Rotating detonation engine pressure signal construction method and device

Abstract

The present invention discloses a method for constructing a pressure signal of a rotating detonation engine. The method comprises: step 1, based on the thermodynamic model of the rotating detonation engine, obtaining the static pressure distribution parameters of the rotating detonation engine under given flight conditions and control quantity input by an iterative solution method; step 2, calculating the change of the pressure value p of any pressure measuring point in the rotating detonation combustion chamber with time t within the signal update period Δt required for one calculation of the thermodynamic model; step 3, constructing the pressure signal of each cross section within the signal update period Δt. The present invention also discloses a device for constructing a pressure signal of a rotating detonation engine. The present invention can simulate the pressure signals and change trends of each cross section of the rotating detonation engine across space and a wide speed range, especially the time-varying characteristics of the periodic pressure distribution signal of the rotating detonation combustion chamber, to meet the simulation and test needs of the development of the rotating detonation engine control system.

Description

Technical Field

[0001] The present invention relates to a method for constructing an aero-engine pressure signal, in particular to a method for constructing a rotating detonation engine pressure signal, and belongs to the technical field of aero-engine simulation. Background Art

[0002] With the development of modern aviation technology, the demand for higher performance in aircraft engines is increasing. Traditional aircraft engines use the Brayton cycle, which continuously performs isobaric combustion within the combustion chamber, resulting in relatively limited thermal efficiency. Detonation combustion combines a combustion wave with a shock wave. Compared to the former, the pressure surges after the detonation wave, resulting in higher thermal efficiency. Therefore, detonation engines based on a Rankine-like cycle are becoming a mainstream research topic both domestically and internationally.

[0003] As the most advanced detonation combustion technology currently available, rotating detonation offers a safer and more efficient design for applying detonation combustion to engine combustion chambers. A rotating detonation engine can sustain detonation after a single detonation, boasts high thermal efficiency and a relatively simple structure. This effectively extends the lower limit of the operating Mach number of conventional subsonic ramjet engines and offers the potential to maintain rotating detonation combustion even under hypersonic flight conditions, making it a highly promising high-speed, supersonic aircraft engine. For conventional turbofan and turbojet engines, rotating detonation combustion technology can be applied to turbine-based rotating detonation engines by replacing the original combustion chamber with a rotating detonation chamber or by adding additional rotating detonation chambers in series or parallel.

[0004] The high-frequency, periodic fluctuations in the combustion chamber pressure of these rotating detonation engines are a key indicator of rotating detonation combustion. Aerospace engines operate within a wide range of speeds and spaces. To ensure safe and stable engine operation throughout this envelope, the control system utilizes various algorithms to provide comprehensive control of the engine, including monitoring and feedback control of the key characteristics of rotating detonation combustion.

[0005] Currently, the engineering development of rotating detonation engines is in its infancy, and there is no effective platform or technology specifically designed for testing and verifying control systems developed for these engines. Furthermore, engine testing is costly and time-consuming, making it difficult to adapt to the rapid testing, iteration, and testing required in the early stages of control system development. Consequently, this hinders the development of controller systems. Therefore, a method tailored to control application requirements and capable of simulating rotating detonation engine pressure signals across the full pressure envelope is urgently needed. This method can provide key signal characteristics and trends within the safety envelope of rotating detonation engines to support control system development and verification.

