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Context 1: Paper Info

Title: Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series
Publish Date: March 7, 2023
Author List: Qianying Cao (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology), Anteng Chang (from College of Engineering, Ocean University of China), Junfeng Du (from College of Engineering, Ocean University of China), Lin Lu (from State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology)

Figure

Fig  1: Procedure to compute the response by a combination of Volterra series and Laguerre polynomials
Fig  2: Linear frequency response function: (a) Modulus of H 1 (ω), (b) phase angle of H 1 (ω)
Fig  6: Comparison of h 1 (t) based on the analytical and reconstructed by Laguerre polynomials
Fig  11: Response for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  18: Comparison of original excitations and reconstructed results: (a) Case 1, (b) Case 2 (c) Case 3
Fig  19: Response to irregular excitation for Case 1: (a) comparison between the proposed method and Runge-Kutta method, (b) contribution of the three components
Fig  23: Input-output dataset used to identify Volterra series: (a) input excitation, (b) output response
Fig  26: Comparison of responses between the predicted and numerical results: (a) response to regular excitation, (b) response to irregular excitation
Parameter values of the irregular excitation

abstract

Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering  To solve these kinds of problems, time-consuming stepby-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used 
In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients  Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods 
Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained  In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated 
For each system, regular excitations and complex irregular excitations with different parameters are studied  Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge-Kutta method 

Introduction

Most real dynamic systems, as encountered in mechanical and civil engineering, are inherently nonlinear and include geometric nonlinearities, nonlinear constitutive relations in material or nonlinear resistances, etc    Nonlinear problems are attracting increasing attention from engineers and scientists 
This work focuses on solving nonlinear system vibration problems, i e , computing transient responses of nonlinear oscillators under arbitrary irregular excitations based on a combination of a pole-residue operation and Volterra series  Because Volterra series are single-valued, the scope of the present study is restricted to nonlinear behaviours without bifurcations  
To analyse nonlinear vibration problems, researchers have performed extensive studies and developed various mathematical methods  Popular methods include step-by-step numerical integration methods in the time domain, such as the Runge-Kutta method  This kind of method not only requires a small time-step resolution for obtaining high-precision solutions but also is prone to numerical instability  
For a long response with small time steps, the time domain methods are very costly in computational time  Volterra series is another widely used method, which is the extension of the Duhamel integral for linear systems   Volterra series can reproduce many nonlinear phenomena, but they are very complex due to higher-dimensional convolution integrals  
Since 1980's, significant progress has been made in the general area of the Volterra series  The reader is referred to Ref  for a quite thorough literature review on the relevant topics  After 2017, most papers focus on Volterra series identification  De Paula and Marques proposed a method for the identification of Volterra kernels, which was based on time-delay neural networks 
Son and Kim presented a method for a direct estimation of the Volterra kernel coefficients  Dalla Libera et al  introduced two new kernels for Volterra series identification  Peng et al  used the measured response to identify the kernel function and performed the nonlinear structural damage detection  Only a few papers concentrated on simplifying the computation of convolution integrals 
Traditional methods for computing convolution integrals involved in the Volterra series have been performed in three distinct domains: time, frequency and Laplace  The time domain method based on Volterra series refers to discrete time convolution methods, which also suffer computational cost problems  
Both the frequency domain method and the Laplace domain method based on the Volterra series consist of three steps: (1) Volterra series are transformed into an algebraic equation in the frequency domain or Laplace domain; the algebraic equation is solved by purely algebraic manipulations; and (3) the solution in Step ( ) is transformed back to the time domain 
Many researchers have used the frequency domain method to compute the responses of nonlinear systems  Billings et al  developed a new method for identifying the generalized frequency response function (GFRF) of nonlinear systems and then predicted the nonlinear response based on these GFRFs  Carassale et al  introduced a frequency domain approach for nonlinear bridge aerodynamics and aeroelasticity 
Ho et al  computed an output frequency domain function of a nonlinear damped duffing system modelled by a Volterra series under a sinusoidal input  Kim et al  identified the higher order frequency response functions by using the nonlinear autoregressive with exogenous input technique and the harmonic probing method 
This type of frequency domain method is much more efficient than the time domain method due to the fast Fourier transform algorithm  However, the frequency domain method not only is limited by frequency resolutions but also suffers from leakage problems due to the use of discrete Fourier transforms  In addition, the frequency domain method calculates only a steady-state response 
A natural response generated by initial conditions and a cross response caused by interactions between a system and an excitation are ignored  In contrast, the Laplace domain method can calculate all response components because initial conditions are considered in the computational procedure  However, it has been restricted to analytical operations for simple excitations, such as sinusoidal excitations and exponential excitations  
The proposed method falls into the category of the Volterra series method computed in the Laplace domain  Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations  Because the proposed method follows a similar path as a pole-residue method for linear systems , the proposed method to solve nonlinear system vibration problems is called the generalized pole-residue method 
The main concept of the pole-residue method developed by Hu et al  was that the poles and residues of the response could be easily obtained from those of the input and system transfer function to obtain the closed-form response solution of linear systems  This method included three steps: (1) writing the system transfer function into pole-residue form; (2) writing the excitation into pole-residue form by the Prony-SS method; (3) computing the poles and residues of the response by an algebraic operation based on those from system and excitation 
Compared to Hu et al  , which was regarded as an efficient tool to compute responses of linear systems, the generalized pole-residue method in this paper is introduced to compute responses of nonlinear systems  The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method 
Compared to the traditional pole-residue method for a linear system, one of the novelties of the generalized pole-residue method is how to deal with the higher-order poles and their corresponding coefficients  Similar to the Taylor series, the Volterra series representation is an infinite series, and convergence conditions are needed to assure that the representation is meaningful 
Because the proposed method is based on the Volterra series, only the system with convergent Volterra series representation can be treated by the proposed method  The paper is organized as follows  In Section 2, the nonlinear response is modelled by a Volterra series, and Volterra kernel functions are decoupled by Laguerre polynomials 
Then, the pole-residue method for computing explicit responses is developed in Section 3  Numerical studies and discussions are given in Section 4  Finally, the conclusions are drawn in Section 5 

Response calculation based on Volterra series

A nonlinear oscillator, whose governing equation of motion is given by where z(t, y, ẏ) represents an arbitrary nonlinear term; m, c, and k are the mass, damping and linear stiffness, respectively; y(t), ẏ(t) and ÿ(t) are the displacement, velocity and acceleration, respectively; and f (t) is the time-dependent excitation 
If the energy of excitation f (t) is limited, the nonlinear response under zero initial conditions (i e , zero displacement and zero velocity) can be represented by the Volterra series : where N is the order of Volterra series and In Eq  3, h 1 (τ ) is called the first-order Volterra kernel function, which represents the linear behaviour of the system; h n (τ 1 ,      
, τ n ) for n > 1 are the higher-order Volterra kernel functions, which describe the nonlinear behaviour of the system  The complete formulation of y(t) includes infinite series where the labour of calculating the n th term increases quickly with the growth of n  Fortunately, the response accuracy may be ensured by the first several order Volterra series 
This is proved here in numerical studies  The commonly known Laguerre polynomials are represented as : where p i is the order of the Laguerre polynomials and a i is the damping rate  The Laguerre polynomials satisfy the orthogonal relationship expressed as: By using Laguerre polynomials, the Volterra kernel function h n (t 1 ,      
, t n ) in Eq  3 can be decoupled as follows : where the coefficient is computed resorting to the orthogonal relationship in Eq  5: Substituting Eq  6 into Eq  3 yields       The above operation that uses the Laguerre polynomials to decouple Volterra higher order kernel functions has been well-developed 
The reader is referred to Refs  for details about the adopted technique  After decoupling Volterra higher order kernel functions in time, one can regroup Eq  8 into:       By denoting Eq  9 becomes The above procedure to compute the nonlinear response by a combination of Volterra series and Laguerre polynomials is schematically shown in Fig   
Volterra kernel functions h n (t 1 ,       , t n ) can be obtained by either an equation of motion or measured input-output signals  To derive a closedform solution of the response, we must obtain a closed-form solution of x i (t) first  In the following presentation, a closed-form solution of the aforementioned x i (t) and y n (t) is derived by using the pole-residue method 
3  Pole-residue method for calculating x i (t) and y n (t) Performing the Laplace transform of x i (t) in Eq  10 yields where in which Eq  13 includes a single pole and several higher-order poles  For k = 0, −a i is a single pole, and b p i (0) is a corresponding coefficient, namely, the residue  For k > 0, −a i are higher-order poles, and b p i (k) are corresponding coefficients 
For an irregular excitation signal f (t) of a finite duration of T , it can always be approximated into a pole-residue form by using the complex exponential signal decomposition method-Prony-SS : where N ℓ is the number of components; α ℓ and λ ℓ are constant coefficients, which either are real numbers or occur in complex conjugate pairs 
We define λ ℓ = −δ ℓ + iΩ ℓ , where Ω ℓ is the excitation frequency and δ ℓ is the damping factor of the ℓ th component  We denote α ℓ = A ℓ e iθ ℓ , where A ℓ is the amplitude and θ ℓ is the sinusoidal initial phase in radians  Taking the Laplace transform of Eq  15 yields Note that the concept of the Prony-SS method is similar to that of a principal component method 
A smooth excitation usually requires just several terms to achieve a good approximation  For high irregular loadings, including more terms would achieve a better approximation  Substituting Eqs  13 and 16 into Eq  12 yields Expressing xi (s) in its pole-residue form yields where λ ℓ are simple poles, and the corresponding residues are easily obtained by
and −a i are higher-order poles, and the corresponding coefficients are firstly derived as: By taking the inverse Laplace transform of Eq  18, a closed-form solution is obtained: Substituting Eqs  11 and 21 into Eq  2 yields Theoretically speaking, the proposed method for deriving the closed-form solution of the nonlinear response is applicable to any order of the Volterra series 
For practical engineering, usually only the first several order responses dominate  By setting up N = 2, Eq  22 can be simplified into three components: where the natural response, which is only related to system poles, is given by and the cross response, which is related to both system poles and excitation poles, is given by
and the forced response, which is related only to excitation poles, is given by The first term in Eq  26 is the first-order forced response governed by the excitation frequency, i e , the imaginary part of the pole λ ℓ   The second term corresponds to the second-order nonlinear forced response, which includes the sum frequency and difference frequency responses governed by λ ℓ + λ j  
Eq  26 straightforwardly offers visible information about the possible nonlinear vibrations by the cooperation of excitation frequencies  Particularly, consider a sinusoidal excitation f (t) = sin ω r t, which can be expressed as f (t) = γe λt + γ * e λ * t , where γ = −0 5i and λ = iω r   Substituting these values into Eq 
26, the second term of Eq  26 is simplified as where the first term is the difference frequency response, and the second term is the sum frequency response 

Numerical studies

In practical engineering, some systems have an accurate equation of motion  Additionally, some systems have difficulty constructing their equations of motion because of complex nonlinear dynamic behaviours and uncertain system parameters  In this article, a system with a known equation of motion is called a known system, and a system with an unknown equation of motion is called an unknown system for simplicity 
In this section, two numerical studies are presented  The first study verifies the proposed method using a known nonlinear oscillator, and the second study demonstrates the applicability of the proposed method to an unknown system  Throughout the numerical studies, the unit system is the metre-kilogramme-second (MKS) system; for conciseness, explicit units for quantities are omitted 

A known nonlinear system

This study chooses a nonlinear oscillator written as: where mass m = 1, damping c = 1, linear stiffness k 1 = 10, quadratic stiffness k 2 = 20 and cubic stiffness k 3 = 20  It is a case that has been studied in a previously published article   The linear natural frequency of the system ω 0 = k 1 /m = 3 16 and the damping ratio ζ = c/(2mω 0 ) = 15 8% 
This kind of oscillator occurs in many engineering problems, such as a model of fluid resonance in a narrow gap between large vessels   In the model, k 1 y represents the linear restoring force of the fluid, and k 2 y 2 and k 3 y 3 are respectively the quadratic and cubic nonlinear restoring forces of the fluid 

Volterra kernel functions

Generally, the first several order responses dominate the total response of a system  Hence, the order of the Volterra series in Eq  22 is chosen to be 3, namely, N = 3  For computing the first three order responses from Eq  22, the first three order Volterra kernel functions need to be known  Since Volterra kernel functions and corresponding frequency response functions are related by a specific Fourier transform pair, we can first write the first three orders of frequency response functions directly from Eq  28 
Then, Volterra kernel functions are obtained by the inverse Fourier transform  Based on the harmonic probing algorithm , the linear frequency response function (LFRF) H 1 (ω), the quadratic frequency response function (QFRF) H 2 (ω 1 , ω 2 ) and the cubic frequency response function (CFRF) H 3 (ω 1 , ω 2 , ω 3 ) are analytically given by:
and Figures show H 1 (ω), H 2 (ω 1 , ω 2 ) and H 3 (ω 1 , ω 2 , ω 3 ), respectively, which agree well with those reported in Ref    As expected, the modulus of H 1 (ω) in Fig  peaks near the linear natural frequency ω 0 , and the phase angle decreases monotonically from 0 to -π with increasing frequency 
Figure shows the sum frequency QFRF, where the energy converges along the line of ω 1 +ω 2 ≈ ω 0   Therefore, when the sum frequency of a two-tone excitation equals the linear resonant frequency, the second-order response may reach its maximum  Additionally, those pairs of excitations in line ω 1 + ω 2 ≈ ω 0 may produce non-negligible vibration magnitudes due to second-order nonlinear effects 
For the difference frequency QFRF in Fig  (b), the energy converges along two main lines, i e , ω 1 ≈ ω 0 and ω 2 ≈ ω 0   Figures show moduli of H 3 (ω, ω, ω) and H 3 (ω, ω, −ω), which are diagonal terms of the sum frequency CFRF and the difference frequency CFRF, respectively  While the modulus of H 3 (ω, ω, ω) peaks near ω ≈ ω 0 /3 and ω 0 , that of H 3 (ω, ω, −ω) peaks near ω ≈ ω 0 with a small hump around ω ≈ ω 0 /2 
Values at ω ≈ ω 0 /3 and ω 0 /2 may be magnified by higher-order stiffness terms in Eq  28  By performing the inverse fast Fourier transform to Eqs  29-31, the corresponding linear impulse response function h 1 (t), quadratic impulse response function h 2 (t 1 , t 2 ) and cubic impulse response function h 3 (t 1 , t 2 , t 3 ) are obtained 
Here, h 1 (t) and h 2 (t 1 , t 2 ) are plotted in Figs  , respectively, and h 3 (t, t, t) is shown in Fig    In the numerical implementation, Eqs  29-31 have been utilized with the frequency interval ∆ω = 0 1, number of frequency components N n = 1025, and cut-off frequencies 102 4 and −102 4  For decoupling Volterra kernel functions by using Laguerre polynomials, the damping rate and number of Laguerre polynomials for each order Volterra kernel function need to be determined (see Eqs  4 and 6) 
In this example, we set a 1 = a 2 = a 3 = 2 and R 1 = R 2 = R 3 = 24 because coefficients c p 1    pn become very small when R n > 24, n = 1, 2, 3  According to Eq  7, the coefficients of the first three order Volterra kernel functions are calculated, which are shown in Figs  9 and 10  For convenience, Fig  plots only c p 1 p 2 p 3 for p 3 = 0 
With the increase of the order of Laguerre polynomials, coefficients in Figs  9 and 10 gradually decrease, which illustrates how the first several orders of Laguerre polynomials dominate all orders of the Volterra kernel function  With the known Laguerre polynomials and corresponding coefficients, Volterra kernel functions are reconstructed by Eq  6 
For comparison, reconstructed Volterra kernel functions are also plotted in Figs    The reconstructed results agree well with the analytical values, which verifies the accuracy of the decomposition 

