 % through topology optimization (Figure 6).

The alternative foundation for the vertically loaded rigid plates is a foundation block with the same width as the rigid plate. The depth is depending on the material volume. The settlements of the cen-trically loaded plate can be reduced up to 29.8 %. The improvement of the deformational behaviour for the plate with eccentric load is limited to the reduc-tion of rotation at greater volume fractions. Rotation of the eccentrically loaded plate can be reduced up to

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90.7 %, whereas the settlements cannot be reduced with the optimized topology.

5 CONCLUSION

This paper presents the application of topology opti-mization to basic geotechnical design. A pile as well as a retaining wall are analysed and they complete the already existing analyses concerning strip foun-dations. All results are evaluated in summary to give an overview on topology optimization in basic geo-technical design.

Topology optimization is applicable on basic geo-technical designs with remarkable improvement of the deformational behaviour. The optimized topolo-gies are better fitted to the applied loads than the al-ternative set-ups that have been presented. Pile-like foundations are best suited for vertical loads. At the load transfer point, horizontally distributed material with diagonally expanding piles in direction of the load are well suited for combined vertical, horizontal and momentum loads. Retaining walls can be sup-ported by material which is located at the height of the excavation surface where the earth wedge begins.

The application of topology optimization needs to be investigated further. The application on three di-mensional geotechnical problems should be investi-gated. Also, the current implementation does not take into account any relative displacements between the optimized structure and the surrounding soil, since the optimized topology is defined as material distri-bution and does not update the existing mesh.

ACKNOWLEDGEMENT

The work is part of the project “Finite-Element based multicriterial numerical optimization of geotechnical structures in the service limit state” (GR-1024-9-3) funded by the German Research Foundation (DFG).

REFERENCES

Bendsøe, M. 1989. Optimal shape design as a material distribution problem, Structural Optimization 1, 193–202. Bendsøe, M. 1995. Optimization of structural topology, shape and material. Springer, Berlin Heidelberg New York.

Čiegis, R. Baravykaité, M. & Belevičius, R. 2006. Parallel global optimization of foundation schemes in civil engineering Applied Parallel Computing. State of the Art in Scientific Computing, Springer, Berlin Heidelberg, 305–312. Ghabraie, K. Xie, Y.M. Huang, X. & Ren, G. 2010. Shape and re-inforcement optimization of underground tunnels, Journal of Computational Science and Technology 4, 51–63. Grabe, J. Kinzler, S. Pucker, T. & Mardfeldt, B. 2010. Unter-suchungen des Tragverhaltens und der Anwendbarkeit nu-merischer Optimierungsverfahren für Kaikonstruktionen. Ta-gungsband der 31. Baugrundtagung 2010 in München (Eds. Deutsche Gesellschaft für Geotechnik DGGT e.V.), 123–129. Grabe, J. & Pucker, T. 2011. Beitrag zum Entwurf und zur Aus-führung von kombinierten Pfahl-Plattengründungen, Bautechnik 88, 828–835. Groenwold, A.A. & Etman, L.F.P. 2010. A quadratic approxima-tion for structural topology optimization, International Journal for Numerical Methods in Engineering 82, 505–524. Kim, K.N. Lee, S.-H. Chung, C.-K. & Lee, H.S. 1999. Optimal pile placement for minimizing differential settlements, KSCE Journal of Civil Engineering 3, 831–839. Kinzler, S. 2011. Zur Parameteridentifikation, Entwurfs- und Strukturoptimierung in der Geotechnik mittels numerischer Ver-fahren. Veröffentlichungen des Institutes für Geotechnik und Baubetrieb der Technischen Universität Hamburg-Harburg, Ham-burg. Pucker, T. 2013. Stoffmodell zur Modellierung von stetigen Mate-rialübergängen im Rahmen der Optimierung geotechnischer Strukturen. Veröffentlichungen des Institutes für Geotechnik und Baubetrieb der Technischen Universität Hamburg-Harburg, Ham-burg. Pucker, T. & Grabe, J. 2011. Structural optimization in geotechni-cal engineering – basics and application, Acta Geotechnica 6, 41–49. Ren, G. Smith, J.V. Tang, J.W. & Xie, Y.M. 2005. Underground excavation shape using an evolutionary procedure, Computers & Geotechnics 32, 122–132. Seitz, K.-F. Pucker, T. & Grabe, J. 2015. Topologieoptimierung in der Geotechnik: Anwendung auf Gründungsstrukturen und Validierung, Geotechnik – to be published. Sigmund, O. 2001. A 99 line topology optimization code written in Matlab, Structural Multidisciplinary Optimization 21, 120–127. Yin, L. & Yang, W. 2000. Topology optimization for tunnel sup-port in layered geotechnical structures, International Journal for Numerical Methods in Engineering 47, 1983–1996. Yin, L. Yang, W. Guo, T. 2000 Tunnel reinforcement via topology optimization, International Journal for Numerical and Analytical Methods in Geomechanics 24, 201–213.

5

0

20 25-5

0

20 25-5

0

20 25-5

0.8

x

u1

u2

ur3

1 % vf 5 % vf 10 % vf

u1 37.3 % 10.3 % 46.5 % u2 64.5 % 54.0 % 84.4 % ur3 73.6 % 83.1 % 88.6 %

Figure 5. Optimized topologies (1, 5 & 10 % vf) for a horizontally and vertically loaded pile; sketch: alternative foundation layout [m]. Deformational behaviour improvement in percentage of alternative model deformation considering u1, u2 and ur3.

0

2.5 7.5-5

0

2.5 7.5-5

0

2.5 7.5-5

3

1x

u1

1 % vf 5 % vf 10 % vf

u1 --* 2.5 % 20.6 % *alternative model not stable

Figure 6. Optimized topologies (1, 5 & 10 % vf) for a retaining wall; sketch: alternative foundation layout [m]. Deformational behav-iour improvement in percentage of alternative model deformation considering u1.

0

0

-32

0

0

-32

0

0

-32

1

xu2

1 % vf 5 % vf 10 % vf

u2 5.2 % 22.2 % 29.8 %

Figure 7. Optimized topologies (1, 5 & 10 % vf) for a centric vertically loaded shallow foundation (Pucker 2013); sketch: alternative foundation layout [m]. Deformational behaviour improvement in percentage of alternative model deformation considering u2.

0

20 25-5

0

20 25-5

0

20 25-5

1

xu2

ur3

1 % vf 5 % vf 10 % vf

u2 -93.3 % -79.2 % -71.5 % ur3 -96.8 % 45.1 % 90.7 %

Figure 8. Optimized topologies (1, 5 & 10 % vf) for an eccentric vertically loaded shallow foundation (Seitz et al. 2015); sketch: alter-native foundation layout [m]. Deformational behaviour improvement in percentage of alternative model deformation considering u2,ur3.

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