[0006] Existing thermodynamic models of rotating detonation engines output the thermodynamic parameters of each cross-section after each dynamic calculation, enabling the effective measurement of discrete pressure signals for each cross-section throughout the model control simulation process. With the exception of the rotating detonation combustor, all engine components already have well-established 0-dimensional component-level modeling methods. These models maintain consistent thermodynamic parameters across the same cross-section, allowing the real-time pressure calculated by the model to be output as the dynamic pressure signal for the entire cross-section. However, the pressure in a rotating detonation combustor exhibits a two-dimensional distribution. Due to the pressure distribution characteristics of the combustor, the pressure at different measurement points on the same cross-section varies significantly depending on the location of the detonation wave: the pressure is highest at the wavefront, where it surges as the incoming flow passes through it; the pressure is lowest at the end of the blockage zone, where it is equal to the incoming flow pressure. As the detonation wave propagates rapidly through the annular combustor, the pressure at a specific point on the chamber wall also varies continuously, reflecting a time-varying pattern of periodic surges and dips. Existing rotating detonation combustion chamber models can only output the static pressure distribution of the detonation wave at the current sampling moment. They lack the ability to directly output the pressure at a fixed point in the combustion chamber, nor can they measure the dynamic pressure signal at a fixed point in the detonation combustion chamber in real time to meet the feedback control requirements of the controller. Therefore, the traditional method of directly feeding back pressure signals based on model calculations is not directly applicable to rotating detonation engines. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to overcome the shortcomings of the existing technology and provide a method for constructing the pressure signal of a rotating detonation engine, which can simulate the pressure signals and changing trends of each section of the rotating detonation engine across space and a wide speed range, especially the time-varying characteristics of the periodic pressure distribution signal of the rotating detonation combustion chamber, to meet the simulation and experimental needs of the development of the rotating detonation engine control system.

[0008] The present invention specifically adopts the following technical solutions to solve the above technical problems:

[0009] A method for constructing a pressure signal of a rotating detonation engine comprises the following steps:

[0010] Step 1: Based on the rotating detonation engine thermodynamic model, the static pressure distribution parameters of the rotating detonation engine under given flight conditions and control variable input are obtained through iterative solution, including the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation wave head as the starting point;

[0011] Step 2: Calculate the change of the pressure value p at any pressure measuring point in the rotating detonation combustion chamber over time t within the signal update period Δt required for one thermodynamic model calculation according to the following formula:

[0012]

[0013] Among them, the wave head position L and the initial wave head position L0 are respectively the current and initial circumferential distances of the detonation wave head relative to the measuring point along the propagation direction of the detonation wave, v d is the detonation wave velocity, C is the circumference of the rotating detonation combustion chamber, t d is the detonation wave period, l is the number of complete detonation wave periods within one signal update period Δt;

[0014] Step 3: Construct the pressure signal of each cross section within the signal update period Δt:

[0015] For each cross section of the non-rotating detonation combustion chamber, the pressure signal is constructed according to the following formula:

[0016] p out =p(qΔt s )@p z ,0≤qΔt s <Δt,q=0,1,2,...

[0017] For each cross section in the rotating detonation combustion chamber, the pressure signal is constructed according to the following formula:

[0018]

[0019] Where, Δt s is the signal sampling period, and z represents the section number.

[0020] Preferably, the detonation wave head height h and the blockage ratio B are taken as guess values, and the pressure distribution p(r) with the detonation wave head as the starting point in the rotating detonation combustion chamber under given flight conditions and control variable input is obtained by iterative solution.

[0021] Based on the same inventive concept, the following technical solutions can also be obtained:

[0022] A rotating detonation engine pressure signal construction device, comprising:

[0023] Thermodynamic model calculation module is used to obtain the static pressure distribution parameters of the rotating detonation engine under given flight conditions and control variable inputs through iterative solution based on the rotating detonation engine thermodynamic model, including the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation wave head as the starting point;

[0024] The rotating detonation combustion chamber pressure calculation module is used to calculate the change of the pressure value p of any pressure measuring point in the rotating detonation combustion chamber with time t within the signal update period Δt required for one calculation of the thermodynamic model according to the following formula:

[0025]

[0026] Among them, the wave head position L and the initial wave head position L0 are respectively the current and initial circumferential distances of the detonation wave head relative to the measuring point along the propagation direction of the detonation wave, v d is the detonation wave velocity, C is the circumference of the rotating detonation combustion chamber, t d is the detonation wave period, l is the number of complete detonation wave periods within one signal update period Δt;

[0027] The pressure signal construction module is used to construct the pressure signal of each cross section within the signal update period Δt:

[0028] For each cross section of the non-rotating detonation combustion chamber, the pressure signal is constructed according to the following formula:

[0029] p out =p(qΔt s )@p z ,0≤qΔt s <Δt,q=0,1,2,...

[0030] For each cross section in the rotating detonation combustion chamber, the pressure signal is constructed according to the following formula:

[0031]

[0032] Where, Δt s is the signal sampling period, and z represents the section number.