Sinusoidal excitation

From Eq  28, we consider a sinusoidal excitation where A and Ω are the amplitude and the frequency, respectively  Five cases of A and Ω are shown in Table   Excitation frequencies in Cases 1 and 2 are larger than the linear natural frequency (ω 0 ≈ 3 16), those in Case 3 are very close to ω 0 , and those in Cases 4 and 5 are smaller than ω 0  
All cases have same amplitudes  The poles of a sinusoidal excitation are λ 1,2 = ±iΩ, and the residues are α 1,2 = ∓iA/2  Numerical values of excitation poles and residues for different cases are listed in Table   Table : Parameter values, poles and residues of the sinusoidal excitation Substituting poles and residues of the excitation, as well as those of the system into Eqs 
20 and 19, response coefficients β p i ,k corresponding to system poles −a i and response coefficients γ p i ,ℓ corresponding to excitation poles λ ℓ are calculated, respectively  According to Eq  22, the first three orders of responses for each case in Table are calculated  Figures )-15(a) show the comparison of responses obtained by the proposed method and the fourth-order Runge-Kutta method with ∆t = 10 −4  
For Cases 1 and 2, the first-order responses agree well with the total responses obtained by the Runge-Kutta method, and the higher-order responses only slightly improve the transient parts  For Cases 3-5, the sum of the first three orders of responses is in good agreement with the Runge-Kutta solution 
When the response nonlinearity increases, higher-order responses need to be considered  In other words, the proposed method can accurately compute the nonlinear responses by choosing a small number N of Volterra series terms  Figures )-15(b) show the contributions of the three response components for the five cases 
In each case, the first-order response is the most dominant component, and the contributions of secondand third-order responses are much less than those of the first-order response  Especially for Cases 1 and 2, whose excitation frequencies are far from the linear natural frequency, second-and thirdorder responses are close to zero 
This may be because the QFRF and CFRF approach zero when the frequency is larger than 4 rad/s (see Figs  )  Furthermore, the mean values of the first-order responses are approximately zero, and those of the second-order responses are always smaller than zero, which are the difference frequency components in Eq  27 
Moreover, it is clearly observed that second-order responses for Cases 3-5 exhibit a periodic oscillation with a period near half of that for the first-order response, which is excited by the sum frequency component of the excitation (see second part of Eq  27)  Compared with steady-state solutions of first-and second-order responses, those of third-order responses in Cases 3-5 are no longer single regular motions 
By performing the FFT, frequency spectra of these three third-order responses are shown in Fig    We find that these three third-order responses are all dominated by their own fundamental harmonic component and the third harmonic (triple frequency) component  Figure shows the computational time to calculate the response of the oscillator for Case 1 by the proposed method, the fourth-order Runge-Kutta method and the convolution method 
The proposed method, which has an explicit solution, is much more efficient in computational time than the latter two methods, which need small time steps to obtain high-precision solutions  In particular, the efficiency of the proposed method increases with the length of the response time  Computation time (sec )
t=0 02s Convolution, t=0 02s Convolution, t=0 001s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the proposed method, the fourth-fifth order Runge-Kutta method and the convolution method regular loading in Case 1

Irregular excitation

In Eq  28, considering an irregular excitation consisting of several cosine functions where N f is the number of cosine components; A n , Ω n and θ n are the amplitude, frequency and phase angle of the n th component, respectively  Table lists three cases of these parameters  In each case, the amplitudes of all components are the same, and phase angles θ n uniformly distributed between 0 and 2π are randomly generated 
To decompose the excitation into a pole-residue form, the Prony-SS method is used, whose concept is similar to that of a principal component method  The readers are referred to Ref  for details  The chosen rank of each case is also shown in Table   Figure shows the comparison of original excitations and reconstructed results of these three cases, which all have excellent agreement 
show the results computed by the fourth-order Runge-Kutta method  In all cases, the sums of the first three orders of responses agree well with those obtained by the Runge-Kutta method  The contributions of the first three orders of responses for each case are plotted in Figs  )-21(b)  Similarly, the system vibration is dominated by the first-order response 
However, the contributions of second-and third-order significantly grow with increasing excitation magnitude and frequency number  Furthermore, when the magnitude of the nonlinear response becomes large, sharp troughs are present  This phenomenon may be induced by the nonlinear stiffness  While the first-order response fails to capture these troughs, the higher-order responses successfully capture these troughs 
Figure plots the computational time to calculate the response of the oscillator for the irregular loading in Case 1 by the proposed method and the fourth-fifth order Runge-Kutta method, respectively  While the fourth-fifth order Runge-Kutta method is more efficient under a small response length, the proposed method becomes much more efficient when the response length is larger than about 130 s 
In addition, the proposed method obtains the explicit response solution, so one can directly obtain the response value at a specific time t p instead of integrating from 0 to t p for traditional numerical methods  Computation time (sec ) Proposed, t=0 02s Runge-Kutta, t=0 001s Fig  : Comparison of computation efficiency of the method and the fourth-fifth order Runge-Kutta method for irregular loading in Case 1

An unknown nonlinear

To check the applicability of the proposed method to an unknown nonlinear system, a known input excitation and its corresponding response are used to identify its Volterra kernel functions  When the Volterra kernel functions are known, we can follow the procedure in Section 4 1 to predict system responses 
In this study, the input excitation is white noise with a constant power spectrum S 0 = 0 001, and the corresponding response is obtained by solving Eq  28 by the fourth-order Runge-Kutta method, which is shown in Fig    From Section 4 1, we determine that the sum of the first two orders of responses agrees well with the total response 
In this study, the order of Volterra series N is chosen to be 2, damping rates of Laguerre polynomials are a 1 = a 2 = 2, and numbers of Laguerre polynomials are R 1 = R 2 = 24  To estimate the first two orders of Volterra kernel functions, a matrix equation is constructed using excitation data and response data 
By using the least square method to solve this matrix equation, coefficients c p 1 and c p 1 p 2 in Eq  8 are identified  Figure plots c p 1 and c p 1 p 2 , respectively, which have good agreement with the exact results in Fig    Then, the first two order Volterra kernel functions are constructed by Eq  6 
Compared with the exact results in Figs  , the identified Volterra kernel functions in Fig  completely agree well with the exact solutions  Note that the white noise excitation, which can excite more frequency components of the response, is chosen to obtain good Volterra kernel functions  A regular excitation f (t) = sin(πt) and an irregular excitation f (t) = N f n=1 A n cos(Ω n t + θ n ) with A n = 0 3 and Ω n varying from 0 to 40 with equal interval 1 are chosen as input excitations 
The predicted responses, along with results obtained by the fourth-order Runge-Kutta method, are shown in Fig    In both cases, the proposed method accurately predicts system responses  As presented in Eq  23, a nonlinear response is the sum of three terms: natural response y s (t), forced response y f (t) and cross response y c (t) 
These individual terms, as well as their sum to two excitations, are shown in Figs  27 and 28, respectively  As shown in Figs  and 28, both first-and second-order responses include the natural response y s (t) and the forced response y f (t), but the cross response y c (t) only exists in second-order responses 
When t becomes larger, both y s (t) and y c (t) diminish due to the presence of system damping, and the total response is entirely governed by y f (t)  Moreover, we notice some features at t = 0 for these components, including y s (0) = −y f (0) for the first-order response and y s (0) + y f (0) = −y c (0) for the second-order response, which are due to imposed zero initial conditions 

Conclusions

Considering arbitrary irregular excitations, an efficient generalized pole-residue method to compute the nonlinear dynamic response modelled by the Volterra series was developed  A core of the proposed method was obtaining poles and corresponding coefficients of Volterra kernel functions, then those of each order response modelled by each order Volterra series 
Once the poles and corresponding coefficients of Volterra kernel functions and excitations were both available, the remaining derivation could follow a similar pole-residue method that had been developed for ordinary linear oscillators  To obtain the poles and corresponding coefficients of Volterra kernel functions, two steps were included: (1) using Laguerre polynomials to decouple higher-order Volterra kernel functions with respect to time and (2) obtaining poles and corresponding coefficients of Laguerre polynomials in the Laplace domain 
Because the proposed method gave an explicit, continuous response function of time, it was much more efficient than traditional numerical methods  Moreover, many meaningful physical and mathematical insights were gained because not only each order response but also the natural response, the forced response and the cross response of each order were obtained in the solution procedure 
To demonstrate that the proposed method was not only suitable for a system with a known equation of motion but also applicable to a system with an unknown equation of motion, two numerical studies were conducted  For each study, regular excitations and complex irregular excitations with different parameters were investigated 
The efficiency of the proposed method was verified by the fourth-order Runge-Kutta method  This paper only computes the response under zero initial conditions  The response under non-zero initial conditions will be investigated in our future work 
Context 2: Paper Info

Title: Specific-heat ratio effects on the interaction between shock wave and heavy-cylindrical bubble: based on discrete Boltzmann method
Publish Date: May 29, 2023
Author List: Yanbiao Gan (from School of Liberal Arts and Sciences, Hebei Key Laboratory of Trans-Media Aerial Underwater Vehicle, North China Institute of Aerospace Engineering), Yudong Zhang (from School of Mechanics and Safety Engineering, Zhengzhou University)

Figure

Figure 1: Research orientation and tasks of DBM 
Figure 2: Sketch of D2V16 model The numbers in the figure represent the index i in Eq  (3) 
Figure 3: The computational configuration of the shock-bubble interaction 
In the figure, results from odd rows are experimental, and the even rows indicate DBM simulation results The typical wave patterns and bubble's main characteristic structures are marked out in the figures Numbers in the pictures represent the time in µs Schlieren images of DBM results are calculated from the density gradient formula, i e , |∇ρ|/|∇ρ| max , with |∇ρ| = (∂ ρ/∂ x) 2 + (∂ ρ/∂ y)2  At t = 0µs, the incident shock wave impacts the upstream interface, and subsequently generates a transmitted shock (TS) propagating downstream in the bubble and a reflected shock wave moving upward in ambient gas The incident shock wave travels downstream contin-
The definitions and the corresponding physical meanings of the common TNE quantities in DBM, where the operator ∑ ix,iy indicates integrating over all the fluid units and multiply the unit area dxdy From a certain perspective, the TNE strength is increasing; While from a different perspective, the TNE strength, on the other hand, may be decreasing It is one of the concrete manifestations of the complexity of non-equilibrium flow behavior 
Figure 4: Snapshots of schlieren images of the interaction between a shock wave and a heavy-cylindrical bubble The odd rows represent experimental results from Ref  [31] with permission, and the even rows are DBM simulation results The typical wave patterns and the bubble's main characteristic structure are marked out in the figures Numbers in the picture represent the time in µs 
Figure 5: The temporal variations of length and width of the bubble The symbols represent DBM results and the lines are experimental The definition of the length and the width of the bubble can be seen in the illustration Experimental results are obtained from Fig  12, in Ref  [31] with permission 
Figure 6: Density contours and particle tracer images at three different moments (i e , t = 0 07,t = 0 11, and t = 0 16) with various specific-heat ratios The odd rows represent density contours, and the even rows are tracer particle images 
Figure 9: Vorticity contours at t = 0 134, with various specific-heat ratios The arrows in the vorticity image point out the apparent difference between case γ = 1 4 and case γ = 1 09 
Figure 11: Density contours (first row) and mixing degree M (second row) at several typical moments 
Figure 13: (a) Temporal evolution of D * 3,1 and D * 4,2  (b) Temporal evolution of D * 2 and D * 3  Lines with different colors represent the cases with various specificheat ratios 
The development of schemes for checking TNE state, extracting TNE information and describing corresponding TNE effects in DBM 

abstract

Specific-heat ratio effects on the interaction between a planar shock wave and a two-dimensional heavy-cylindrical bubble are studied by the discrete Boltzmann method  Snapshots of schlieren images and evolutions of characteristic scales, being consistent with experiments, are obtained  The specific-heat ratio effects on some relevant dynamic behaviors such as the bubble shape, deformation process, average motion, vortex motion, mixing degree of the fluid system are carefully studied, as well as the related Thermodynamic Non-Equilibriums (TNE) behaviors including the TNE strength, entropy production rate of the system 
Specifically, it is found that the influence of specific-heat ratio on the entropy production contributed by non-organized energy flux (NOEF) is more significant than that caused by non-organized momentum flux (NOMF)  Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary 
The effects of specific-heat ratio on various TNE quantities show interesting differences  These differences consistently show the complexity of TNE flows which is still far from clear understanding 

