Titanium gas turbine disks are subject to a rare but not insignificant probability of fracture due to metallurgical defects, particularly hard $α$. A probabilistic methodology has been developed and implemented in concordance with the Federal Aviation Administration (FAA) Advisory Circular 33.14-1 to compute the probability of fracture of gas turbine titanium disks subject to low-frequency metallurgical (hard $α$) defects. This methodology is further developed here to ensure that a robust, converged, accurate calculation of the probability is computed that is independent of discretization issues. A zone-based material discretization methodology is implemented, then refined locally through further discretization using risk contribution factors as a metric. The technical approach is akin to “h” refinement in finite element analysis; that is, a local metric is used to indicate regions requiring further refinement, and subsequent refinement yields a more accurate solution. Supporting technology improvements are also discussed, including localized finite element refinement and onion skinning for zone subdivision resolution, and a restart database and parallel processing for computational efficiency. A numerical example is presented for demonstration.

Multicomplex and multidual numbers are two generalizations of complex numbers with multiple imaginary axes, useful for numerical computation of derivatives with machine precision. The similarities between multicomplex and multidual algebras allowed us to create a unified library to use either one for sensitivity analysis. This library can be used to compute arbitrary order derivates of functions of a single variable or multiple variables. The storage of matrix representations of multicomplex and multidual numbers is avoided using a combination of one-dimensional resizable arrays and an indexation method based on binary bitwise operations. To provide high computational efficiency and low memory usage, the multiplication of hypercomplex numbers up to sixth order is carried out using a hard-coded algorithm. For higher hypercomplex orders, the library uses by default a multiplication method based on binary bitwise operations. The computation of algebraic and transcendental functions is achieved using a Taylor series approximation. Fortran and Python versions were developed, and extensions to other languages are self-evident.

Solving transient heat transfer equations is required to understand the evolution of temperature and heat flux. This physics is highly dependent on the materials and environmental conditions. If these factors change with time and temperature, the process becomes nonlinear and numerical methods are required to predict the thermal response. Numerical tools are even more relevant when the number of parameters influencing the model is large, and it is necessary to isolate the most influential variables. In this regard, sensitivity analysis can be conducted to increase the process understanding and identify those variables. Here, we combine the complex-variable differentiation theory with the finite element formulation for transient heat transfer, allowing one to compute efficient and accurate first-order sensitivities. Although this approach takes advantage of complex algebra to calculate sensitivities, the method is implemented with real-variable solvers, facilitating the application within commercial software. We present this new methodology in a numerical example using the commercial software Abaqus. The calculation of sensitivities for the temperature and heat flux with respect to temperature-dependent material properties, boundary conditions, geometric parameters, and time are demonstrated. To highlight, the new sensitivity method showed step-size independence, mesh perturbation independence, and reduced computational time contrasting traditional sensitivity analysis methods such as finite differentiation.

Phononic metamaterials (PMs) exhibit frequency ranges at which elastic waves are attenuated called band gaps. However, this phenomenon is highly sensitive to geometrical variations and the unit cell’s mechanical properties. It is useful to have accurate sensitivity information to identify the variables that produce the highest impact on band gaps and guide the design of PMs with a desired wave propagation behavior. Current methodologies for sensitivity analysis in PMs, such as the finite difference method (FDM), are computationally inefficient, subjected to subtraction cancelation errors, and their accuracy is highly dependent on the magnitude of the perturbation step size. In this study, we introduce a new computational methodology to perform parameter sensitivity in the dynamic behavior of PMs using the multicomplex Taylor series expansion (ZTSE) coupled with Bloch’s theorem. The methodology allows one to obtain arbitrary-order sensitivities with high accuracy. In contrast to FDM, this methodology is computationally more efficient, eliminates the step size selection issue, and is not subjected to subtractive cancelation errors. Also, we show how the method can be applied using real algebra solvers. We limit our analysis to linear undamped PMs. The methodology using ZTSE with Bloch’s theorem is presented in numerical examples for the diatomic lattice and a 2D square lattice, where we compute up to third-order sensitivities. The results show a maximum normalized root-mean-squared deviation in the order of 10⁻⁹ for the diatomic lattice and in the order of 10⁻⁸ for the 2D square lattice when compared to the analytical solutions.