Topology optimization of viscoelastic structures using a time-dependent adjoint method

We present a mathematical framework for producing optimal designs of structures that exhibit viscoelastic creep deformation. Using a linear viscoelastic finite element model, we implement a corresponding time-dependent adjoint sensitivity formulation. The resulting formula is then incorporated into a computational topology optimization framework in order to achieve optimal topologies based on the expected lifespan or operating cycle of the structure. Designs are optimized for minimum mass subject to a constraint on the maximum local deflection. We validate the method using several two-dimensional numerical examples that include design-dependent gravitational loading (self-weight) and time-dependent applied loads. Creep plots are used to quantify the impact of the viscoelastic optimization method. The results show that the design of the optimal structure is highly dependent on the load duration and the complete load history. This result supports the design premise that it is necessary to account for the full viscoelastic response of the structure when designing for optimal long-term performance.