Introduction

The applications of shock-accelerated inhomogeneous flows (SAIFs) are of significant value in biomedicine, energy utilization, and astrophysics fields, including but not limited to scenarios such as the impact of shock waves on kidney stones, the interaction between shock waves with foams, the impacting of detonation wave with burning flames in supersonic combustion systems, the formation of supernova remnants, etc  
Shock-bubble interaction (SBI) is one of the most fundamental problems in the research of SAIFs  Its applications and academic research are interdisciplinary  Generally, there are two kinds of problems encountered in SBI research: (i) The geometry of shock waves, the shape of material interfaces, and the structure of container are complex in the actual scene 
They will result in various wave patterns and significantly affect the flow morphology and bubble's evolution  (ii) There usually exist multi-physics coupling problem in the engineering application of SBI  Such as the supersonic combustion machines  When the shock waves passing through the reactants, it may lead to phase transition and chemical reactions, making the flow morphology more complex and inducing small structure (or fast-changing pattern)  
In an underwater explosion experiment, the interaction between shock waves and bubbles may refer to the cavitation and annihilation effects  The other scene is the inertial confinement fusion (ICF), in which the laser ablation, electron heat conduction, self-generated electromagnetic field, radiation, and many other factors may complicate the investigation of hydrodynamic instabilities  
Commonly, research on SBI mainly includes three methods: theoretical derivation, experiment, and numerical simulation  As a fundamental research method, theoretical research can provide a clear understanding of physical processes  In 1960, Rudinger et al  developed a theory that permits computing the response of bubbles to accelerations  
In order to describe the formation and evolution processes of vortex structure quantitatively, many scholars have developed circulation models   However, theoretical works provide limited information  Meanwhile, in the late stage of SBI evolution, the bubble deformation and flow morphology dominated by the developed Richtmyer-Meshkov instability (RMI) and Kelvin-Helmholtz instability (KHI) are difficult to be predicted accu-rately 
As the research method closest to engineering application, the experimental results are often regarded as standard results to verify the rationality and accuracy of theoretical and numerical works  To study the SBI process accurately, the scholars have made a series of improvements to experimental equipment or technique, including the generation techniques of different types of shock waves, interface formation methods, schlieren facilities, and image recognition techniques  
Among these, two of important and valuable works are performed by Ding et al   Based on the soap film technique, they formed kinds of initial interfaces with different curvatures through the wire-restriction method and captured the wave patterns and interface evolution with high-speed schlieren photography  
Other works, such as evolutions of a spherical gas interface under reshock conditions , developments of a membrane-less SF 6 gas cylinder under reshock conditions , and interactions of a cylindrical converging shock wave with an initially perturbed gaseous interface , are also performed by many other scholars 
However, we know that the experimental studies mainly depend on the experimental platform  When studying some complex and demanding condition problems, it takes much work to build the experimental platform  In this situation, numerical simulation research becomes an option  Generally, there are three kinds of physical modeling methods (or models) for SBI numerical research, i e , the macroscopic, mesoscopic, and microscopic modeling methods 
Most of the existing numerical researches on SBI are related to the macroscopic modeling methods (such as the Euler and Navier-Stokes (NS) models) based on the continuous hypothesis (or equilibrium and nearequilibrium hypothesis)   For example, presented the computational results on the evolution of the shock-accelerated heavy bubbles through the multi-fluid Eulerian equation  
There also exist a few SBI works based on the mesoscopic modeling method, such as the Direct Simulation Monte Carlo method   The microscopic modeling methods such as the Molecular dynamics (MD) simulation, is capable of capturing much more flow behaviors but restricted to smaller spatiotemporal scales because of its huge computing costs 
In the numerical research on SBI, three points need to be concerned  (i) Investigation of kinetic modeling that describes the non-continuity/non-equilibrium flows  Most of the current researches are based on macroscopic models  However, there exist abundant small structure (and fast-changing patterns) behaviors and effects such as the shock wave, boundary layer, material defects, etc 
For cases with small structures, the mean free path of molecules cannot be ignored compared to the characteristic length, i e , the non-continuity (discreteness) of the system is pronounced, which challenge the rationality and physical function of the macroscopic models based on the continuity hypothesis 
For cases with fast-changing patterns, the system dose not have enough time to relax to the thermodynamic equilibrium state, i e , the system may significantly deviate from the thermodynamic equilibrium state  Therefore, the rational-ity and physical function of the macroscopic models based on the hypothesis of thermodynamic equilibrium (or near thermodynamic equilibrium) will be challenged 
(ii) Improvement of method that describes the evolution characteristics of bubbles and flows morphology  Most of the studies describe bubble characteristics and flows morphology from a macroscopic view  The mesoscopic characteristics such as the kinetic effects which help understand the kinetic process, are rarely to be studied 
(iii) Further studies of effects of specific-heat ratio on SBI process  The specific-heat ratio is an essential index for studying the compressibility of the gas  Research from Igra et al  has shown that the differences in the specific-heat ratio of bubbles would cause various wave patterns and pressure distribution inside the bubbles during the interaction process  
Besides, many works on hydrodynamic instability have also demonstrated the importance of investigating the specific-heat ratio effect   Among these, Chen et al  investigated the specific-heat ratio effects on temperature gradient and the TNE characteristics of compressible Rayleigh-Taylor (RT) system  
For the above three points, in this work we apply the recently proposed discrete Boltzmann method (DBM)   The Lattice Boltzmann Method (LBM) research has two complementary branches   One aims to work as a kind of new scheme for numerical solving various partial differential equation(s)  The other aims to work as a kind of new method for constructing kinetic model to bridge the macro and micro descriptions 
The two branches have different goals and consequently have different rules  The current DBM is developed from the second branch of LBM and focusing more on the Thermodynamic Non-Equilibrium (TNE) behaviors that the macro modeling generally ignore  It breaks through the continuity and near-equilibrium assumptions of traditional fluid modeling, discards the lattice gas image of standard LBM, and adds various methods based on phase space for checking, exhibiting, describing and analyzing the non-equilibrium state and resulting effects 
More information extraction technologies and analysis methods for complex field are introduced with time  The numerical simulation includes three parts, as shown in Fig    (1) Physical modelling, (2) Algorithm design, (3) Numerical experiments and analysis of complex physical fields  The research of equation algorithm corresponds to the part (2) of the above three parts 
The DBM aims at parts (1) and (3) of the three mentioned above  It belongs to a physical model construction method rather than a numerical solution for the equations  The tasks of DBM are to: (i) Ensure the rationality of the physical model (theoretical model) and balance the simplicity for the problem to be studied; (ii) Try to extract more valuable physical information from massive data and complex physical fields 
Based on the coarse-grained modeling method of nonequilibrium statistical physics, the DBM aims to solve the following dilemma: (i) The traditional hydrodynamic modelings are based on the continuous hypothesis (or near-equilibrium hypothesis)  They only concern the evolution of three conserved kinetic moments of the distribution function, i e  the density, momentum and energy, so their physical functions are insufficient 
(ii) The situation that the MD can be used is restricted to too small spatial-temporal scales  The physical requirement for the modeling is that except for the Hydrodynamic Non-Equilibriums (HNE), the most related TNE are also needed to be captured  Theoretically, the Boltzmann equation is suitable for all-regime flows, including the continuum regime, slip regime, transition regime, and free molecule flow regime 
Based on the Chapman-Enskog (CE) multiscale analysis , through retaining various orders of Kn number (or considering different orders of TNE effects), the Boltzmann equation can be reduced to the various orders of hydrodynamic equations  They can be used to describe the hydrodynamic behaviors, i e , the conservations of mass, momentum and energy , in corresponding flow regimes 
Because what the traditional hydrodynamic equations describe are only the conservation laws of mass, momentum and energy  Consequently, it should be pointed out that, the information lost in the traditional hydrodynamic equations increases sharply with increasing the Kn number  With increasing the Kn number, to ensure the describing capability not to decrease significantly, the more appropriate hydrodynamic equations should be the Extended Hydrodynamic Equations (EHEs) which include not only the evolution equations of conserved kinetic moments but also the most relevant nonconserved kinetic moments of distribution function 
For convenience of description we refer the modeling method that derives EHEs from the fundamental kinetic equation to Kinetic Macroscopic Modeling (KMM) method  It is clear that, the complex process of CE expansion is necessary and the simulation is still based on the macroscopic equations in KMM  As a comparison, the DBM is a kind of Kinetic Direct Modeling (KDM) method 
In DBM modeling, the CE analysis is only used to quickly determine which kinetic moments should keep values unchanged, the final EHEs are not needed, and the simulation is not based on the complicated EHEs  As the TNE degree of the flow to be described rises gradually, the complexity of the derivation process and difficulty of numerical simulation in the KMM method increase sharply 
However, in the DBM method, to describe flows in a one-order more deeper depth of TNE, only two more related kinetic moments need to be added  Since without needing to derive and solve the EHEs, as the TNE degree deepens, the complexity of the DBM approach increases much slower than that of KMM method 
The core step in DBM modeling is to provide a feasible scheme for detecting, describing, presenting, and analyzing TNE effects and behaviors beyond traditional macroscopic modeling  Based on the non-equilibrium statistical physics, we can use the non-conservative moments of ( f − f eq ) to describe how and how much the system deviates from the thermodynamic equilibrium state and to check corresponding effects due to deviating from the thermodynamic equilibrium 
The non-conservative moments of ( f − f eq ) open a phase space, and this space and its subspaces provide an intuitive geometric correspondence for describing complex TNE system properties  The development of schemes for checking TNE state, extracting TNE information and describing corresponding TNE effects in DBM are seen in Table  
Actually, this set of TNE describing methods has been applied in many kinds of complex fluid systems such as hydrodynamic instability system , combustion and detonation systems , multiphase flow system , plasma system , etc  Besides the scheme for detecting, describing, presenting, and analyzing TNE effects and behaviors, the DBM incorporates other methods for analyzing the complex physical field 
One of them is the tracer particle method  The introduction of the tracer particle method makes the gradually blurred interface appear clearly   The rest of the paper is structured as follows  Section 2 shows Year Scheme for investigating TNE effects and behaviors Before 2012 Two classes of LBMs did not show a significant difference in physical function 
2012 Use the non-conservative moments of ( f − f eq ) to check and describe TNE   This is the starting point of current DBM approach  2015 Open TNE phase space based on non-conservative moments of ( f − f eq ) and define a TNE strength using the distance from a state point to the origin  This is the starting point of the phase space description method  
2018 Extend the distance concepts in phase space to describe the TNE difference/similarity of TNE states and kinetic processes   2021 Further extend the phase space description methodology to any set of system characteristics   the modeling method  Then, the numerical simulations and results are presented in Section 3, which includes two subsections 
Section 4 concludes the paper  Other complementary information is given in the Appendix 

Model construction

Based on the Bhatnagar-Gross-Krook (BGK) singlerelaxation model, a two-fluid DBM with a flexible specific-heat ratio is presented in this part  From the origin Boltzmann equation to a DBM, four fundamental steps are needed: (i) Simplification and modification of the Boltzmann equation according to the research requirement 
(ii) Discretization of the particle velocity space under the condition that the reserved kinetic moments keep their values unchanged  (iii) Checking the TNE state and extracting TNE information  (iv) The selection/design of the boundary conditions 

Simplification and modification of the Boltzmann equation

As we know, the collision term in the original Boltzmann contains high dimensional distribution functions  Therefore, the direct solution to it needs too much computing consumption  The most common method to simplify the collision operator is to introduce a local equilibrium distribution function ( f eq ) and write the complex collision operator in a linearized form, i e , the original BGK collision operator − 1 τ ( f − f eq ), where τ is the relaxation time  
The original BGK operator describes the situation where the system is always in the quasi-equilibrium state  Namely, it characterizes only the situation where the Kn number of the system is small enough and f ≈ f eq   The currently used BGK operator for non-equilibrium flows in the field is a modified version incorporating the meanfield theory description  
Based on the above considerations, the simplified Boltzmann equation describing the SBI process is where the two-dimensional equilibrium distribution function is ) where ρ, T , v, u, I, R, and η are the mass density, temperature, particle velocity vector, flow velocity vector, the number of the extra degrees of freedom including molecular rotation and vibration inside the molecules, gas constant, and a free parameter that describes the energy of the extra degrees of freedom, respectively 
The specific-heat ratio is flexible by adjusting parameter I, i e , γ = (D + I + 2)/(D + I), where D = 2 represents the two-dimensional space 

Discretization of the particle velocity space and determination of f σ ,eq i

The continuous Boltzmann equation should be discretized for simulating  Specifically, the continuous velocity space can be replaced by a limited number of particle velocities  So that the values of continuous kinetic moments can be obtained from the summation form of kinetic moments  In this process, it requires the reserved kinetic moments, which are used to characterize the system behaviors, keep their values unchanged after discretizing the velocity space 
Namely, f the reserved kinetic moments  According to the CE analysis, f can be expressed by f eq   Therefore, in the process of discretization, the reserved kinetic moments of f eq should keep their values unchanged, i e , where i represents the kind of discrete velocities and α (α = x or y) is the direction in cartesian coordinate 
To simulate the interaction between two different fluids, a two-fluid DBM should be constructed  Based on the singlerelaxation model, the discrete two-fluid Boltzmann equation can be written as : where σ represents the types of material particle and f σ ,eq i = f σ ,eq i (ρ σ , u, T)  In two-fluid DBM, the macroscopic quantities of the mixture and each component are
where ρ σ and u σ are the mass density and flow velocity of the component σ , respectively  ρ and u represent the mass density and flow velocity of the mixture, respectively  There exist two kinds of temperature (internal energy) definitions in two-fluid DBM because the definition of temperature (internal energy) depends on the flow velocity to be chosen as a reference 
The first definition is by choosing the velocity of the mixture to be a reference, i e , So that the expressions of temperature of component σ and mixture are where We can also choose the flow velocity of component as a reference, i e , , where u σ is the flow velocity of component σ   The corresponding definitions of temperature for component σ and the mixture are
where ∆E * I is It is clear to see that these two definitions of temperature for mixture are the same, but for temperature of component σ are different  We choose the first definition in this manuscript  To solve the Eq  ( ), it is necessary to determine the values of f σ ,eq i   Its values depend on the reserved kinetic moments which characterize the main system behaviors 
In DBM modeling, the CE multiscale analysis is used to determine quickly the reserved kinetic moments  Specifically, when constructing a DBM which only the first order term of Kn number is retained (i e , only the first order TNE effects are retained), seven kinetic moments should be reserved, i e , the M 0 , M 1 , M 2,0 , M 2 , M 3,1 , M 3 , M 4,2  
Two more kinetic moments ( M 4 and M 5,3 ) are needed when the second order TNE is considered   However, it should be noted that the function of CE analysis in DBM modeling is only to determine the kinetic moments that need to be preserved  Whether or not to derive the hydrodynamic equations does not affect the DBM simulation 
The kinetic moments used in our physical modeling are shown in the Appendix B  Their expressions can be obtained by integrating v and η with continuous-form f eq   For better understanding, the Appendix C gives the two-fluid hydrodynamic equations recovered from the Boltzmann equation  The kinetic moments in Appendix B can be written in matrix form, i e , C • f σ ,eq = fσ,eq , (
where C is the matrix of discrete velocity and feq represents the kinetic moments  A proper discrete velocity model is needed to confirm the values of f σ ,eq i   The f σ ,eq can be obtained by solving the inverse matrix, i e , f σ ,eq = C −1 • fσ,eq , where C −1 is the inverse matrix of C  It is very convenient to obtain the inverse matrix of C through some mathematical softwares such as Mathematica, etc 
The D2V16 model is chosen in this paper, its sketches can be seen in Fig    The specific values of D2V16 are given in the following equations: where "cyc" indicates cyclic permutation and c is an adjustable parameter of the discrete velocity model  The sketch of η in D2V16 is η i = η 0 for i = 1 − 4, and η i = 0 for i = 5 − 16 

