XFEM level set-based topology optimization considering damage

Abstract

In various engineering fields it is required to design material layouts with improved performance or tai- lored properties. Hereby, the structural integrity should not be compromised, such that material failure is avoided. Topology optimization enables the systematic design of structures presenting enhanced performance with respect to criteria, such as mass, volume, stiffness, or eigenfrequencies. In the majority of research works in topology optimization, material failure is addressed by constraining or minimizing the stress state in simplified analyses, in which the material is modeled as isotropic linear elastic. However, brittle and quasi-brittle materials, such as concrete and rock, progressively degrade on a micro-scale, before material rupture occurs macroscopically. Using continuum damage mechan- ics, the state of material degradation may be modeled through one or multiple internal variables that are thermodynamically irreversible. However, modeling damage leads to a computationally intensive, non-linear and path-dependent optimization problem. The limited previous research suggests, though, that accounting for material degradation throughout the design phase can significantly improve the per- formance and reliability of the resulting designs. In this work, a level set-based topology optimization approach is proposed to generate steel- reinforced concrete layouts with enhanced damage resistance. Additionally, a thorough comparison of addressing material failure in a linear stress-based and a non-linear damage-based TO approach is conducted. The geometry of the designs is represented implicitly through a level set function. The level set and state fields are discretized using the extended finite element method, which alleviates the need to remesh in each optimization iteration. Assuming isotropic damage, strain softening degradation in the concrete material is modeled through a single scalar damage variable. The steel as well as the un- damaged concrete are modeled as isotropic linear elastic materials. To avoid instability and localization of deformation, non-locality is introduced through a gradient-enhanced damage model. Boundary and interface conditions are imposed weakly using Nitsche’s method. Numerical instabilities due to small material subdomains are mitigated through face-oriented ghost stabilization. The optimization prob- lems are solved through mathematical programming using the globally convergent method of moving asymptotes. Accounting for the irreversibility of damage, the required shape-sensitivities are evaluated semi-analytically through an adjoint approach. The proposed optimization approach is applied to two-dimensional numerical problems from the literature. The results show that resorting to a simplified linear analysis is insufficient to generate struc- turally reliable designs of brittle material. When modeling the non-linear damaged material behavior throughout the optimization process, though, resulting layouts show an enhanced load-bearing capacity and resistance to damage. It is further demonstrated that the maximum damage level in steel-reinforced concrete layouts can be controlled, through an optimized distribution of reinforcing steel material.