[0033] Preferably, the thermodynamic model of the rotating detonation engine uses the detonation wave head height h and the blockage ratio B as guess values, and obtains the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation wave head as the starting point under given flight conditions and control variable input through iterative solution.

[0034] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0035] (1) High applicability: The present invention can calculate the pressure of each cross section of a rotating detonation engine within a wide speed range and a large flight envelope across space, and can meet the signal construction requirements of rotating detonation engine control simulation for current control applications.

[0036] (2) High confidence and simple calculation: The construction of the pressure signal of the present invention is based on the aerodynamic thermodynamics theory of the rotating detonation engine, the engine working principle, etc., combined with the essential principles and working characteristics of the rotating detonation combustion chamber, thereby ensuring the high confidence of the calculated pressure signal value, and can calculate the pressure change at the corresponding position in each model update cycle according to the measuring point position as the dynamic pressure signal output of the measuring point; the pressure signal acquisition of the present invention is completely based on the model, and there is no need to perform numerical calculations such as CFD, which greatly reduces the calculation amount of the pressure signal generation, improves the real-time performance of the calculation, and reduces the difficulty of the calculation.

[0037] (3) Versatility and portability: The present invention is implemented based on component-level modeling and signal construction based on model calculation results. It is applicable to ramjet engines, turbojets, turbofans, turboshafts, turboprops, variable cycle engines, and combined cycle engines with rotating detonation combustion chambers. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 This is a structural diagram of a rotating detonation ramjet engine;

[0039] Figure 2 A flow chart is constructed for the pressure signal of the rotating detonation engine of the present invention;

[0040] Figure 3 This is a schematic diagram of the location of pressure measurement points in the rotating detonation combustion chamber;

[0041] Figure 4 The pressure diagram of the measuring point under signal generation condition 1; (a), (b), and (c) are the pressure P of the measuring point within a unit detonation wave cycle. d , the pressure of the measuring point P within the signal update cycle, the pressure signal of the measuring point of the combustion chamber inlet section P out ;

[0042] Figure 5 The pressure diagram of the measuring point under signal generation condition 2; (a), (b), and (c) are the pressure P of the measuring point within a unit detonation wave cycle. d , the pressure of the measuring point P within the signal update cycle, the pressure signal of the measuring point of the combustion chamber inlet section P out ;

[0043] Figure 6 The pressure diagram of the measuring point under signal generation condition 3; (a), (b), and (c) are the pressure P of the measuring point within a unit detonation wave cycle. d , the pressure of the measuring point P within the signal update cycle, the pressure signal of the measuring point of the combustion chamber inlet section P out . DETAILED DESCRIPTION

[0044] To address the inability of existing technologies to construct and feedback pressure measurement signals for the controlled object required for developing a rotating detonation engine control system, the present invention establishes aerodynamic thermodynamic models of various components based on principles of aerodynamic thermodynamics, aeroengine principles, and rotor dynamics. Based on flight conditions and engine control variable inputs, the system calculates the pressure parameter distribution of each cross section, specifically the two-dimensional distribution and dynamic variation of combustion chamber pressure. Furthermore, a signal simulation device is used to generate realistic sensor signals. Applicable simulation targets include aerospace engines that include a rotating detonation combustion chamber and are composed of multiple components that work together to complete a thermodynamic cycle, including but not limited to turbojets, turbofans, turboshafts, turboprops, variable cycle engines, ramjets, and combined cycle engines.

[0045] To facilitate public understanding, the technical solution of the present invention will be described in detail below using a rotating detonation ramjet engine as an example.

[0046] Figure 1 Schematic diagram of the rotating detonation ramjet engine structure; section 0 in the figure is the far front flow, section 1 is the inlet inlet, section 2 is the inlet outlet, section 3 is the isolation section outlet and the rotating detonation combustion chamber inlet, section 4 is the rotating detonation combustion chamber outlet, section 8 is the tail nozzle throat section, and section 9 is the tail nozzle outlet.