Checking the TNE state and extracting TNE information

Many physical quantities can characterize the degree of TNE in a fluid system, such as relaxation time, Kn number, viscosity, heat conduction, the gradients of macroscopic quantity, etc  They are all helpful to characterize the TNE strength and describe the TNE behaviors of a fluid system from their perspectives 
But it is not enough only relying on these quantities  Besides the above physical quantities describing the TNE behaviors, in DBM modeling, we can also use the non-conservative moments of ( f − f eq ) to characterize the TNE state and extract TNE information from the fluid system  Fundamentally, four TNE quantities can be defined in a firstorder DBM, i e , ∆ σ * 2 , ∆ σ * 3,1 , ∆ σ * 3 , and ∆ σ * 4,2  
Their definitions can be seen in Table , where v * i = v i − u represents the central velocity and u is the macro flow velocity of the mixture  Physically, ∆ σ * 2 = ∆ σ * 2,αβ e α e β and ∆ σ * 3,1 = ∆ σ * 3,1 e α represent the viscous stress tensor (or non-organized momentum flux, NOMF) and heat flux tensor (or non-organized energy flux, NOEF), respectively 
The e α (e β ) is the unit vector in the α (β ) direction  The later two higher-order TNE quantities contain more condensed information  Specifically, and it indicates the flux information of ∆ σ * 2   To describe the TNE strength of the whole fluid system, some TNE quantities contained more condensed information are also defined, i e ,
Other TNE quantities can be defined based on specific requirements  All the independent components of TNE characteristic quantities open a highdimensional phase space, and this space and its subspaces provide an intuitive image for characterizing the TNE state and understanding TNE behaviors   It should be emphasized that: (i) The TNE strength/intensity/degree is the most basic parameter of non-equilibrium flow description; And any definition of non-equilibrium strength/intensity/degree depends on the research perspective 
(ii) The physical meaning of D * m,n is the TNE strength of this perspective  (iii) From a certain perspective, the TNE strength is increasing; While from a different perspective, the TNE strength, on the other hand, may be decreasing  It is normal, one of the concrete manifestations of the complexity of non-equilibrium flow behavior 
Strictly speaking, those TNE intensity and effect descriptions that do not account for the research perspective are not correct  Do not explain the research perspective, the corresponding is not dependent on the research perspective 

Numerical simulations and results

In this section, we first validate the DBM code by comparing the DBM results with experimental results  Then, the effects of specific-heat ratio on the dynamic process and TNE behaviors on SBI are investigated 

Comparison with experimental results

In the following part, we use a first-order two-fluid DBM to simulate the interaction between a planar shock wave with a 2-D heavy-cylindrical bubbles, and compare the DBM results with the experimental results from Ref    The computational configuration can be seen in Fig    In a flow field which is filled with Air, there is a static bubble composed of 26% Air and 74% SF 6  
A shock with Ma = 1 2 would pass through the bubble from left to right  The initial conditions of ambient gas are ρ 0 = 1 29kg/m 3 , T 0 = 293K, p 0 = 101 3kPa  Ignoring the pressure difference between interior gas and ambient gas, the initial parameters of the bubble are ρ bubble = 4 859kg/m 3 , p bubble = 101 3kPa,
and T 0 = 293K  For simulating, these actual physical quantities should be transferred to dimensionless parameters  This process can refer to the Appendix A  The dimensionless conditions of macroscopic quantities of the fluid field in initial time are (ρ, T, u x , u y ) bubble = (4 0347, 1 0, 0 0, 0 0), (ρ, T, u x , u y ) 1 = (1 3416,
1 128, 0 3616, 0 0), (ρ, T, u x , u y ) 0 = (1 0, 1 0, 0 0, 0 0), where the subscript "0" ("1") represents downstream (upstream) region  In two-fluid DBM code, the distribution function f Air is used to describe the ambient gas, i e , Air  The f bubble characters the bubble which is a mixture that composed of Air and SF 6  
The grid number is N x × N y = 800 × 400, where the N x and N y are grid number in x and y direction, respectively  This grid size has passed the mesh convergence test  The below results also show that it is sufficient to meet the requirements of the following research problem  Other parameters used for the simulation are: c = 1 0, η Air = η bubble = 10 0,
I Air = 3, I bubble = 15, ∆x = ∆y = 1 2 × 10 −4 and ∆t = 1 × 10 −6   The viscosity effect is feeble compared to the shock compression effect, so it does not significantly affect the deformation of the bubble  Therefore, in this part, the relaxation time τ is set sufficiently small  The inflow (outflow) boundary condition is used in the left (right) boundary, and the periodic boundary is adopted in the y direction 
The first-order forward difference scheme is used to calculate the temporal derivative, and the second-order nonoscillatory nonfree dissipative scheme is adopted to solve the spatial derivative in Eq  ( )   Two quantitative comparisons between experimental results and DBM simulations are shown in the following part, including snapshots of schlieren images and evolutions of characteristic scales for the bubble 
The first is shown in Fig  non-organized momentum flux (NOMF) uously to form a diffracted shock (DS)  As TS propagates, it will split into three branches due to the considerable pressure perturbations caused by the gradual decay of the DS strength   Afterward, as shown in the subfigure at about t = 128µs, two high pressure regions (ROH) generate because of the interaction of these branches 
Subsequently, at about t = 148µs, the two ROHs meet, causing the shock focusing  On the one hand, at about t = 168µs, the shock focusing causes the generation of downstream-propagating second transmitted shock (STS) and upward-moving rarefaction wave  On the other hand, it will produce high pressure region inside the bubble, which later leads to a jet structure, as shown at about t = 288µs 
At about t = 428µs, due to the deposited vorticity, there will produce a pair of counter-rotating vortexes at the pole region of the bubble  The further development of the vortex pair and the effect of viscosity decrease the amplitude of the jet  Finally, the jet structure disappears  The second quantitative comparison is the interface structure described by the length and width of the bubble, as shown in Fig   
The experimental data are extracted from Fig  , in Ref    Quantitative agreements between DBM simulation and experimental results are seen  For the profile of bubble width, there are mainly two stages  At an early time (t < 150µs), it decreases to a minimum value because of the shock compression effect 
After the shock wave passes through the bubble (t > 150µs), the developed vortex pair caused by the deposited vorticity gradually dominates the growth of bubble width  Different from width evolution, the temporal variation of length experiences three stages  In the early stages (t < 150µs), it decreases quickly due to the shock compression effect 
Then, the jet structure emerges, which results in a growth in length (150µs < t < 250µs)  Because the upstream interface moves faster than the downstream interface, the bubble length would decrease at 250µs < t < 500µs  In the third stage (t > 500µs), the vortex pair forms and then leads to a continuous development of bubble length 
Both the length and width experience oscillations in the later stages due to complex wave patterns  The quantitative agreements between DBM simulation and experimental results indicate the following two facts: (i) the order of TNE considered in the current DBM is sufficient, (ii) the choosing of discrete velocities and spatial-temporal steps and simulation parameters like the relaxation times is suitable for characterizing the deformation of bubble, wave patterns, main characteristics of flow morphology 

Effects of specific-heat ratio on SBI

The major of current works on SBI research have not focused on specific-heat ratio effects  In this part, the simulation parameters are fine-adjusted based on the parameters in Section 3 1 to highlight the influence of specific-heat ratio  Through adjusting the extra degree of freedom I, five cases with various specific-heat ratios of the bubble are simulated, i e , γ = 1 4,
1 28, 1 18, 1 12, and 1 09  Two kinds of analysis methods, including tracer particle method and two-fluid model, are used to characterize qualitatively the macroscopic behaviors such as the shape, deformation process, mixing degree, etc  The related TNE behaviors are also studied 

Effects of specific-heat ratio on jet shape, deformation process, and average motion

We first observe the specific-heat ratio effect on the bubble shape from the view of density contour and images of particle tracer visually  As shown in Fig  , pictures with three typical moments are plotted, i e , t = 0 07,t = 0 11, and t = 0 16  The odd rows represent density contours and the even rows are tracer particle images 
It can be seen that the specific-heat ratio significantly affects the length and shape of the jet structure  The smaller the specific-heat ratio is, the stouter the jet structure can be seen  The reason is that the specific-heat ratio significantly changes the propagation speed of shock waves and wave patterns inside the bubble 
The specific-heat ratio also influences the vortex structure in early stage but contributes little effects to it in later stage  In the later stage, for cases with different specific-heat ratios, the differences in vortex pairs are almost invisible  Then, the effects of specific-heat ratio on deformation process are analyzed 
Shown in Fig  are the evolutions of characteristic scales which used to describe the bubble size, i e , width and length  It can be seen that the smaller the specific-heat ratio of bubble, the smaller the bubble width and length  For the fluid with smaller specific-heat ratio, it is easier to be compressed 
Therefore, the characteristic scales of bubbles with smaller specific-heat ratio tend to be compressed smaller  It can also be seen that the case with the largest specific-heat ratio reaches the minimum characteristic scales firstly  The reason is that the shock wave propagates faster in case with larger specific-heat ratio 
Through the method of tracer, information on the average motion of the bubble is easy to obtain  Shown in Fig  are the average position and average velocity of the bubble, with different specific-heat ratios  It is found that, in the shock compression stage (t < 0 03), the effect of specific-heat ratio contributes little to the average motion 
However, when the shock wave passes through the bubble (t > 0 03), a larger specific-heat ratio speeds up the average motion of the bubbles  The reason is that the bubbles with smaller specific-heat ratio need more energy to compress their size, so their translational energy is smaller 

Effects of specific-heat ratio on vortex motion

Vorticity is one of the most important physical quantities in describing the vortex motion  In the 2-D case, the vorticity can be calculated by the following equation: The positive (negative) value of ω represents the positive (negative) direction along the z axis  Vorticity contours at t = 0 134, with various specific-heat ratios, are shown in Fig   
The discernable difference between cases with various specific-heat ratios can be observed  The arrows in the vorticity images point out the obvious differences around the interface between case γ = 1 4 and case γ = 1 09  That is to say, there exists influences of specific-heat ratio on the rotational motion of the bubble 
The strength of vorticity is described by circulation Γ, where Γ = ∑ ω∆x∆y  Γ + = ∑ ω| ω>0 ∆x∆y is the positive circulation and Γ − = ∑ ω| ω<0 ∆x∆y represents the negative circulation  Figure shows the temporal evolution of circulations on SBI process  It can be seen that the values of Γ are equal to zero all the time because the values of Γ + and Γ − are the same 
But they point in the opposite direction  In the shock compression stage (t < 0 03), the specific-heat ratio effect contributes little to the circulation of the bubble  When the shock wave sweeps through the bubble (t > 0 03), the specific-heat ratio affects the value of circulation obviously  The cases with a smaller specific-heat ratio experiences a larger range of amplitude of change, which is caused by its good compressibility 

Effects of specific-heat ratio on mixing degree

The mixing process is a fundamental research content on SBI  In two-fluid DBM, the mixing degree at each fluid unit can be defined as M = 4 • M A M B , where M σ represents the mass fraction of component σ   The higher the value of M, the higher the mixing amplitude  Images of density (first cow) and mixing degree M (second row) at several typical moments are shown in Fig   
As can be seen, the mass mixing occurs in the region where two media contact  In addition, the mixing degree M g described the whole fluid field can be defined where the symbol "−" indicates integrating the M σ over the whole fluid field and then dividing the grid size N x • N y   Shown in Fig  is the temporal evolution of the global mixing degree M g  
As can be seen, temporal profiles of the global mixing degree show two stages: t < 0 03 and t > 0 03  When t < 0 03, there is almost no difference between cases with various specific-heat ratios  But for t > 0 03, the stronger the specific-heat ratio effect, the larger the mixing degree  Actually, there are mainly two indicators that measure the global mixing degree: the amplitude of mixing and the area of the mixing zone between two fluids 
At the stage t > 0 03, the shock compression dominates the mix by enhancing the mixing amplitude and increasing the area of the mixing zone simultaneously  In this stage, the specific-heat ratio effect contributes little to the mix  However, when the shock passes through the bubble, the deformation of interface and the evolution of vortex core both significantly raise the area of the mixing zone 
As can be seen in Fig  , the smaller specific-heat ratio of bubble, the stronger global mixing degree of fluid field  Intuitively, for the fluid with smaller specific-heat ratio, it is easier to deform and compress, which is beneficial for the fluid mixing  It can also be explained by the diffusion formula, i e ,  
The specific-heat ratio affects both the temperature T and the gradient of density simultaneously  Therefore, these two aspects comprehensively influence the material diffusion between the two fluids  Due to the complex reflected shock wave, the global mixing degree shows a tendency for oscillating growth 

