A study on the computational effort of hyper-dual numbers to evaluate derivatives in geometrically nonlinear hyperelastic trusses

We report the use of hyper-dual numbers as a derivative tool for stress and tangent modulus calculation in hyperelastic models. By using the analytical expressions of Ogden's model as a reference, we confirmed the accuracy of this tool for both first- and second-order derivatives. Moreover, the hyper-dual implementation considering the operator overloading technique corroborated the interesting generality properties of this method; by using virtually the same code syntax, the material behavior was entirely described by the hyper-dual representation of the strain energy function. Therefore, the use of the hyper-dual procedure brings an important advantage in this field since the development of the expressions concerning stress and tangent modulus becomes unnecessary; only the potential function of a given model is implemented. In this context, due to its specific set of arithmetic operations, a detailed study on the efficiency of the hyper-dual scheme in a finite element analysis is demanded. In this research, we proposed a comparative study regarding the effects of the hyper-dual procedure on the analysis processing time. Using a specific mesh generator, we evaluated a wide range of model sizes for a beam structure made of hyperelastic trusses. As a result, when compared to an ordinary finite element execution using analytical expressions for the constitutive model, we found corresponding hyper-dual performance in the larger models. Particularly, we identified the model size from which the hyper-dual approach becomes competitive; considering this element type, this model contains approximately 30,000 elements and 25,000 degrees of freedom. Furthermore, as for the computational time related to the material subroutine and the stiffness matrix, we found that the hyper-dual scheme increased the computational time by a factor of 4 and 2, respectively. We demonstrated that the hyper-dual procedure combines interesting characteristics in terms of accuracy, generality and computational costs. Hence, this numerical strategy for derivative calculation potentially represents an important tool for the development and application of new constitutive models in structural mechanics.