Topology optimization with mixed finite elements: methods and formulations

Topology optimization is nowadays a fertile area of research concerned with the topical issue of defining the best design that solves an assigned physical problem with prescribed requirements or
restraints. This general concept may be applied to different industrial applications, where, already in an early stage of the design process, questions as finding an optimal layout in terms of
static or dynamic stiffness, cost, structural performances and so on must be affordably answered. Since the pioneering paper [1], where the topology optimization concept was introduced as an innovative and powerful approach to structural design, many steps have been taken in
several directions. Within this rich scenario, the aim of this contribution is to present alternative formulations for topology
optimization by distribution of isotropic material exploiting features peculiar to bidimensional mixed finite element schemes.

The variational principle of Hellinger-Reissner is adopted and two dual formulations are considered. The first one uses classical
polynomial finite elements to approximate a regular displacement field and a piecewise discontinuous stress field. Conversely, the dual
formulation, often referred to as “truly-mixed” in the literature [2], interpolates displacements with discontinuous functions while regular
ones are used for stresses. The adoption of these finite element techniques within a topology optimization framework has nice consequences on the numerical stability of the method, with special regard to the so-called “checkerboard problem”. Moreover, mixed schemes have two important properties that may be usefully exploited in topology optimization, i.e.:
-accuracy in the evaluation of the stresses (post-processing techniques peculiar to displacement-based finite elements are not
needed); -capability of passing the inf-sup condition, even in the case of incompressible materials (differently from standard displacement-based discretizations).

Advantage of the first feature is taken to deal with the still open
problem of finding the optimal topology with local/global stress constraints on the material strength [3]. The so-called “singularity
problem” [4], a numerical trouble that may prevent from convergence to the expected global minima, is also faced and a novel methodology of
solution, referred to as “qp-approach”, is presented.

The second of the above two main features is conversely exploited to find optimal designs for incompressible materials, providing the
numerical robustness needed to handle the incompressibility property within an optimization context. Since these materials have recently
been used in different applications mainly concerned with vibration issues and aseismic design, alternative topology optimization
formulations are presented and tested not only in a static framework but also within eigenvalue-based methodologies for dynamic design.
The capability of the “truly-mixed” method to pass the “inf-sup” condition even in presence of incompressible material may also be used
to model fluid phases with the aim of solving pressure-load problems, moving from the approach recently proposed in [5].

The contribution also outlines recent applications of truly-mixed finite elements in the field of cohesive crack propagation for
quasi-brittle materials.

[1] Bendsøe M., Kikuchi N., Generating optimal topologies in structural design using a homogeneization method, Computer Methods in
Applied Mechanics and Engineering, 1988, 71(2): 197-224.

[2] Brezzi, F., Fortin, M., Mixed and Hybrid Finite Element Methods,1991, Springer-Verlag, New York.

[3] Duysinx P., Bendsøe M., Topology optimization of continuum structures with local stress constraints. International Journal for
Numerical Methods in Engineering, 1998, 43:1453-1478.

[4] Rozvany G.I.N., Difficulties in truss topology optimization with stress, local buckling and system stability constraints, Structural
Optimization, 1996, 11: 213-217.

[5] Sigmund O., Clausen P.M., Topology optimization using a mixed formulation: An alternative way to solve pressure load problems,
Computer Methods in Applied Mechanics and Engineering, 2007, 196: 1874-1889.