Effects of specific-heat ratio on TNE behaviors

The investigation of TNE behaviors is of great importance for understanding the kinetics process on SBI  These TNE quantities describe the fluid system deviating from the thermodynamic state from their own perspectives  The effects of specific-heat ratio on global TNE strength, i e , D * 2 , D * 3 , D * 3,1 , and D * 4,2 , are shown in Fig   
It can be seen that the effects of specific-heat ratios on various TNE quantities are different  Theoretically, the influence of specific-heat ratio on the non-equilibrium effect is reflected in two aspects: transport coefficient and macroscopic quantity gradient  For example, on the one hand, the specific-heat ratio reduces heat conductivity, while on the other hand, it enhances the temperature gradient 
Therefore, the effect of specific heat ratio on NOEF is the comprehensive result of the competition between the two aspects  As shown in Fig  , the smaller the specific-heat ratio, the stronger strength of D * 3,1   It indicates that the specific-heat ratio increase the strength of D * 3,1 by raising the heat conductivity  
For the strength of D * 3 , as shown in Fig  (b), it is seen that it decreases as the specific-heat ratio becomes small  The reason is that a smaller specific-heat ratio decreases the temperature gradient  Effects of specific-heat ratio on D * 4,2 show two-stage  In the shock compression stage (t < 0 03), the smaller specificheat ratio, the larger the strength of D * 4,2  
But the situation is reversed at the stage t > 0 03  For strength of D * 2 , the specificheat effects are more significant in later stage 

Effects of specific-heat ratio on entropy production rate

and entropy production The concepts of entropy are commonly used in complex flows   In DBM, there are two kinds of en- tropy production rates, i e , ṠNOEF and ṠNOMF   They are key factors in compression science field  The former is induced by temperature gradient and the NOEF (∆ * 3,1 )  The latter is affected by velocity gradient and the NOMF (∆ * 2 ) 
The entropy production rates are defined by the following formulas : Integrating the ṠNOEF and ṠNOMF over time t, the entropy generations over this period of time are obtained, i e , S NOEF = t 0 ṠNOEF dt and S NOMF = t 0 ṠNOMF dt  Plotted in Fig  ) and 14(b) are the temporal evolution of ṠNOMF and ṠNOEF , respectively 
The evolution of entropy generation rate is related to two aspects: (i) the propagation of the shock wave, and (ii) the deformation of the bubble  The former generates a macroscopic quantity gradient, and the latter makes the contact interface wider, longer, and deformed  Depending on the location of the shock wavefront, there exist two critical moments in this SBI process: (i) at around t = 0 03, the shock wave just sweeps through the bubble, and (ii) at t = 0 06, the shock wave exits the flow field 
Therefore, the temporal evolution of the entropy production rate shows three stages, i e , t < 0 03, 0 03 < t < 0 06, and t > 0 06  At the stage t < 0 03, the shock compression stage, the shock effects compress the bubble  It generates the large macroscopic quantity gradients, resulting in a quick increase of ṠNOMF  
At around t = 0 03, the shock wave passed through the bubble  So the values of ṠNOMF decreases  The values of ṠNOMF would continue to decrease due to the gradually wider contact interface caused by the diffusion effect  At around t = 0 06, the shock wave comes out of the flow field so that the values of ṠNOMF drops rapidly 
In the third stage, i e , t > 0 06, because of the diffusive effect, the general trend of ṠNOMF is downward  However, it shows an oscillatory trend due to the influence of various reflected shock waves  The specific heat ratio indirectly changes the value of ṠNOMF by changing the velocity gradient  The smaller the specific-heat ratio, the larger ṠNOMF  
Different understanding can be seen in Fig  , where the temporal evolution of ṠNOEF is plotted  In the first stage (t < 0 03), cases with different specific-heat ratios show various trends  At the stage where the bubble deformation is not very large, i e , 0 03 < t < 0 06, values of ṠNOEF fluctuate near the average value 
In the third stage (t > 0 06), evolutions of ṠNOEF in cases with larger specific-heat ratios show an apparent growing tendency  Differently, the values of ṠNOEF in cases with smaller specific-heat ratios remain almost unchanged  The influence of specific heat ratio on the ṠNOEF , similar with the effect on NOEF, is also affected by the heat conductivity and the temperature gradient 
It can be seen that, except for the case of γ = 1 09, the larger the specific-heat ratio, the higher entropy production rate ṠNOEF   The temporal evolutions of ṠNOEF of case γ = 1 09 and case γ = 1 12 are very similar  Consequently, the specific-heat ratio increases the ṠNOEF by raising the temperature gradient 
Further understanding can be seen in Fig  , where the entropy productions over this period are plotted  For convenience, the sum and difference between S NOMF and S NOEF are also plotted in the figure  The variation range of S NOEF is larger than that of S NOMF   It indicates that the influence of specific-heat ratio on S NOEF is more significant than that on S NOMF  
Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary  Specifically, it can be seen that the entropy production contributed by NOMF increases with re- duced specific-heat ratio  But the entropy production caused by NOEF first reduces with decreasing specific-heat ratio and then approaches to a saturation value 
The S NOEF in case γ = 1 09 is almost the same with it in case γ = 1 12  When the specificheat ratio γ is smaller than a threshold value γ c (γ c ≈ 1 315), the entropy production induced by NOEF is more significant than that caused by NOMF  However, in the case of γ > γ c , the situation reverses  The temporal evolution of the total entropy production (S NOMF +S NOEF ) is similar to the S NOEF profile 
The difference between S NOMF and S NOEF increases with decreasing specific-heat ratio 

