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Konstantinos G. Kyprianidis Module Development iteratively solve the non-linear equation system formed. The Newton-Raphson method was selected as the starting point for developing LISIS. In LISIS, the Jacobian may be approximated by forward or central differences, or complex-step differentiation [153,154]. Central differences are more accurate than forward differences but at a significant computational penalty i.e. twice as many function evaluations are required to approximate the Jacobian matrix. Complex-step differentiation is the most accurate of the three options and it’s accuracy is independent of the differentiation step chosen. Despite the fact that complex-step differentiation requires the same number of function evaluations as forward differences, the function evaluations themselves are computationally more expensive since complex calculus is required i.e. all real variables in the function need to be transformed into complex variables as described by Martins et al. [153]. When function evaluations are expensive, cheap approximations to the Jacobian (or better said, cheap updates to the inverted Jacobian) may be introduced in order to reduce computational time. The main idea here is that the Jacobian matrix is determined using a differentiation method only during the first iter- ation; for consecutive iterations cheap approximations (updates) are used for the inverted Jacobian. A whole class of algorithms have been developed that implement this idea, and are known as quasi-Newton or secant methods [155]. Through a careful study of the variations of the inverted Jacobian between engine steady state operating points that are not far away from each other (for example top of climb and max cruise), one can make a useful observation [156]: the inverted Jacobian matrix does not vary significantly and may still be reused successfully. The idea of reusing past information - avoiding costly Jacobian evaluations in each iteration - is not a new one. Stamatis et al. [157] describe the use of this idea for real-time transient performance simulations. Unfortunately, little detail is provided on the potential benefits for steady state simulations and the exact implementation carried out by the authors. With respect to steady state simulations, a further improvement is therefore proposed here to the class of quasi-Newton methods and has been implemented successfully in the LISIS library: 1. The Jacobian matrix is determined using one of the differentiation methods described above only during the simulation of the very first point. 67PDF Image | Multi-disciplinary conceptual design of future jet engine systems
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