Solid Mechanics

Faculty of Engineering, LTH

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Structural Optimization

Structural optimization has over the past decades qualified as an important tool in the design process. The method can be grouped into topology, size and shape optimization. The objective of the optimization can be to minimize the stresses weight or compliance for a given amount of material and boundary conditions. The method can be utilized to design engineering structures but it can also be used to tailor mircostructures.

The most widely used numerical scheme for topology optimization is the Solid Isotropic Material with Penalization (SIMP) scheme where the density is approximated as constant within each element. The objective of the optimization is in the present work to find a design that maximizes the stiffness for a given amount of material. The advantage of using the stiffness as the objective, or rather its complement the compliance, is that it is a global measure and thus can be represented by a scalar value. Moreover, the constraint on the volume is also particular simple since it is linear and monotone which in most cases gives rise to a robust numerical algorithm. The SIMP procedure is based on a sequence of convex approximations and the algorithm is simple to implement and at the same time numerically efficient. A distinct black/white solution is obtained by penalizing gray designs via a scaling of the elastic constitutive relation. This formulation has been shown to be ill-posed due to the length scale missing in the formulation. The remedy to this problem regularize the problem via e.g. a filter.

Phase-field based topology optimization

Several procedures for regularization of the topology optimization problem has been proposed, these can be local such that the smallest size of a segment is limited, or it can restrict the total length of the perimeter. Such methods are known as filtering techniques, perimeter control, gradient control and recently also phase-field based approaches. All these methods has in common that they introduce a length scale in the formulation and thus restricting the smallest characteristic size either locally or globally. In [1] the Cahn-Hilliard approach was used to solve the topology optimization problem. This procedure has the advantage of being able to inherently preserve the material. The method is, however, computational expensive since a fourth order PDE is governing the optimality.

In Fig.1 one example of an optimal design derived in [1] is shown. Initial a rectangular domain if filled with material, in a random manner, the sequence of snap-shots then shown how the material is redistributed during the iteration process. The final figure then shows the optimal structure.

Figure 1. Cahn-Hilliard approach to topology optimization

To to improve the computational efficiency use can be made of the Allen-Cahn approach. In [2], the Allen-Cahn approach was combined with the Howard's policy iteration schemer. The advantage of the Howard policy iteration scheme is that the optimality is defined by a second order PDE which is significantly more efficient than the Cahn-Hilliard approach. In [3] the possibility by including boundary control directly into the objective functional was investigated.

Most topology optimization is performed for the assumptions that the strains are small. If finite strains are considered the optimal design depends on the load level, i.e. different optimal designs will be found for different load levels. In [4], the phase-field approach was extended to take finite strains into account. 

An example of a phase-field based topology optimization for large deformations can be seen in Fig. 2. Starting with an initial homogenous material distribution the sequence clearly shows that the structure becomes stiffer when the material is redistributed.

Figure 2. Phase-field based topology optimization for finite strains


  1. Optimal topologies derived from a phase-field method, M.Wallin, M.Ristinmaa and H. Askfelt, Structural and Multidiciplinary Optimization, Vol. 45, 171-183, (2012)
  2. Howard's algorithm in a phase-field topology optimization approach, M. Wallin and M. Ristinmaa, International Journal for Numerical methods for Numerical Methods in Engineering, Vol. 94, 43-59 (2013)
  3. Boundary Effects in a Phase-Field approach to Topology Optimization, M. Wallin and M. Ristinmaa, Comp. Methods Appl. Mech, Engrg., Vol. 278, pp145-159
  4. Finite strain topology optimization using phase-field regularization, M, Wallin and M. Ristinmaa, Structural and Multidiciplinary Optimization, In press (2014) doi: 10.1007/s00158-014-1141-8
  5. Topology optimization utilizing inverse motion based form finding, M. Wallin and M. RistinmaaAccepted for publication in Comp. Methods Appl. Mech, Engrg.
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