Conclusions

Specific-heat ratio effects on the interaction between a planar shock wave and a 2-D heavy-cylindrical bubble are studied by a two-fluid DBM which has a flexible specific-heat ratio and includes several schemes for analyzing the complex physical fields  Besides the HNE that NS easily describes, the DBM pays more attention to the related TNE that NS is not convenient to describe 
First, both the snapshots of schlieren images and evolutions of characteristic scales from DBM simulation are compared with those from experiment  The quantitative agreements between them indicate the following two facts: (i) the order of TNE considered in the current DBM is sufficient, (ii) the choosing of discrete velocities, spatial-temporal steps, and simulation parameters like the relaxation times are suitable for the following physical researches 
Then, five cases with various specific-heat ratios are simulated  Several analysis methods for complex physical fields, including the description scheme of TNE behaviors, tracer particle method, and two-fluid model, are used to characterize the effects of specific-heat ratio on the bubble shape, deformation process, average motion, vortex motion, mixing degree of the fluid system, TNE strength, and entropy production 
Specifically, for bubble shape, bubbles with different specific-heat ratios display various jet structures  The smaller the specific-heat ratio is, the stouter the jet structure can be seen  For the case with smaller specific-heat ratio, the fluid is easier to be compressed  So, the characteristic scales of bubbles with smaller specific-heat ratio tend to be compressed smaller 
For the bubble, the smaller the specific-heat ratio, the slower average motion  In the shock compression stage, the specific-heat ratio contributes little effects to the vortex motion  Differently, after the shock passes through the bubble, it significantly influences the vorticity around the interface and the corresponding amplitude of circulation due to the development of KHI 
The larger the difference in specific-heat ratio between the bubble and ambient gas, the higher the degree of material mixing  Effects of specific-heat ratio on various TNE quantities are different  These differences consistently show the complexity of TNE flows which is still far from a clear understanding 
In addition, it is found that the temporal evolution of the entropy production rates ṠNOMF and ṠNOEF both show three stages because of the influence of the shock wave location  The smaller the specific-heat ratio, the larger the velocity gradient, which indirectly enhances the strength of ṠNOMF   The specific-heat ratio increases the ṠNOEF by raising the temperature gradient 
The influence of specific-heat ratio on S NOEF is more significant than that on S NOMF   Effects of specific-heat ratio on entropy production caused by NOMF and NOEF are contrary  Specifically, the entropy production contributed by NOMF increases with reduced specific-heat ratio  But the entropy production caused by NOEF first reduces with decreasing specific-heat ratio and then approaches to a saturation value 
When the specific-heat ratio γ is smaller than a threshold value γ c (γ c ≈ 1 315), the entropy production induced by NOEF is more significant than that caused by NOMF  However, in the case of γ > γ c , the situation reverses  The fundamental research in this paper helps to understand the interaction mechanism between shock waves and bubbles in ICF, supersonic combustors, underwater explosions, etc 
The effects of viscosity and heat conduction on the interaction between shock waves and bubbles will be studied in the following work  where the subscript "m, n" means that the m-order tensor is contracted to n-order tensor  According to the CE multiscale analysis, the Boltzmann-BGK equation can be reduced to the hydrodynamic equations 
In the following part, the derivation process from Boltzmann-BGK equation to a two-fluid hydrodynamic equation are shown  More details can see the reference presented by Zhang et al    The discrete Boltzmann equation for component σ is (C 1) In Eq  C 1 there are two equilibrium distribution functions, i e , f σ ,seq = f σ ,seq (ρ σ , u σ , T σ ) and f σ ,eq = f σ ,eq (ρ σ , u, T ) 
For convenience, S σ i is defined as We perform the CE expansion around the f σ ,seq   That is, the distribution function f σ i can be expanded as where ε is a coefficient referring to Knudsen number  The partial derivatives of time and space can also be expanded to Substituting the above four equations into Eq 
(C 1), we can obtain (C 6) When retaining to ε terms, the following equation is obtained When retaining to ε 2 terms, we can obtain where M 2,αβ ( f σ ,(1) ) = ∑ i v iα v iβ f σ ,(1) i and M 3,1,α ( f σ ,(1) ) =   Substituting Eq  (C 14) into the above three equations, and replacing the time derivatives with the space derivatives, we obtain It should be noted that the ability to recover the corresponding level of macroscopic fluid mechanics equations is only part of the physical function of DBM 
The corresponding to the physical functions of DBM is the EHEs, which, in addition to the conserved moments evolution equations corresponding to the three conservation laws of mass, momentum and energy, also includes some of the most closely related nonconserved moments evolution equations  We refer the EHEs derivation based on kinetic equation to as KMM 
The necessity of the expanded part, the evolution equations of the relevant non-conserved moments, increases rapidly as increasing the degree of non-continuity/non-equilibrium  As the degree of non-continuity/non-equilibrium increases, the complexity will rapidly make KMM simulation studies, deriving and solving EHE, impossible 
Context 3: Do you know the difference between V T  and T V?
Like any exclusive club, heart disease has its own jargon, understandable only by other members of the club, particularly by cardiac care providers  For example, I remember lying in my CCU bed (that’s the Coronary Intensive Care Unit), trying to memorize the letters LAD (that’s the Left Anterior Descending, the large coronary artery whose 99% blockage had caused my MI (myocardial infarction – in my case, the so-called ‘widowmaker’ heart attack) 
To help others needing simultaneous translation of this new lingo in your research or in your own medical records, here’s a helpful list of some of the most common acronyms/terms you’ll likely find around the cardiac ward 
NOTE from CAROLYN: This entire patient-friendly, jargon-free glossary (all 8,000 words!) is also part of my book “A Woman’s Guide to Living with Heart Disease“ (Johns Hopkins University Press, November 2017) 
AA – Anti-arrhythmic: Drugs used to treat patients who have irregular heart rhythms 
Ablation – See Cardiac Ablation 
ACE Inhibitor – Angiotension Converting Enzyme inhibitor: A drug that lowers blood pressure by interfering with the breakdown of a protein-like substance involved in regulating blood pressure 
ACS – Acute Coronary Syndrome: An emergency condition brought on by sudden reduced blood flow to the heart  The first sign of acute coronary syndrome can be sudden stopping of your heart (cardiac arrest) 
AED – Automatic External Defibrillator: A portable defibrillator for use during a cardiac emergency; it can be used on patients experiencing sudden cardiac arrest by applying a brief electroshock to the heart through electrodes placed on the chest 
AF or Afib – Atrial Fibrillation: An irregular and often rapid heart rate that can cause poor blood flow to the body  Afib symptoms include heart palpitations, shortness of breath, weakness or fainting  Episodes of atrial fibrillation can come and go, or you may have chronic atrial fibrillation 
AFL – Atrial Flutter: A type of arrhythmia where the upper chambers of the heart (the atria) beat very fast, causing the walls of the lower chambers (the ventricles) to beat inefficiently as well 
A-HCM – Apical Hypertrophic Cardiomyopathy: Also called Yamaguchi Syndrome or Yamaguchi Hypertrophy, a non-obstructive form of cardiomyopathy (a disease of the heart muscle that leads to generalized deterioration of the muscle and its pumping ability) in which a portion of the heart muscle is hypertrophied (thickened) without any obvious cause although there may be a genetic link  It was first described in individuals of Japanese descent 
AI – Aortic Insufficiency: A heart valve disease in which the aortic valve does not close tightly, leading to the backward flow of blood from the aorta (the largest blood vessel) into the left ventricle (a chamber of the heart) 
AIVR – Accelerated Idioventricular Rhythm: Ventricular rhythm whose rate is greater than 49 beats/min but less than 100 beats/min, usually benign  (Ventricles are the two main chambers of the heart, left and right) 
Angina (stable) – A condition marked by distressing symptoms typically between neck and navel that come on with exertion and go away with rest, caused by an inadequate blood supply to the heart muscle typically because of narrowed coronary arteries feeding the heart muscle  Also known as Angina Pectoris  Unstable angina (UA) occurs when fatty deposits (plaques) in a blood vessel rupture or a blood clot forms, blocking or reducing flow through a narrowed artery, suddenly and severely decreasing blood flow to the heart muscle  Unstable angina is not relieved by rest; it’s dangerous and requires emergency medical attention 
Antiplatelet drugs – Medications that block the formation of blood clots by preventing the clumping of platelets (examples: Plavix, Effient, Brillinta, Ticlid, etc)  Heart patients, especially those with implanted stents after PCI, are often prescribed dual antiplatelet therapy (DAPT) which includes one of these prescribed meds along with daily low-dose aspirin 
Aorta – The main artery of the body, carrying blood from the left side of the heart to the arteries of all limbs and organs except the lungs 
Aortic Stenosis: A disease of the heart valves in which the opening of the aortic valve is narrowed  Also called AS 
Aortic valve – One of four valves in the heart, this valve allows blood from the left ventricle to be pumped up (ejected) into the aorta, but prevents blood from returning to the heart once it’s in the aorta 
AP – Apical Pulse: A central pulse located at the apex (pointy bottom) of the heart 
Apex – the lowest (pointy) tip of the heart that points downward at the base, forming what almost looks like a rounded point 
Apical Hypertrophic Cardiomyopathy (A-HCM): Also called Yamaguchi Syndrome or Yamaguchi Hypertrophy, a non-obstructive form of cardiomyopathy (a disease of the heart muscle that leads to generalized deterioration of the muscle and its pumping ability) in which a portion of the heart muscle is hypertrophied (thickened) without any obvious cause  There may be a genetic link  It was first described in people of Japanese descent 
Arrhythmia – A condition in which the heart beats with an irregular or abnormal rhythm 
AS – Aortic Stenosis: A disease of the heart valves in which the opening of the aortic valve is narrowed 
ASD – Atrial Septal Defect: See Septal Defect 
Atrial Flutter – A heart rhythm problem (arrhythmia) originating from the right atrium, most often involving a large circuit that travels around the area of the tricuspid valve (between the right atrium and the right ventricle (this is called typical atrial flutter)  Less commonly, atrial flutter can also result from circuits in other areas of the right or left atrium that cause the heart to beat fast (called atypical atrial flutter) 
Atrial Septum, the membrane that separates the left and the right upper chambers of the heart (the atria) 
Atrium – A chamber of the heart that receives blood from the veins and forces it into a ventricle or ventricles  Plural: atria 
AV – Atrioventricular: A group of cells in the heart located between the upper two chambers (the atria) and the lower two chambers (the ventricles) that regulate the electrical current that passes through it to the ventricles  Also Atrioventricular Block: An interruption or disturbance of the electrical signal between the heart’s upper two chambers (the atria) and lower two chambers (the ventricles)  Also Aortic valve: The valve that regulates blood flow from the heart into the aorta 
AVNRT – Atrioventricular Nodal Re-entry Tachycardia: a heart rhythm problem that happens when there’s an electrical short circuit in the centre of the heart, one of the most common types of SVT, most often seen in people in their twenties and thirties, and more common in women than in men 
BAV – Bicuspid Aortic Valve: The most common malformation of the heart valves in which the aortic valve has only two cusps instead of three 
BB – Beta Blocker: A blood pressure-lowering drug that limits the activity of epinephrine, a hormone that increases blood pressure 
BBB – Bundle Branch Block: – A condition in which parts of the heart’s conduction system are defective and unable to normally conduct the electrical signal, causing an irregular heart rhythm (arrhythmia) 
BMI – Body mass index: A number that doctors use to determine if you’re overweight  BMI is calculated using a formula of weight in kilograms divided by height in meters squared (BMI =W [kg]/H [m2])  Better yet, just click here to figure out your own BMI 
BNP blood test – BNP (B-type Natriuretic Peptide) is a substance secreted from the ventricles or lower chambers of the heart in response to changes in pressure that happen when heart failure develops and/or worsens  The level of BNP in the blood increases when heart failure symptoms worsen, and decreases when the heart failure condition is stable 
BP – Blood Pressure: The force or pressure exerted by the heart in pumping blood; the pressure of blood in the arteries  See also hypertension 
BrS – Brugada Syndrome: Brugada syndrome is a genetic heart disease that is characterized by distinctively abnormal electrocardiogram (EKG/ECG) findings and an increased risk of sudden cardiac arrest 
CAA – Coronary artery anomaly: A congenital defect in one or more of the coronary arteries of the heart 
CABG – Coronary Artery Bypass Graft: A surgical procedure that reroutes blood flow around a diseased or blocked blood vessel that supplies blood to the heart by grafting either a piece of vein harvested from the leg or the artery from under the breastbone 
CA – Coronary Artery: The arteries arising from the aorta that arch down over the top of the heart and divide into branches  They provide blood to the heart muscle 
CAD – Coronary Artery Disease: A narrowing of the arteries that supply blood to the heart  The condition results from a plaque rupture/blood clot or spasm and greatly increases the risk of a heart attack 
Cardiac Ablation – A procedure performed by an Electrophysiologist (EP) – a cardiologist with specialized training in treating heart rhythm problems – that typically uses catheters — long, flexible tubes inserted through a vein in the groin and threaded to the heart — to correct structural problems in the heart that cause an arrhythmia  Cardiac ablation works by scarring or destroying the tissue in your heart that triggers an abnormal heart rhythm 
Cardiac Arrest – Also known as Sudden Cardiac Arrest: The stopping of the heartbeat, usually because of interference with the electrical signal that regulates each heartbeat (often associated with coronary heart disease)  Can lead to Sudden Cardiac Death 
Cardiac Catheterization – An invasive procedure in which a catheter is inserted through a blood vessel in the wrist/arm or groin with x-ray guidance  This procedure can help provide information about blood supply through the coronary arteries, blood pressure, blood flow throughout the chambers of the heart, collection of blood samples, and x-rays of the heart’s ventricles or arteries  It’s typically performed in the cath lab during angiography 
Cardiac Resynchronization Therapy (CRT) also called bi-ventricular pacemaker: an electronic pacing device that’s surgically implanted in the chest to treat the delay in heart ventricle contractions that occur in some people with heart failure 
Cardiac Tamponade – Pressure on the heart that occurs when blood or fluid builds up in the space between the heart muscle (myocardium) and the outer covering sac of the heart (pericardium)  Also called Tamponade 
Cardiomyopathy – a chronic disease of the heart muscle (myocardium), in which the muscle is abnormally enlarged, thickened, and/or stiffened 
Cardioversion – A medical procedure in which an abnormally fast heart rate (tachycardia) or cardiac arrhythmia like atrial fibrillation is converted to a normal rhythm using electricity or drugs  Synchronized electrical cardioversion uses a therapeutic dose of electric current to the heart at a specific moment in the cardiac cycle  Chemical cardioversion uses medications to convert to normal rhythm 
Cath lab – the room in the hospital/medical clinic where cardiac catheterization procedures take place (for example, when a stent is implanted into a blocked coronary artery) 
CCB – Calcium Channel Blocker: A drug that lowers blood pressure by regulating calcium-related electrical activity in the heart 
CDS – Cardiac Depression Scale: A scale that can help assess the effects of depression occurring as a result of a heart disease diagnosis 
CHF – Heart Failure (also called Congestive Heart Failure): A condition in which the heart cannot pump all the blood returning to it, leading to a backup of blood in the vessels and an accumulation of fluid in the body’s tissues, including the lungs 
CM – Cardiomyopathy: A disease of the heart muscle that leads to generalized deterioration of the muscle and its pumping ability 
CO – Cardiac Output: The amount of blood the heart pumps through the circulatory system in one minute 
Collateral arteries – These extra coronary blood vessels are sometimes able to bypass a blockage in an artery in order to supply enough oxygenated blood to enable the heart muscle to survive when in danger of being damaged because of blockage(s) 
Collateral arteries – Blood vessels that provide an alternative arterial supply of blood to an area of the heart that’s in danger of being deprived of oxygenated blood because of one or more blocked arteries 
Congenital heart defect – one of about 35 different types of heart conditions that happen when the heart or the blood vessels near the heart don’t develop normally before a baby is born (in about 1% of live births)  Because of medical advances that treat babies born with heart defects, there are now for the first time more adults with congenital heart disease than children 
Congestive heart failure (CHF) – a chronic progressive condition that affects the pumping power of your heart muscle  Often referred to simply as heart failure, CHF specifically refers to the stage in which fluid builds up around the heart and causes it to pump inefficiently 
COPD – Chronic Obstructive Pulmonary Disease: A lung disease defined by persistently poor airflow as a result of breakdown of lung tissue (known as emphysema) and dysfunction of the small airways Often associated with smoking, it typically worsens over time 
Coronary Microvascular Disease – A heart condition that causes impaired blood flow to the heart muscle through the small vessels of the heart  Also called Microvascular Disease or Small Vessel Disease 
Coronary Reactivity Test – An angiography procedure specifically designed to examine the blood vessels in the heart and how they respond to different medications  Physicians use these images to distinguish different types of blood vessel reactivity dysfunction (such as Coronary Microvascular Disease) 
Costochondritis– the cause of severe chest pain, but NOT heart-related; it’s an inflammation of the cartilage that connects a rib to the breastbone 
Coumadin – A drug taken to prevent the blood from clotting and to treat blood clots  Coumadin is believed to reduce the risk of blood clots causing strokes or heart attacks  See also Warfarin 
Cox Maze procedure – A complex “cut-and-sew” surgical procedure done to treat atrial fibrillation through a complicated set of incisions made in a maze-like pattern on the left and right atria (the upper chambers of the heart) to permanently interrupt the abnormal electrical signals that are causing the irregular heartbeats of Afib  See also: Mini-Maze 
CP – Chest Pain (may also be felt as squeezing, pressure, fullness, pressure, heaviness, burning or tightness in the chest) 
CPR – Cardiopulmonary Resuscitation: An emergency procedure in which the heart and lungs are made to work by manually compressing the chest overlying the heart and forcing air into the lungs, used to maintain circulation when the heart stops pumping during Cardiac Arrest  Current guidelines suggest hands-only CPR  See also AED 
CQ10 – Co-enzyme Q10: A dietary supplement sometimes recommended for heart patients taking statin drugs 
CRP – C-reactive protein: A byproduct of inflammation, produce by the liver, found in the blood in some cases of acute inflammation 
CRT – Cardiac Resynchronization Therapy also called bi-ventricular pacemaker: an electronic pacing device that’s surgically implanted in the chest to treat the delay in heart ventricle contractions that occur in some people with heart failure 
CT – Computed tomography (CT or CAT scan): An x-ray technique that uses a computer to create cross-sectional images of the body 
CTA – Computerized Tomographic Angiogram: An imaging test to look at the arteries that supply the heart muscle with blood  Unlike a traditional coronary angiogram, CT angiograms don’t use a catheter threaded through your blood vessels to your heart but instead rely on a powerful X-ray machine to produce images of your heart and heart vessels 
CV – Coronary Vein: One of the veins of the heart that drain blood from the heart’s muscular tissue and empty into the right atrium 
CV – Cardiovascular: Pertaining to the heart and blood vessels that make up the circulatory system 
DBP – Diastolic blood pressure: The lowest blood pressure measured in the arteries  It occurs when the heart muscle is relaxed between beats 
DCM – Dilated Cardiomyopathy: A disease of the heart muscle, primarily affecting the heart’s main pumping chamber (left ventricle)  The left ventricle becomes enlarged (dilated) and can’t pump blood to your body with as much force as a healthy heart can 
DDI – Drug-drug interaction: A situation in which a medication affects the activity of another medication when both are administered together 
DIL – Diltiazem: A calcium channel blocker drug that acts as a vasodilator; used in the treatment of angina pectoris, hypertension, and supraventricular tachycardia 
Diuretic – A class of drugs used to lower blood pressure  Also known as “water pills” 
Dobutamine stress echocardiography: This is a form of a stress echocardiogram diagnostic test  But instead of exercising on a treadmill or exercise bike to stress the heart, the stress is obtained by giving a drug that stimulates the heart and makes it “think” it’s exercising  The test is used to evaluate your heart and valve function if you are unable to exercise  It is also used to determine how well your heart tolerates activity, and your likelihood of having coronary artery disease (blocked arteries), and it can evaluate the effectiveness of your cardiac treatment plan  See also TTE and Stress Echocardiogram 
Dressler’s syndrome – Happens to a small number of people three to four weeks after a heart attack  The heart muscle that died during the attack sets the immune system in motion, calling on lymphocytes, one of the white blood cells, to infiltrate the coverings of the heart (pericardium) and the lungs (pleura)  It also starts generating antibodies, which attack those two coverings  Chest pain (CP) is the predominant symptom; treated with anti-inflammatory drugs 
Dual Antiplatelet Therapy – Medications that block the formation of blood clots by preventing the clumping of platelets (examples Plavix, Effient, Brillinta, Ticlid, etc ) are often prescribed along with aspirin as part of what’s known as dual antiplatelet therapy, especially to patients who have undergone PCI and stent implantation 
DVT – Deep Vein Thrombosis: A blood clot in a deep vein in the calf 
ECG / EKG – Electrocardiogram: A test in which several electronic sensors are placed on the body to monitor electrical activity associated with the heartbeat 
Ectopic beats – small changes in an otherwise normal heartbeat that lead to extra or skipped heartbeats, often occurring without a clear cause, most often harmless 
EF – Ejection Fraction: A measurement of blood that is pumped out of a filled ventricle  The normal rate is 50-60% 
EKG/ECG – Electrocardiogram: A test in which several electronic sensors are placed on the body to monitor electrical activity associated with the heartbeat 
Endothelium: A single-cell layer of flat endothelial cells lining the closed internal spaces of the body such as the inside of blood vessels  Endothelial dysfunction affects the ability of these cells to help dilate blood vessels, control inflammation or prevent blood clots  The endothelium is associated with most forms of cardiovascular disease, such as hypertension, coronary artery disease, chronic heart failure, peripheral vascular disease, diabetes, chronic kidney failure, and severe viral infections 
Enhanced External Counterpulsation – EECP is an FDA-approved non-invasive, non-drug treatment for angina  It works by promoting the development of collateral coronary arteries  The therapy is widely used in prominent heart clinics such as the Cleveland Clinic, Mayo Clinic and Johns Hopkins – especially for patients who are not good candidates for invasive procedures such as bypass surgery, angioplasty or stenting



ground_truth: The paper aims to solve nonlinear system vibration problems efficiently.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How does the specific-heat ratio affect the average motion of the bubble?

Read the following text and answer briefly.