[0047] The pressure signal construction process of the rotating detonation ramjet engine is as follows: Figure 2 As shown, the specific steps include:

[0048] Step 1: Based on the rotating detonation engine thermodynamic model, the static pressure distribution parameters of the rotating detonation engine under given flight conditions and control variable input are obtained through iterative solution, including the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation wave head as the starting point;

[0049] The rotating detonation ramjet engine studied in this example primarily consists of four components: the inlet duct, the isolator, the rotating detonation combustion chamber, and the tail nozzle. Therefore, these four components require modeling. In particular, since this application example lacks a rotor component, rotor dynamics modeling is not provided. However, for rotating detonation engines (turbine-based engines) with rotor components, rotor dynamics modeling is still required to calculate the speed variation pattern.

[0050] The three major components, the inlet duct, the isolator, and the tail nozzle, were modeled using conventional component-level modeling methods, which will not be elaborated here. For the rotating detonation combustion chamber, a two-dimensional combustion chamber model was established based on the CJ theory. In practice, the combustion chamber model can also be constructed using existing HJ theory, ZND theory, and other methods, not limited to CJ theory.

[0051] (1) Calculation of detonation wave velocity v based on CJ theory d

[0052] v d =f(k CJ ,R CJ ,c p,3 ,c p,CJ ,T3,P3,v3,f3,H f ,η g ) (1)

[0053] Where k represents the specific heat ratio, R is the gas constant, and c p is the constant pressure specific heat capacity, T3 represents the static temperature at the combustion chamber inlet, P3 represents the static pressure at the combustion chamber inlet, v3 represents the flow velocity at the combustion chamber inlet, f3 represents the fuel consumption rate of the combustion chamber, H f Indicates the calorific value of fuel, η g represents combustion efficiency, subscript CJ represents parameters under CJ theory, and subscript 3 represents section 3, i.e., the inlet section of the detonation combustion chamber.

[0054] (2) Calculate the shock wave angle β for the calculation of parameters after the detonation wave expansion:

[0055] β=f(k CJ ,Ma d ) (2)

[0056] In the formula, Ma d Represents the Mach number of the airflow after passing through the blast wave.

[0057] (3) Calculate the detonation wave pressure distribution p in the annular combustion chamber based on the shock wave angle β and the detonation wave head height h bz (r):

[0058] p bz (r)=f(k CJ ,P CJ ,C,r,h,sinβ) (3)

[0059] Where, P CJ is the post-detonation pressure under CJ theory, calculated based on CJ theory and omitted for clarity. C is the combustion chamber circumference, r is the position relative to the detonation wave head on the circumference, r∈[0,B*C], and B is the intake blockage ratio. The static pressure in the non-blocked portion of the detonation chamber is equal to the static pressure at the detonation chamber inlet. Combined with the detonation wave pressure distribution, the overall pressure distribution p(r) of the rotating detonation chamber, starting from the detonation wave head and gradually decreasing to the throughflow pressure, can be obtained:

[0060]

[0061] The common working equations are set according to the engine's operating characteristics: Since there is no rotor component in this embodiment, no power balance common working equation is set. In combination with the operating characteristics of the four components of the rotating detonation ramjet engine, two common working equations for flow balance and static pressure balance are set at the combustion chamber inlet:

[0062] (1) According to the gas flow balance entering the combustion chamber, set

[0063]

[0064] Where, W3 is the actual gas flow rate at the inlet of the detonation combustion chamber, which can be calculated based on the blockage ratio; W d is the flow rate after the detonation wave.

[0065] (2) According to the static pressure balance of the combustion chamber inlet, set

[0066]

[0067] The detonation wave head height h and the blockage ratio B are selected as guess values ​​for iterative calculation.

[0068] Based on the above steps, a complete rotating detonation ramjet engine thermodynamic model including the air inlet, isolation section, rotating detonation combustion chamber, and tail nozzle components can be established.

