Form-finding of thermal-adaptive pin-bar assemblies based on eigenvalue modification

Abstract

Lattice structures with tunable expansion properties have been investigated in multidisciplinary fields to control the temperature effects of structures or materials. The expected thermal adaptivity can be achieved by optimizing the structural geometry. A novel method for the form-finding of thermal-adaptive pin-bar assemblies is developed in this paper by considering the control of structural temperature effects as the minimization of the potential energy of the system. Based on the stationarity condition of the potential energy with respect to the nodal coordinates, the compatibility relationship between the thermal elongations of members and the target nodal displacements is proven to be the sufficient and necessary condition for structural thermal adaptivity. The solvability of the compatibility equation is determined by the rank equality between the compatibility matrix and its augmented form, which can be measured by the number of nonzero eigenvalues of its Gramian matrix. The analytical relationship between the eigenvalues of the Gramian matrix and the nodal coordinates is established using the matrix perturbation theory. A numerical strategy based on Newton’s method is proposed in which the eigenvalues are gradually modified by adjusting the nodal coordinates until the rank equality is satisfied. To address the existence of multiple solutions with structural thermal adaptivity, structural symmetry and periodicity constraints are introduced to narrow the solution space. The thermal-adaptive configurations of three illustrative pin-bar assemblies are analyzed using the proposed form-finding method, and the expected thermal deformations are verified for the obtained configurations using the finite element software ABAQUS. Comparing the results obtained by the proposed method with those obtained by nonlinear programming and the genetic algorithm validates the advantages of the proposed method in terms of computational time, optimality of the obtained configuration and applicability to complex structural geometries.