Structural optimization in geotechnical engineering: basics and application
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DOI: 10.1007/s11440-011-0134-7
- Cite this article as:
- Pucker, T. & Grabe, J. Acta Geotech. (2011) 6: 41. doi:10.1007/s11440-011-0134-7
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Abstract
Structural optimization methods are used for a wide range of engineering problems. In geotechnical engineering however, only limited experience exists with these methods. The difficulties in applying such methods to geotechnical problems are discussed in this paper, and the adaption of the commonly known SIMP-method to geotechnical problems is introduced for a special case. An application example is used to present the potential of topology optimization methods, and the application to geotechnical engineering is evaluated.
Keywords
Geotechnic Hypoplasticity Optimization Topology1 Introduction
Optimization of structural topology, shape and material is state-of-the-art in a wide area of engineering applications. Methods of topology optimization are especially used to reduce weight, costs and material in the automotive and aircraft industries. The methods of structural optimization allow an automatic development of the optimal design of construction elements. Depending on the particular application, it is possible to generate an optimized structure from an unshaped block element. The coupling of structural optimization methods with numerical simulation software, such as the Finite Element Method, allows the simulation and estimation of almost any imaginable geometry. In structural engineering, the development and application of such optimization methods have only recently been initiated. This due to the differences between structural and mechanical engineering. Mass productions are aimed at mechanical engineering, where a small reduction in costs can enable large savings. In structural engineering, however, the product typically is a individual one-off design. The application of optimization methods in geotechnics is uncommon to date. Until now, optimization methods are used in geotechnics e.g. to determine slope stability (Baker and Gaber [2]). Another application of optimization is the inverse soil parameter determination (Zhang et al. [21] and Meier et al. [11]). An application of optimization af a geotechnical design process can be found in Kinzler [8]. The potential of these methods is shown by the results of the application in other engineering areas. In geotechnical engineering, these methods are able to support the engineer in achieving economic designs, even if the problem is very difficult and complex. In the strategies of optimization in automotive and aircraft industries, merely the structure of homogeneous materials is optimized. This contrasts with geotechnical engineering, where two different materials, structure and soil, have to be considered. In this paper, the development and application of structural optimization methods in geotechnics are demonstrated. Several methods to solve strutural optimization problems have been developed. A review of the most common formulations can be found by Arora and Wang [1]. The SIMP-Method of Sigmund [17], a combined method of topology and shape optimization, is described and used. This method is very robust and the link-up to a finite element programm can be realised simply.
2 Terminology
Structural optimization is divided in topology optimization, shape optimization and dimension optimization. Most of the structural optimization methods are developed especially for one of these three groups.
3 SIMP-method
The goal of the optimization method is to minimize the compliance of the structure in the design domain \(\Upomega\). Thus, the stiffness is maximized.
Sigmund [17] uses a filter to ensure a unique solution of the optimization task. The filter modifies the element sensitivity, so the solution does not depend on the discretization level. Further detail can be found in Sigmund [15, 16].
4 Adaptation of the SIMP-method to geotechnical engineering
The main assumption of the SIMP-method is the homogeneity of the material in the structure. The elements in the design domain with no material are disabled by assigning very soft material properties. The homogeneous material is modelled generally with linear elastic material behaviour. In geotechnics, at least two different material types, the soil and the structure, have to be considered. The material behaviour of soils is highly nonlinear and cannot be reproduced adequately with a linear elastic or a linear elastic - ideal plastic constitutive model.