Context 3: 
EP – Electrophysiologist: A cardiologist who has additional training in diagnosing/treating heart rhythm disorders 
EPS – Electrophysiology Study: A test that uses cardiac catheterization to study patients who have arrhythmias (abnormal hear rhythm)  An electrical current stimulates the heart in an effort to provoke an arrhythmia, which is immediately treated with medications  EPS is used primarily to identify the origin of the arrhythmia and to test the effectiveness of medications used to treat abnormal heart rhythms 
EVH – Endoscopic Vessel Harvesting: To create the bypass graft during CABG open heart surgery, a surgeon will remove or “harvest” healthy blood vessels from another part of the body, often from the patient’s leg or arm  This vessel becomes a graft, with one end attaching to a blood source above and the other end below the blocked area  See CABG 
Exercise stress test – An exercise test (walking/running on a treadmill or pedalling a stationary bike) to make your heart work harder and beat faster  An EKG is recorded while you exercise to monitor any abnormal changes in your heart under stress, with or without the aid of drugs to enhance this assessment  See also: MIBI, Echocardiogram, Nuclear Stress Test 
Familial hypercholesterolemia (FH) – A genetic predisposition to dangerously high cholesterol levels  FH is an inherited disorder that can lead to aggressive and premature cardiovascular disease, including problems like heart attacks, strokes, or narrowing of the heart valves 
Femoral Artery: a major artery in your groin/upper thigh area, through which a thin catheter is inserted, eventually making its way into the heart during angioplasty to implant a stent; currently the most widely used angioplasty approach in the United States, but many other countries now prefer the Radial Artery access in the wrist 
FFR – Fractional Flow Reserve: A test used during coronary catheterization (angiogram) to measure pressure differences across a coronary artery stenosis (narrowing or blockage) defined as as the pressure behind a blockage relative to the pressure before the blockage 
HC – High Cholesterol: When fatty deposits build up in your coronary arteries 
HCTZ – Hydrochlorothiazide: A drug used to lower blood pressure; it acts by inhibiting the kidneys’ ability to retain water  Used to be called “water pills” 
Heart Failure – a chronic progressive condition that affects the pumping power of your heart muscle  Sometimes called Congestive Heart Failure (CHF) 
Holter Monitor – A portable monitoring device that patients wear for recording heartbeats over a period of 24 hours or more 
HTN – Hypertension: High blood pressure, the force of blood pushing against the walls of arteries as it flows through them 
Hypokinesia – Decreased heart wall motion during each heartbeat, associated with cardiomyopathy, heart failure, or heart attack  Hypokinesia can involve small areas of the heart (segmental) or entire sections of heart muscle (global)  Also called hypokinesis 
ICD – Implantable Cardioverter Defibrillator: A surgically implanted electronic device to treat life-threatening heartbeat irregularities 
IHD – Ischemic Heart Disease: heart problems caused by narrowing of the coronary arteries, causing a decreased blood supply to the heart muscle  Also called coronary artery disease and coronary heart disease 
INR – International Normalized Ratio: A laboratory test measure of blood coagulation, often used as a standard for monitoring the effects of the anti-coagulant drug, warfarin (coumadin) 
IST – Inappropriate sinus tachycardia: A heart condition seen most often in young women, in which a person’s resting heart rate is abnormally high (greater than 100 bpm), their heart rate increases rapidly with minimal exertion, and this rapid heart rate is accompanied by symptoms of palpitations, fatigue, and/or exercise intolerance 
Interventional cardiologist – A cardiologist who is trained to perform invasive heart procedures like angiography, angioplasty, percutaneous coronary intervention (PCI), implanting stents, etc 
IVS – Interventricular Septum: The stout wall that separates the lower chambers (the ventricles) of the heart from one another 
IVUS – Intravascular Ultrasound: A form of echocardiography performed during cardiac catheterization in which a transducer (a device that can act as a transmitter (sender) and receiver of ultrasound information) is threaded into the heart blood vessels via a catheter; it’s used to provide detailed information about the blockage inside the blood vessels 
LAD – Left Anterior Descending coronary artery: One of the heart’s coronary artery branches from the left main coronary artery which supplies blood to the left ventricle 
LAFB – Left Anterior Fascicular Block: A cardiac condition,distinguished from Left Bundle Branch Block because only the anterior half of the left bundle branch is defective and more common than left posterior fascicular block 
LAHB – Left Anterior Hemiblock: The Left Bundle Branch divides into two major branches – the anterior and the posterior fascicles  Occasionally, a block can occur in one of these fascicles 
Left Circumflex Artery – The artery carries oxygenated blood from the heart to the body; it’s a branch of the Left Main Coronary Artery after the latter runs its course in between the aorta and the Main Pulmonary Artery 
Left Main Coronary Artery – The artery that branches from the aorta to supply oxygenated blood to the heart via the Left Anterior Descending Artery (LAD) and the Left Circumflex Artery 
Lipids – fat-like substances found in your blood and body tissues; a lipid panel is a blood test that measures the level of specific lipids in blood to help assess your risk of cardiovascular disease, measuring four types of lipids: total cholesterol, HDL cholesterol, LDL cholesterol, and triglycerides 
Lipoprotein-a or Lp(a) – molecules made of proteins and fat, carrying cholesterol and similar substances through the blood  A high level of Lp(a) is considered a risk factor for heart disease; detectable via a blood test 
Long QT syndrome (LQTS): A heart rhythm disorder that can potentially cause fast, chaotic heartbeats that may trigger a sudden fainting spell or seizure  In some cases, the heart may beat erratically for so long that it can cause sudden death 
LV – Left Ventricle – One of four chambers (two atria and two ventricles) in the human heart, it receives oxygenated blood from the left atrium via the mitral valve, and pumps it into the aorta via the aortic valve 
LVAD – Left ventricular assist device: A mechanical device that can be placed outside the body or implanted inside the body  An LVAD does not replace the heart – it “assists” or “helps” it pump oxygen-rich blood from the left ventricle to the rest of the body, usually as a bridge to heart transplant 
LVH – Left Ventricular Hypertrophy: A thickening of the myocardium (muscle) of the Left Ventricle (LV) of the heart  
Lumen – The hollow area within a tube, such as a blood vessel 
Main Pulmonary Artery – Carries oxygen-depleted blood from the heart to the lungs 
MIBI – Nuclear Stress Test/Cardiac Perfusion Scan/Sestamibi: tests that are used to assess the blood flow to the heart muscle (myocardium) when it is stressed by exercise or medication, and to find out what areas of the myocardium have decreased blood flow due to coronary artery disease  This is done by injecting a tiny amount of radionuclide like thallium or technetium (chemicals which release a type of radioactivity called gamma rays) into a vein in the arm or hand 
Microvascular disease – a heart condition that causes impaired blood flow to the heart muscle through the small blood vessels of the heart  Symptoms mimic those of a heart attack  Also called Coronary Microvascular Disease or Small Vessel Disease  I live with this diagnosis and have written more about it here, here and here 
Mini-Maze – a surgical procedure to treat atrial fibrillation, less invasive than what’s called the Cox Maze III procedure (a “cut-and-sew” procedure), and performed on a beating heart without opening the chest 
Mitral Valve: One of four valves in the heart, the structure that controls blood flow between the heart’s left atrium (upper chamber) and left ventricle (lower chamber)  The mitral valve has two flaps (cusps)  See also MV and/or Valves 
Mitral valve prolapse: a condition in which the two valve flaps of the mitral valve don’t close smoothly or evenly, but instead bulge (prolapse) upward into the left atrium; also known as click-murmur syndrome, Barlow’s syndrome or floppy valve syndrome 
MR – Mitral regurgitation: (also mitral insufficiency or mitral incompetence) a heart condition in which the mitral valve does not close properly when the heart pumps out blood  It’s the abnormal leaking of blood from the left ventricle, through the mitral valve and into the left atrium when the left ventricle contracts 
MRI – Magnetic Resonance Imaging: A technique that produces images of the heart and other body structures by measuring the response of certain elements (such as hydrogen) in the body to a magnetic field  An MRI can produce detailed pictures of the heart and its various structures without the need to inject a dye 
MS – Mitral Stenosis: A narrowing of the mitral valve, which controls blood flow from the heart’s upper left chamber (the left atrium) to its lower left chamber (the left ventricle)  May result from an inherited (congenital) problem or from rheumatic fever 
MUGA – Multiple-Gated Acquisition Scanning: A non-invasive nuclear test that uses a radioactive isotope called technetium to evaluate the functioning of the heart’s ventricles 
Murmur – Noises superimposed on normal heart sounds  They are caused by congenital defects or damaged heart valves that do not close properly and allow blood to leak back into the originating chamber 
MV – Mitral Valve: The structure that controls blood flow between the heart’s left atrium (upper chamber) and left ventricle (lower chamber) 
Myocardial Infarction (MI, heart attack) – The damage or death of an area of the heart muscle (myocardium) resulting from a blocked blood supply to the area  The affected tissue dies, injuring the heart 
Myocardium – The muscular tissue of the heart 
New Wall-Motion Abnormalities – Results seen on an echocardiogram test report (see NWMA, below) 
Nitroglycerin – A medicine that helps relax and dilate arteries; often used to treat cardiac chest pain (angina)  Also called NTG or GTN 
NSR – Normal Sinus Rhythm: The characteristic rhythm of the healthy human heart  NSR is considered to be present if the heart rate is in the normal range, the P waves are normal on the EKG/ECG, and the rate does not vary significantly 
NSTEMI – Non-ST-segment-elevation myocardial infarction: The milder form of the two main types of heart attack  An NSTEMI heart attack does not produce an ST-segment elevation seen on an electrocardiogram test (EKG)  See also STEMI 
Nuclear Stress Test – A diagnostic test that usually involves two exercise stress tests, one while you’re exercising on a treadmill/stationary bike or with medication that stresses your heart, and another set while you’re at rest  A nuclear stress test is used to gather information about how well your heart works during physical activity and at rest  See also: Exercise stress test, Nuclear perfusion test, MIBI 
Open heart surgery – Any surgery in which the chest is opened and surgery is done on the heart muscle, valves, coronary arteries, or other parts of the heart (such as the aorta)  See also CABG 
Pacemaker – A surgically implanted electronic device that helps regulate the heartbeat 
PAD – Peripheral Artery Disease: A common circulatory problem in which narrowed arteries reduce blood flow to the limbs, usually to the legs  Symptoms include leg pain when walking (called intermittent claudication) 
PAF – Paroxysmal Atrial Fibrillation: Atrial fibrillation that lasts from a few seconds to days, then stops on its own  See also Atrial Fibrillation 
Palpitations – A noticeably rapid, strong, or irregular heartbeat due to agitation, exertion or illness 
Paroxysmal Atrial Fibrillation – An unusual heart arrhythmia of unknown origin, at one time believed to be associated with an unusual sensitivity to alcohol consumption 
PDA – patent ductus arteriosus: A persistent opening between two major blood vessels leading from the heart  The opening is called ductus arteriosus and is a normal part of a baby’s circulatory system before birth that usually closes shortly after birth  But when it remains open, it’s called a patent ductus arteriosus  If it’s small, it may never need treatment, but a large PDA left untreated can allow poorly oxygenated blood to flow in the wrong direction, weakening the heart muscle and causing heart failure or other complications 
Pericardium: two thin layers of a sac-like tissue that surround the heart, hold it in place and help it work 
PET – Positron Emission Tomography: A non-invasive scanning technique that uses small amounts of radioactive positrons (positively charged particles) to visualize body function and metabolism  In cardiology, PET scans are used to evaluate heart muscle function in patients with coronary artery disease or cardiomyopathy 
PFO – Patent Forman Ovale: An opening between the left and right atria (the upper chambers) of the heart  Everyone has a PFO before birth, but in 1 out of every 3 or 4 people, the opening does not close naturally as it should after birth 
Plaque – A deposit of fatty (and other) substances in the inner lining of the artery wall; it is characteristic of atherosclerosis 
POTS – Postural Orthostatic Tachycardia Syndrome: A disorder that causes an increased heart rate when a person stands upright 
PPCM – Post-partum cardiomyopathy: A form of cardiomyopathy that causes heart failure toward the end of pregnancy or in the months after delivery, in the absence of any other cause of heart failure 
Preeclampsia – a late-pregnancy complication identified by spikes in blood pressure, protein in the urine, possible vision problems  Women who experience pregnancy complications like preeclampsia are at significantly higher risk for heart disease 
Prinzmetal’s Variant Angina – Chest pain caused by a spasm in a coronary artery that supplies blood to the heart muscle 
PSVT – Paroxysmal Supraventricular Tachycardia: – An occasional rapid heart rate (150-250 beats per minute) that is caused by events triggered in areas above the heart’s lower chambers (the ventricles)  “Paroxysmal” means from time to time  See also supraventricular tachycardia (SVT) 
Pulmonary Valve: One of the four valves in the heart, located between the pulmonary artery and the right ventricle of the heart, moves blood toward the lungs and keeps it from sloshing back into the heart 
PV – Pulmonary Vein: A vein carrying oxygenated blood from the lungs to the left atrium of the heart 
PVC – Premature Ventricular Contraction: An early or extra heartbeat that happens when the heart’s lower chambers (the ventricles) contract too soon, out of sequence with the normal heartbeat  In the absence of any underlying heart disease, PVCs do not generally indicate a problem with electrical stability, and are usually benign 
RA – Right Atrium: The right upper chamber of the heart  The right atrium receives de-oxygenated blood from the body through the vena cava and pumps it into the right ventricle which then sends it to the lungs to be oxygenated 
Radial Artery: the artery in the wrist where a thin catheter is inserted through the body’s network of arteries in the arm and eventually into the heart during a procedure to implant a stent  Doctors may also call this transradial access, the transradial approach, or transradial angioplasty  Because it’s associated with fewer complications, this is increasingly considered the default access approach in most countries, except in the U S  where the traditional Femoral Artery (groin) approach is still the most popular access 
RBBB – Right Bundle Branch Block: A delay or obstruction along the pathway that electrical impulses travel to make your heart beat  The delay or blockage occurs on the pathway that sends electrical impulses to the right side of your heart  See also Left Bundle Branch Block 
RCA – Right Coronary Artery: An artery that supplies blood to the right side of the heart 
Restenosis – The re-closing or re-narrowing of an artery after an interventional procedure such as angioplasty or stent placement  Sometimes called “stent failure” 
RHD – Rheumatic Heart Disease: Permanent damage to the valves of the heart caused especially by repeated attacks of rheumatic fever 
RM – Right Main coronary artery: A blood vessel that supplies oxygenated blood to the walls of the heart’s ventricles and the right atrium 
RV – Right Ventricle: The lower right chamber of the heart that receives de-oxygenated blood from the right atrium and pumps it under low pressure into the lungs via the pulmonary artery 
SA – Sinus node: The “natural” pacemaker of the heart  The node is a group of specialized cells in the top of the right atrium which produces the electrical impulses that travel down to eventually reach the ventricular muscle, causing the heart to contract 
SB – Sinus Bradycardia: Abnormally slow heartbeat 
SBP – Systolic Blood Pressure: The highest blood pressure measured in the arteries  It occurs when the heart contracts with each heartbeat  Example: the first number in 120/80 
SCAD – Spontaneous Coronary Artery Dissection: A rare emergency condition that occurs when a tear forms in one of the blood vessels in the heart, causing a heart attack, abnormalities in heart rhythm and/or sudden death  SCAD tends to strike young healthy women with few if any cardiac risk factors 
SD – Septal defect: A hole in the wall of the heart separating the atria (two upper chambers of the heart) or in the wall of the heart separating the ventricles (two lower chambers) 
Sestamibi stress test – See MIBI 
Short QT intervals (SQT): An abnormal heart rhythm where the heart muscle takes a shorter time to recharge between beats  It can cause a variety of complications from fainting and dizziness to sudden cardiac arrest 
Sick Sinus Syndrome (also known as sinus node dysfunction) is caused by an electrical problem in the heart; a group of related heart conditions that can affect how the heart beats, most commonly in older adults, although it can be diagnosed in people of any age  “Sick sinus” refers to the sinoatrial node (see below)  In people with sick sinus syndrome, the SA node does not function normally



ground_truth: The specific-heat ratio affects the average motion of the bubble. The bubbles with smaller specific-heat ratios have slower average motion.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the purpose of an ICD?