[0069] By iteratively solving the thermodynamic model, the specific value of the pressure signal to be constructed can be obtained, which is then used by the signal generator to generate the signal. The iterative solution of the above thermodynamic model can be achieved by using various existing iterative algorithms such as the Newton method and the quasi-Newton method. The following is a brief explanation using the Newton-Raphson method as an example:

[0070] (1) Initialize the established rotating detonation ramjet thermodynamic model and the initial guess values ​​of the detonation wave head h and the blockage ratio B;

[0071] (2) Given the current flight conditions and control variable input; in this embodiment, the flight conditions include altitude H and Mach number Ma, and the control variable input is the fuel flow rate m fb ;

[0072] (3) Use the Newton-Raphson method to iteratively solve:

[0073] X=XJ -1 E (7)

[0074] Where X = [h B] T , E=[e1 e2] T , J is the Jacobian matrix of E with respect to X;

[0075] Iterate the system of co-working equations until the residual e of each co-working equation is less than 10 -6 , thereby obtaining the solution to the equation group, namely the accurate detonation wave height h and blockage ratio B. By substituting the detonation wave head height h and blockage ratio B into the established thermodynamic model, the temperature and pressure values ​​of each cross section and the static pressure distribution at the inlet of the rotating detonation combustion chamber are calculated.

[0076] (4) Rotor dynamics calculation: Since the object of this embodiment is a ramjet engine without a rotor component, rotor dynamics calculation is not performed. Instead, it is directly determined whether the dynamic calculation process is completed.

[0077] Step 2: Calculate the change of the pressure value p at any pressure measuring point in the rotating detonation combustion chamber over time t within the signal update period Δt required for one calculation of the thermodynamic model:

[0078] Assuming that continuous dynamic, multi-cycle signal construction is performed, the signal update period is consistent with the control period of a typical engine control system to meet the signal acquisition and feedback requirements of the control system. In this embodiment, the update period Δt is set to 20ms. In particular, it is pointed out that the signal update period referred to here is the sampling step length for updating the various thermal parameters of the engine after the model is calculated once, rather than the update period of the signal generation system's external output signal (the update period of the signal generation system's external output signal can be as short as microseconds or even nanoseconds, depending on the actual signal construction needs and the signal generation capability of the signal generation system itself). It is worth noting that when the corresponding control system control period is selected as a smaller value, the signal update period of the present invention can be synchronously reduced to meet the actual control system's signal acquisition and feedback requirements.

[0079] First calculate the current position L of the detonation wave head:

[0080]

[0081] L=mod(s,C) (9)

[0082] Where s represents the distance traveled by the detonation wave, Δt represents the signal update period, N represents the number of signal update periods calculated up to the current time, L0 represents the initial wave head position, that is, the detonation wave initiation position, and mod represents the remainder operation. Note that the wave head positions L and L0 here are the circumferential distances of the detonation wave head relative to the measuring point along the propagation direction of the detonation wave.

[0083] Assuming that the pressure distribution p(r) of the rotating detonation combustion chamber with the detonation wave head as the starting point is obtained after iterative solution, the initial pressure value p(L) of the pressure measuring point is obtained according to the pressure distribution; at the same time, the detonation wave frequency f is calculated according to the detonation wave velocity re , detonation wave period t d :

[0084]

[0085] Therefore, in each detonation wave period t d During this time, the pressure value at the pressure measuring point is

[0086]

[0087] Therefore, within one signal update cycle, there are l complete detonation wave cycles

[0088]

[0089] In the formula, ceil means rounding down.

[0090] Then, within a signal update cycle, the pressure value curve of the pressure measuring point is:

[0091]

[0092] Step 3: Construct the pressure signal of each cross section within the signal update period Δt:

[0093] Assuming that the signal generating system outputs the pressure signal at a specific sampling rate S, the signal sampling period Δt s for

[0094]

[0095] Special note on the difference between the signal update period and the signal sampling period: the signal update period refers to the time to maintain the pressure distribution law unchanged, and the signal sampling period refers to the period in which the signal generating system outputs the pressure once at the sampling rate S.

[0096] Therefore, for the cross-sectional pressure of the non-combustion chamber, the pressure signal can be considered to be consistent with the model output pressure; the signal generation system constructs the pressure signal within the signal update period according to formula (16):

[0097] p out =p(qΔt s )@p z ,0≤qΔt s <Δt,q=0,1,2,... (16)

[0098] Where z represents the section number.

[0099] For any cross section in the rotating detonation combustion chamber, the signal generation system constructs the pressure signal within the signal update period according to formula (17):

[0100]

[0101] In order to demonstrate the effectiveness of the technical solution of the present invention, pressure signals are continuously generated under different engine sizes, flight conditions, and control variable inputs. The following results take the pressure signal of a rotating detonation combustion chamber as an example, and the signal update period is 20ms. Without loss of generality, it is assumed that the combustion chamber pressure measurement point is located at the combustion chamber inlet at 0° of the circumference, as shown in Figure 2. Figure 3 Table 1 shows the different signal generation conditions. Table 2 shows the characteristic parameters of the detonation wave under different signal conditions.

