A fast approach to analysis and optimization of viscoelastic beams

A new truly-mixed finite element for the analysis of viscoelastic beams is presented that is based on the additive decomposition of the bending moment in a viscoelastic and a purely elastic contribution. Bending moments are the primary variables that belong to H2(0,ℓ) whereas the kinematic variables (that are the velocities and not the displacements as usual) are globally discontinuous and elementwise linear. As for the peculiarities of the proposed finite element, results from relaxation and creep numerical tests are presented in much detail and a quadratic convergence assessed for all the variables involved. In the second part of the paper, a fast approach to structural (sizing) optimization, set as a topology optimization problem, of such viscoelastic beams is presented in the presence of time-dependent objective functions. Within a gradient-based minimization scheme that is solved via the method of moving asymptotes (Svanberg, 1987), a dual sensitivity analysis approach is derived and representative numerical results presented and discussed in much detail.