Read the following text and answer briefly.

Context 3: 
Sinoatrial node (SA): also commonly called the sinus node; it’s a small bundle of neurons situated in the upper part of the wall of the right atrium (the right upper chamber of the heart)  The heart’s electrical impulses are generated there  It’s the normal natural pacemaker of the heart and is responsible for the initiation of each heartbeat 
Spontaneous Coronary Artery Dissection (SCAD) – A rare emergency condition that occurs when a tear forms in one of the blood vessels in the heart, causing a heart attack, abnormalities in heart rhythm and/or sudden death  SCAD tends to strike young healthy women with few if any cardiac risk factors 
SSS – Sick Sinus Syndrome: The failure of the sinus node to regulate the heart’s rhythm 
ST – Sinus Tachycardia: A heart rhythm with elevated rate of impulses originating from the sinoatrial node, defined as greater than 100 beats per minute (bpm) in an average adult  The normal heart rate in the average adult ranges from 60–100 bpm  Also called sinus tach or sinus tachy 
Statins – Any of a class of drugs that lower the levels of low-density lipoproteins (LDL) – the ‘bad’ cholesterol in the blood – by inhibiting the activity of an enzyme involved in the production of cholesterol in the liver  Examples of brand name statins: Lipitor, Crestor, Zocor, Mevacor, Levachol, Lescol, etc  Also available as a cheaper generic form of the drug 
STEMI – ST-elevation heart attack (myocardial infarction)  The more severe form of the two main types of heart attack  A STEMI produces a characteristic elevation in the ST segment on an electrocardiogram (EKG)  The elevated ST segment is how this type of heart attack got its name  See also NSTEMI 
Stent – An implantable device made of expandable, metal mesh (looks a bit like a tiny chicken wire tube) that is placed (by using a balloon catheter) at the site of a narrowing coronary artery during an angioplasty procedure  The stent is then expanded when the balloon fills, the balloon is removed, and the stent is left in place to help keep the artery open  TRIVIA ALERT: the coronary stent was named after Charles Stent (1807-1885), an English dentist who invented a compound to produce dentures and other things like skin grafts and hollow tubes (essentially what a metal coronary stent is)  His real claim to fame occurred when he suggested using his material to coat underwater trans-Atlantic cable, which had broken several times as a result of corrosion by seawater  You’re welcome 
Stint – a common spelling mistake when what you really mean is the word “stent” (see above) 
Stress Echocardiography – A standard echocardiogram test that’s performed while the person exercises on a treadmill or stationary bicycle  This test can be used to visualize the motion of the heart’s walls and pumping action when the heart is stressed, possibly revealing a lack of blood flow that isn’t always apparent on other heart tests  The echocardiogram is performed just before and just after the exercise part of the procedure  See also TTE 
Sudden Cardiac Arrest – The stopping of the heartbeat, usually because of interference with the electrical signal (often associated with coronary heart disease)  Can lead to Sudden Cardiac Death 
Takotsubo Cardiomyopathy – A heart condition that can mimic a heart attack  Sometimes called Broken Heart Syndrome, it is not a heart attack, but it feels just like one, with common symptoms like severe chest pain and shortness of breath  It sometimes follows a severe emotional stress  Over 90% of reported cases are in women ages 58 to 75  Also referred to as Broken Heart Syndrome, stress cardiomyopathy, stress-induced cardiomyopathy or apical ballooning syndrome 
TAVR – Transcatheter aortic valve replacement: A minimally invasive procedure to repair a damaged or diseased aortic valve  A catheter is inserted into an artery in the groin and threaded to the heart  A balloon at the end of the catheter, with a replacement valve folded around it, delivers the new valve to take the place of the old  Also called TAVI (Transcatheter aortic valve implantation) 
Tetralogy of Fallot – A rare condition caused by a combination of four heart defects that are present at birth, affecting the structure of the heart and causing oxygen-poor blood to flow out of the heart and into the rest of the body  Infants and children with Tetralogy of Fallot usually have blue-tinged skin because their blood doesn’t carry enough oxygen  Often diagnosed in infancy, but sometimes not until later in life depending on severity 
Tg – Triglycerides: The most common fatty substance found in the blood; normally stored as an energy source in fat tissue  High triglyceride levels may thicken the blood and make a person more susceptible to clot formation  High triglyceride levels tend to accompany high cholesterol levels and other risk factors for heart disease, such as obesity 
TIA – Transient Ischemic Attack: A stroke-like event that lasts only for a short time and is caused by a temporarily blocked blood vessel 
TEE – Transesophageal echocardiogram: This test involves an ultrasound transducer inserted down the throat into the esophagus in order to take clear images of the heart structures without the interference of the lungs and chest 
Treadmill Stress Test – See Exercise Stress Test 
troponin – a type of cardiac enzyme found in heart muscle, and released into the blood when there is damage to the heart (for example, during a heart attack)  A positive blood test that shows elevated troponin is the preferred test for a suspected heart attack because it is more specific for heart injury than other blood tests, especially the newer high sensitivity troponin tests (hs-cTnT) 
TTE – Transthoracic Echocardiogram: This is the standard echocardiogram, a painless test similar to X-ray, but without the radiation, using a hand-held device called a transducer placed on the chest to transmit high frequency sound waves (ultrasound)  These sound waves bounce off the heart structures, producing images and sounds that can be used by the doctor to detect heart damage and disease 
TV – Tricuspid Valve: One of four one-way valves in the heart, a structure that controls blood flow from the heart’s upper right chamber (the right atrium) into the lower right chamber (the right ventricle) 
UA or USA – Unstable Angina: Chest pain that occurs when diseased blood vessels restrict blood flow to the heart; symptoms are not relieved by rest; considered a dangerous and emergency crisis requiring immediate medical help 
Valves: Your heart has four one-way valves that keep blood flowing in the right direction  Blood enters the heart first through the tricuspid valve, and next goes through the pulmonary valve (sometimes called the pulmonic valve) on its way to the lungs  Then the blood returning from the lungs passes through the mitral (bicuspid) valve and leaves the heart through the aortic valve 
Vasodilator: A drug that causes dilation (widening) of blood vessels 
Vasospasm: A blood vessel spasm that causes sudden constriction, reducing its diameter and blood flow to the heart muscle  See also Prinzmetal’s Variant Angina 
VB – Ventricular Bigeminy: A heart rhythm condition in which the heart experiences two beats of the pulse in rapid succession 
Vena Cava – a large vein that carryies de-oxygenated blood into the heart  There are two in humans, the inferior vena cava (carrying blood from the lower body) and the superior vena cava (carrying blood from the head, arms, and upper body) 
Ventricle – each of the two main chambers of the heart, left and right 
VF – Ventricular Fibrillation: A condition in which the ventricles (two lower chambers of the heart) contract in a rapid, unsynchronized fashion  When fibrillation occurs, the ventricles cannot pump blood throughout the body  Most sudden cardiac deaths are caused by VF or ventricular tachycardia (VT) 
VLDL – Very Low Density Lipoprotein: Molecules made up of mostly triglycerides, cholesterol and proteins  VLDL, also known as the “very bad” cholesterol, carries cholesterol from the liver to organs and tissues in the body  It may lead to low density lipoproteins (LDL), associated with higher heart disease risks  VLDL levels are tricky to measure routinely, and are usually estimated as a percentage of your triglyceride levels  By reducing triglycerides, you are usually also reducing your VLDL levels 
Warfarin – A drug taken to prevent the blood from clotting and to treat blood clots  Warfarin is believed to reduce the risk of blood clots causing strokes or heart attacks  Also known as Coumadin 
Widowmaker heart attack – The type of heart attack I survived, since you asked  A nickname doctors use to describe a severely blocked left main coronary artery or proximal left anterior descending coronary artery of the heart  This term is used because if the artery gets abruptly and completely blocked, it can cause a massive heart attack that will likely lead to sudden cardiac death  Please note the gender imbalance here: despite the number of women like me who do experience this type of cardiac event, doctors are not calling this the widowermaker, after all 
WPW – Wolff-Parkinson-White Syndrome: A condition in which an extra electrical pathway connects the atria (two upper chambers) and the ventricles (two lower chambers)  It may cause a rapid heartbeat 
NOTE FROM CAROLYN: I was very happy when we were able to include this entire glossary in my book, “A Woman’s Guide to Living with Heart Disease“ (Johns Hopkins University Press, 2017) 
Are we missing any important heart acronyms/terms from this list? Let me know!
Please can someone explain something for me  I am a 53 yr old woman and generally fit and healthy  Had 2 ECG’s due to a one off dizzy spell during a stressful time dealing with my fathers terminal diagnosis  The 2nd ECG request did give me concern as i did not know why i had to have one  On 24/01/19 at my doctors appointment she explained that on 3 the leads it showed inverted T waves  And she explained that it may suggest angina  I was so shocked  Wasn’t expecting that  She gave me a GNT (nitroglycerin) spray in case I do get pain and take 75Mg of aspirin  I’m now waiting for a Cardiology referral 
I am so stressed and consumed by what might be wrong  My maternal grandmother had angina and valve issues  Her 3 brothers all had double bypasses  Could I have inherited this? I am not overweight at 63kg and 5 ft 9  I walk 20-25 miles a week at work and general walking here and there  I started HRT (patches evorol 25 -50) in July as menopause pain was making me feel like I was 90 and was getting me down 
I am worried so much now and analysing every ache/ twinge I get  I feel like a hypochondriac at the moment  I’m worried what will happen at the cardiologist and what the test will entail and tell me  I am waiting on cholesterol test which I had on 25/01/19  Can I have inverted T waves and be fine  Please help I am so scared and crying far too much 
Hello Colleen – the first thing is: please take a big deep breath before you read another word here! I’m not a physician so of course cannot comment on your specific case, but I can tell you generally that the definition of “angina” (as this glossary lists above) is “distressing symptoms”, typically chest pain that gets worse with exertion, and goes away with rest  That’s classic stable angina… typically caused by something that’s reducing blood flow to the heart muscle (causing the chest pain of angina) 
A family history that might make a difference for you personally is only in what’s called your ‘first degree’ relatives: for example, if your mother or sister were diagnosed with heart disease before age 65, or if your Dad or brother were diagnosed before age 55, then doctors would consider that you have a family history as a risk factor for heart disease  There’s little if any scientific evidence that a grandparent or uncle’s heart disease history has any effect on your own risk 
It is a very good thing that you’re having further tests and a referral to a cardiologist, if only to ease your mind  There are many reasons for inverted T-waves, ranging from cardiac issues to completely benign conditions  One way of looking at this is choosing to believe that seeing a cardiologist will ease your mind one way or the other – so this is something to look forward to, not dread  If the cardiologist spots something suspicious, a treatment plan will be created  If not, you can wave goodbye and go back to happily living your life 
Try thinking of this cardiology appointment just as you would if your car were making some frightening noises and you were bringing it to your mechanic for a check up  You could work yourself into a complete state worrying ahead of time if the car trouble is going to be serious, or you could look at this appointment as the solution – at last! – to figuring out what’s wrong so the mechanic can recommend the next step 
Thank you for this list of so many definitions provided in plain English  what a valuable resource this is  THANK YOU, I have been looking for translations FOR PATIENTS not med school graduates– like this for three years 
My family doctor had me wear a 24 hr EKG  After reading the results, she has scheduled a scope to look inside my heart by a specialist  Completely forgoing a stress test  Said I have major changes in the EKG, what type of changes could they be looking at? Had LAD STENT INSERTED 7 YRS AGO – WHAT COULD THEY BE LOOKING FOR?
This is a great wealth of information, Carolyn! I looked and did not see my diagnosis, which is aortic stenosis  I looked under aortic as well as stenosis  Did I just miss it somehow?
I learned some new information, I am a bit familiar now, but not when I had my MI, it was like learning a new language  But, my favorite part was seeing SCAD on this list! Thank you 
Thanks and welcome! I was thinking of editing that SCAD definition actually: I suspect that that it isn’t so much that SCAD is “rare”, but it’s more that it’s “rarely correctly diagnosed” 
I totally agree that SCAD is not as rare as I believed for many years  Once awareness is spread to all medical staff, I believe many lives will be saved  Hoping for a brighter future for all SCAD patients 
I hope so too, Cathy  Perhaps when more SCAD studies (like Mayo Clinic’s) are published and read by more and more MDs, it will no longer be “rarely correctly diagnosed” 
It’s great to see IST on here  I was diagnosed with it 9 years ago and the lack of awareness is frustrating 
What a great resource for heart patients and their families!
Thanks so much, Ashley  I recently updated my original 2011 list after the world-famous Cleveland Clinic tweeted their glossary recently and I noticed that their list had a few glaring omissions (like SCAD and Brugada Syndrome) so this made me wonder what my list might be missing, too  Let me know if there’s anything else you think should be included, okay?
How is your health these days? How are you feeling?
New for me too  I have just been diagnosed with A-HCM: Apical Hypertrophic Cardiomyopathy 
I’ll add that one to my list, Kathleen – thanks!
Just saw this, Carolyn, and you’ve compiled a great resource  One note on A-HCM: Present thinking is that it’s due to a genetic modification  Runs in families though sometimes occurs spontaneously  I have not as yet done genetic testing, though it’s been offered 
Thanks Kathleen – like many cardiac diagnoses, it sounds like a moving target… Good luck to you!
This list is great  I’ve just been diagnosed and am utterly overwhelmed  Even in the WomenHeart online support community, I often have no clue most days what others are talking about with all these initials about their heart tests and specific disease  This is VERY helpful, thank you SO MUCH  Love your website which has been a godsend since my diagnosis 



ground_truth: Implantable Cardioverter Defibrillator (ICD) is a surgically implanted electronic device to treat life-threatening heartbeat irregularities.