Typically, when instabilities for gas turbine operation are modelled, these calculations are solved by means of a mathematical factor known as 'eigenvalue'. But given the complexity of interactions within a turbine, this approach can quickly become 'computationally heavy'.
System savings though adjoint surrogate approach
“The adjoint approach allows us to perform Monte Carlo simulations of the Helmholtz equation by means of mere Matrix-Vector multiplications...."TUM researchers said in a research paper, adding "therefore, we replaced the effort of solving a full eigenvalue problem (given by the Helmholtz equation) by only a few matrix-vector multiplications."
Traditionally, the 'eigenvalue' appears under nonlinear terms with exponentials such as time delays related to the flame model but Dr. Silva and his team have instead proposed to simplify this calculation using matrix-vector multiplications.
“Let's assume we want to obtain the output (in terms of one eigenmode growth rate) of 10,000 eigenvalue problems... [With our approach] it is enough to perform two eigenvalue problems…. [and] 10000-30000 matrix vector multiplications,” he explained. The method, applied at TUM, is particularly well-suited to large systems, such as the modelling of turbines, where there are hundreds of thousands to millions of degrees of freedom.
Replacing Monte Carlo simulations with Uncertainty Quantification
With the results of the initial phase of research delivering a promising decrease in computing time, the team are now focused on refining the approach. “The idea now is to replace Monte Carlo simulations by a more efficient Uncertainty Quantification method. There are two methods in view: Method of moments and Polynomial Chaos Expansion (PCE). Instead of modeling a "discrete" stochastic field (each realization is a point in the probability space), we want to model a "continuous" stochastic field,” the paper explains.
To do this, researchers plan to replace deterministic variables with stochastic ones. It is expected that adjoint surrogate method will allow for easier implemented as compared with the traditional eigenvalue problem formulation. The researchers hope that the findings from their research can readily be integrated into existing design tools and software to allow rapid commercial adoption of the technology.
“Flexible tools (like Comsol) should allow an easy implementation of the method, so that in the future we will talk about 'Adjoint Helmholtz solvers' as something commonly found in standard tools,” researchers concluded.