PSI - Issue 19

A. Halfpenny et al. / Procedia Structural Integrity 19 (2019) 150–167 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Where the structure responds dynamically and steady-state, then the ROM can represented by a harmonic frequency response function H(ω) as illustrated in Equation (2): �� ( ) = ∑ � �� ( ) ⋅ � ( ) (2) Where �� ( ) is the Fourier transform of the complex stress tensor, � ( ) is the Fourier transform of the load time history for load case , and �� ( ) is the harmonic transfer function obtained from FEA. ω is the frequency expressed in rad ⁄s. Random PSD loading Where the structure is analysed using random dynamic loads expressed as Power Spectral Density (PSD) functions, the ROM is also represented by the harmonic frequency response function H(ω) as illustrated in Equation (3) and described by (Halfpenny, 1999). �� ( ) = ∑ ∑ � ( ) ⋅ � ( ) ⋅ �� ( ) ���� ���� (3) Where ��(�) is the single-sided PSD stress tensor, �� ( ) is the cross-power spectral density function of loading between load cases and , and � ( ) and � ( ) are the harmonic transfer function and its complex conjugate respectively.  Non-linear FEA In the case of non-linear FEA models, the ROM may become more difficult to define. In the worst case the entire FEA model, along with the fatigue model, will have to be solved repeatedly within the Deterministic Process shown in Fig. 2. This might necessitate having to reduce the number of simulations because of runtime/cost implications. However, it should be remembered that simulations are still much more cost-effective than physical prototyping, and the Latin Hypercube sampling approach is optimized for low sample sizes. In other cases, the non-linear ROM may be modelled using a non-linear regression technique. These are beyond the scope of this paper. For more information on nonlinear FEA, refer to [HIN 92]. For information on non-linear structural dynamics and response, refer to [WOR 01], [MAS 05] and [MOH 92]. 2.5. Exploring the extremities of design space Whereas the Monte Carlo with Latin Hypercube sampling technique is used to determine the reliability of a design, other techniques are available that consider the robustness of a design. A robust design considers permutations of the stochastic input parameters that lead to the most extreme responses. These combinations may have a low likelihood of occurring, however they will result in the worst case conditions. Simulation is required to determine whether the component will survive these conditions, or whether their probability of occurrence is acceptably low. An excellent introduction to robust design is given by [TAG 05] and also [REL 15a]. There are multiple approaches to this problem. They usually proceed in the following steps: Calculate an initial design matrix using factorial sampling and run the simulation models Generate a response surface fitting the simulation results Identify the most likely critical permutations from the response surface

Rerun the simulations based on the revised estimate of the critical permutations Iterate through steps 2-4 until all critical permutations have been resolved Factorial sampling An illustration of a 2-level factorial analysis is shown in Fig. 7.