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.

我们报告使用超对偶数作为超弹性模型中应力和切线模量计算的导数工具。通过使用Ogden模型的解析表达式作为参考，我们确认了该工具对于一阶和二阶导数的准确性。此外，考虑到运算符重载技术的超双重实现证实了该方法有趣的通用性。通过使用几乎相同的代码语法，材料的行为完全由应变能函数的双偶表示来描述。因此，由于不需要有关应力和切线模量的表达式的开发，因此使用双偶过程在该领域带来了重要的优势。仅实现了给定模型的潜在功能。在这种情况下，由于其特定的算术运算集，需要对有限元分析中超对偶方案的效率进行详细研究。在这项研究中，我们提出了关于超双重程序对分析处理时间的影响的比较研究。使用特定的网格生成器，我们评估了由超弹性桁架制成的梁结构的各种模型尺寸。结果，与使用本构模型的解析表达式执行普通有限元分析相比，我们在较大的模型中发现了相应的超双性能。尤其是，我们确定了模型的大小，从这个模型中，超对偶方法变得具有竞争力。考虑到此元素类型，此模型包含大约30,000个元素和25,000个自由度。此外，关于与材料子程序和刚度矩阵有关的计算时间，我们发现超对偶方案将计算时间分别增加了4倍和2倍。我们证明了超对偶过程结合了准确度，通用性和计算成本方面的有趣特征。因此，这种用于导数计算的数值策略可能代表了一种新的本构模型在结构力学中的开发和应用的重要工具。