Research into thermoacoustic instabilities for gas turbines may allow designers to produce calculations up to ten times faster, according to Camilo Silva, researcher at the Technical University of Munich (TUM). “Results from the adjoint surrogate model (matrix-vector multiplication) were obtained around ten times faster than the method involving the full solution of the eigenvalue problem. The larger the matrix, the larger the difference of computing time between one matrix vector product and one eigenvalue problem,” he told Gas to Power Journal.
System savings though adjoint surrogate approach
The adjoint surrogate model proposed by Silva aims to replace the computational load of traditional models. Typically, when modeling instabilities for gas turbine operation the calculations are solved by means of a mathematical known as an eigenvalue problem but given the complexity of interactions within a turbine these can quickly become computationally heavy.
“The adjoint approach allows us to perform Monte Carlo simulations of the Helmholtz equation by means of mere Matrix-Vector multiplications... We are therefore replacing the effort of solving a full eigenvalue problem (given by the Helmholtz equation) by only a few matrix-vector multiplications,” Silva explained.
Traditionally, the eigenvalue appears under nonlinear terms with exponentials such as time delays related to the flame model but 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,” Silva said. The researchers noted that the method 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.
New research into continuous stochastic field
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,” Silva said.
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.
“I believe that 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,” Silva concluded.