4.1 SIMP-method adaption requirements
4.2 Material change-over in geotechnics
Material parameters for the hypoplastic constitutive model with intergranular strain
Parameter | Description |
---|---|
\(\varphi_c\) | Critical state friction angle |
h_{s} | Granular hardness |
n | Exponent |
e_{d0} | Minimal void ratio |
e_{i0} | Critical void ratio |
e_{c0} | Maximal void ratio |
α | Exponent |
β | Exponent |
R | Maximum value of intergranular strain |
m_{R} | Stiffness ratio at a change of direction of 180° |
m_{T} | Stiffness ratio at a change of direction of 90° |
β_{R} | Exponent |
χ | Exponent |
e_{0} | Void ratio at a stress state of σ = 0 kN/m^{2} |
Hypoplastic material parameters for the examples of material change-over
Parameter | \(\varphi_c\) | h_{s} [MPa] | n | e_{d0} | e_{i0} | e_{c0} | α | β |
---|---|---|---|---|---|---|---|---|
Nonlinear material change-over | ||||||||
Soft material | 36 | 3.2e7 | 0.18 | 0.53 | 0.73 | 0.80 | 0.08 | 1.80 |
Stiff material | 33 | 1.5e6 | 0.28 | 0.55 | 0.95 | 1.05 | 0.25 | 1.05 |
Linear material change-over | ||||||||
Soft material | 30 | 5.8e6 | 0.28 | 0.53 | 0.84 | 1.00 | 0.13 | 1.05 |
Stiff material | 45 | 1.5e6 | 0.28 | 0.55 | 0.95 | 1.05 | 0.50 | 3.05 |
Parameter | m_{R} | m_{T} | R | β_{R} | χ | e_{0} |
---|---|---|---|---|---|---|
Nonlinear material change-over | ||||||
Soft material | 2.0 | 1.0 | 1e-4 | 0.5 | 6.0 | 0.65 |
Stiff material | 2.0 | 1.0 | 1e-4 | 0.5 | 6.0 | 0.65 |
Linear material change-over | ||||||
Soft material | 2.0 | 5.0 | 1e-4 | 0.5 | 6.0 | 0.65 |
Stiff material | 2.0 | 5.0 | 1e-4 | 0.5 | 6.0 | 0.65 |
4.3 Implementation
The SIMP-method is coupled here with the commercial finite element method program Abaqus/Standard, Version 6.8. The method is implemented with an external stand-alone program routine. The routine calculates the distribution of the relative density and executes the finite element calculation. The compliance is evaluated after each successful calculation, and a new distribution of the relative density is defined. In this way the system is optimized iteratively.
Abaqus/Standard offers the possibility to use material behaviour depending on the actual temperature. The material parameters can be defined regarding different temperature states. This option is used to obtain the correlation between the material parameters and the relative density. The distribution of the relative density is set equal to a virtual temperature state T(x) in the finite element calculation. The temperature can reach values between 0 K < T(x) < 1 K. The material properties of elements with a temperature of T = 0 K relate to the soft material and elements with T = 1 K relate to the stiff material. The hypoplastic constitutive model is implemented in Abaqus via a user subroutine, which has to be extended for a temperature-depending soil parameter definition. The material change-over occurs according to Eq. 10.
5 Application
The estimated settlement and inclination of a structure affect mainly its usability and thus usually describe the main boundary conditions in geotechnics. In this application, the potential of the supposed topology optimization method is demonstrated using the example of a simple strip foundation. An optimized foundation structure topology has to be developed under an already existing foundation. The objective function is given in Eq. 5. A resulted settlement reduction can be shown with this simple example. The soil and all structural parts are discretized with 4-node volume elements with reduced integration and linear approach. The contact between the existing structural parts and the soil is modelled as hard contact in the normal direction and frictionless in the tangential direction. The contact surface between soil and developing foundation cannot be identified exactly with the optimization process; therefore, no contact formulation is considered. It is assumed that the foundation structure does not move relatively to the soil.
The behaviour of the already existing foundation is simulated with a linear elastic constitutive model. The Young’s modulus is \(E=3.2 \cdot 10^7\hbox{ kN/m}^2\) and the Poisson’s ratio ν = 0.2. Both soil and new foundation structure are simulated with the hypoplastic constitutive model of von Wolffersdorff [19] with the extension of intergranular strain of Niemunis and Herle [12]. The material parameters for this model are listed in Table 2. Parameters of the linear material change-over for the soft material correspond to the parameters of the soil and the stiff material correspond to the parameters of the foundation structure.
6 Practical realisation
7 Conclusion
The applicability of structural optimization methods, especially of the topology optimization in geotechnical engineering, can be demonstrated in this paper. The optimized topology of the application example can be explained with classical theory of soil mechanics. The realization of such topologies is partially possible with the current construction methods, such as the jet grouting. Further investigation is required for the interpretation of optimized structures regarding the application to other construction methods due to the costs of complex topologies. Nevertheless, the potential of the structural optimization methods has been demonstrated. These methods provide significant opportunities in the design process of geotechnical engineering.