[0102] Table 1 Signal generation conditions

[0103]

[0104] Table 2 Detonation wave characteristic parameters

[0105]

[0106] Figures 4-6 The pressure P of the measuring point within a unit detonation wave cycle under the above three signal generation conditions are given respectively. d , the pressure of the measuring point P within the signal update cycle, the pressure signal of the measuring point of the combustion chamber inlet section P out Among them, the pressure P d ,P,P out As can be seen from the figure, the present invention can effectively simulate the pressure of the rotating detonation combustion chamber and has the ability to continuously output the pressure signal of each cross section of the rotating detonation ramjet engine for controlling the simulation.

Claims

1. A method for constructing a pressure signal of a rotating detonation engine, characterized in that: The following steps are involved: Step 1: Based on the rotating detonation engine thermodynamic model, the static pressure distribution parameters of the rotating detonation engine under given flight conditions and control variable input are obtained through iterative solution, including the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation wave head as the starting point; Step 2: Calculate the change of the pressure value p at any pressure measuring point in the rotating detonation combustion chamber over time t within the signal update period Δt required for one thermodynamic model calculation according to the following formula: Among them, the wave head position L and the initial wave head position L0 are respectively the current and initial circumferential distances of the detonation wave head relative to the measuring point along the propagation direction of the detonation wave, v d is the detonation wave velocity, C is the circumference of the rotating detonation combustion chamber, t d is the detonation wave period, l is the number of complete detonation wave periods within one signal update period Δt; Step 3: Construct the pressure signal of each cross section within the signal update period Δt: For each cross section of the non-rotating detonation combustion chamber, the pressure signal is constructed according to the following formula: p out =p(qΔt s )@p z ,0≤qΔt s <Δt,q=0,1,2,... For each cross section in the rotating detonation combustion chamber, the pressure signal is constructed according to the following formula: Where Δt s is the signal sampling period, and z represents the section number.

2. The method for constructing a pressure signal of a rotating detonation engine according to claim 1, wherein: Taking the detonation wave head height h and the blockage ratio B as guess values, the pressure distribution p(r) with the detonation wave head as the starting point in the rotating detonation combustion chamber under given flight conditions and control variable input is obtained through iterative solution.

3. A rotating detonation engine pressure signal construction device, characterized in that: include: Thermodynamic model calculation module is used to obtain the static pressure distribution parameters of the rotating detonation engine under given flight conditions and control variable inputs through iterative solution based on the rotating detonation engine thermodynamic model, including the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation wave head as the starting point; The rotating detonation combustion chamber pressure calculation module is used to calculate the change of the pressure value p of any pressure measuring point in the rotating detonation combustion chamber with time t within the signal update period Δt required for one calculation of the thermodynamic model according to the following formula: Among them, the wave head position L and the initial wave head position L0 are respectively the current and initial circumferential distances of the detonation wave head relative to the measuring point along the propagation direction of the detonation wave, v d is the detonation wave velocity, C is the circumference of the rotating detonation combustion chamber, t d is the detonation wave period, l is the number of complete detonation wave periods within one signal update period Δt; The pressure signal construction module is used to construct the pressure signal of each cross section within the signal update period Δt: For each cross section of the non-rotating detonation combustion chamber, the pressure signal is constructed according to the following formula: p out =p(qΔt s )@p z ,0≤qΔt s <Δt,q=0,1,2,... For each cross section in the rotating detonation combustion chamber, the pressure signal is constructed according to the following formula: Where Δt s is the signal sampling period, and z represents the section number.

4. The rotating detonation engine pressure signal construction device according to claim 3, characterized in that: The thermodynamic model of the rotating detonation engine takes the detonation head height h and the blockage ratio B as guess values, and obtains the pressure distribution p(r) in the rotating detonation combustion chamber with the detonation head as the starting point under given flight conditions and control variable input through an iterative solution.

Images