In this conceptual study, we introduce the novel EPTD (Exceptional Point-based Thermoacoustic Design)-method for the design of stable thermoacoustic systems: dampen the entire thermoacoustic spectrum by shifting the EP towards smaller growth rates. For this, we first study a generic, linear parametric eigenvalue problem (LPEP) of a non-hermitian matrix featuring an EP. We derive analytical solutions for the eigenvalues and the EP as a function of arbitrary system parameters in the LPEP. Subsequently, we formulate a method for shifting the eigenvalue spectrum by relocating the EP in the complex plane. For this method, we show that not only the location of the EP in the complex plane is of relevance, but also the exceptional parameters, as they enforce the relations of the EP and the eigenvalues in the spectrum. When fixing these while shifting the EP, we demonstrate that the entire eigenvalue spectrum preserves its original shape, such that all eigenvalues only shift with the exact amount the EP was shifted. The derivation and the analysis of the relation between EP and spectrum generally holds for arbitrary system parameter combinations. Additionally, we showcase results for an exemplary configuration, i.e., a configuration with arbitrarily chosen and fixed parameters. We subsequently introduce a generic, thermoacoustic system, a Rijke tube, and study it by means of (semi-) analytical methods to confirm that the findings from the analysis of the LPEP also transfer to thermoacoustic systems. The configuration is modeled as a thermoacoustic network that allows to trace back the full system modes to their respective origins, which are either of acoustic or intrinsic thermoacoustic (ITA) nature. After identifying an EP that is formed by acoustic and ITA driven mode branches, we study its impact on the eigenfrequency spectrum and how this shift impacts the EP’s relation to the mode origins and the remaining eigenvalues. Similar to the analysis of the LPEP, the results show that not only the EP’s location in the CP is of relevance, but also its exceptional parameters: shifting the EP without consideration of the exceptional parameters leads to unsatisfactory shifts of the associated spectrum that becomes distorted. With this, we formulate the EPTD-method: shift the EP towards smaller growth rates while enforcing a well-defined relation between the mode origins and the EP by keeping the exceptional parameters constant. We show that this constraint preserves the relation between EP and mode origins when shifting the EP and finally demonstrate the practicality of the proposed method. For this, we shift the EP of the reference Rijke tube configuration towards smaller growth rates and simultaneously show that the resulting spectrum preserves its overall shape. While this paper covers the derivation and demonstration of the proposed EPTD-method based on a generic configuration, we consider a more complex combustion test rig, featuring a turbulent confined swirl flame, in part II of this study (M. Casel & A. Ghani, Combust. Flame, 2024), where we elaborate on consequences regarding the larger system parameter dimension as well as the existence of two EPs in the spectrum.Novelty and Significance statement:Thermoacoustic instabilities are actively researched for more than half a century. Despite this effort, scientifically sound strategies to design thermoacoustically stable combustors do not exist today. We present a novel method on a conceptual level that takes into account the latest discoveries in our field (e.g. ITA modes between 2014/2015 and Exceptional Points in 2018), connects them into one fast and robust numerical framework and opens the path to design the thermoacoustic stability map. This framework does not require tuning parameters or the like, which translates to an interpretable, parsimonious and computationally cheap design strategy. We demonstrate the Exceptional Point-based Thermoacoustic Design method (called EPTD-method) on two well-known experimental setups with increasing complexity and carefully perform parametric tests to unveil the key aspects of this novel design strategy. Finally, we turn the initially unstable setups into stable ones and explain why and how this was achieved.
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