Minimum energy adaptive structures – All-In-One problem formulation

•Design of adaptive structures through Total Energy Optimization (TEO) based on MINLP.•Minimization of whole-life energy (material embodied + operational energy during service).•Structural sizing, actuator layout and control commands are obtained simultaneously.•TEO-MINLP produces similar solutions to those obtained by the method TEO-Nested.•Structural adaptation is particularly beneficial for stiffness governed design. This paper presents a new All-In-One (AIO) implementation of an existing formulation to design adaptive structures through Total Energy Optimization (TEO). The method implemented in previous work is a nested optimization process, here named TEO-Nested. Numerical simulations and experimental testing have shown that the TEO-Nested method produces structures that embody and use significantly lower energy compared to passive designs. However, TEO-Nested does not guarantee solution optimality. The formulation presented in this paper is an AIO optimization based on Mixed Integer Nonlinear Programming (MINLP), here named TEO-MINLP. Element cross-section areas, internal forces, nodal displacements and control commands are treated as continuous variables while the actuator positions as binary variables. Stress and displacement limits are included in the optimization constraints. Case studies of reticular structures are employed to benchmark the solutions with those produced by the TEO-Nested method. Results have shown that both formulations produce similar solutions which are only marginally different in energy terms thus proving that the TEO-Nested method tends to converge to optimal (local) solutions. However, the computation time required by TEO-Nested is only a fraction of that required by TEO-MINLP, which makes the former more suitable for structures of complex layout that are made of many elements.