Method for low-frequency estimation of a flow rate of recycled exhaust gases at the intake of an internal combustion engine

Abstract

According to this method of calculating the exhaust gas recirculation flow rate at the intake of an internal combustion engine: We measure the minimum Πmin and maximum Πmax values ​​of an expansion ratio Π, We measure the average value of the expansion ratio Πmoy, We calculate the Barré de Saint-Venant function applied to the average value of the expansion ratio, We calculate the second derivative of the Barré de Saint-Venant function, and We calculate the exhaust gas recirculation flow rate. Figure for the abstract: Figure 2

Description

Title of the invention: Method for low-frequency estimation of a flow rate of recycled exhaust gases at the intake of an internal combustion engine Technical field

[0001] The present invention relates to internal combustion engines of motor vehicles of the spark-ignition type (petrol) or compression-ignition type (diesel), and relates in particular to a circuit for partial recirculation of exhaust gases at the exhaust of such an engine, called an EGR circuit, for "Exhaust Gas Recirculation" in Anglo-Saxon terms or "Recirculation of Exhaust Gases" in French.

[0002] It relates more particularly to a method and a system for low-frequency calculation of the flow rate of recycled exhaust gases in such an EGR circuit, also known as EGR flow rate. Previous techniques

[0003] Conventionally, exhaust gas recirculation is a process for limiting the production of nitrogen oxides (NOx) in the combustion gases of a diesel engine, or for reducing the fuel consumption of gasoline engines. This EGR process consists of taking gases from the exhaust and sending them to the intake, for example downstream of an engine air flow control valve. On diesel engines, the action on the formation of NOx can be up to 50% overall reduction and is interpreted by a reduction in the temperatures of the combustion gases due to dilution and a consequent slowing down of the kinetics of formation of the pollutant.In gasoline engines, the reduction in fuel consumption is due to a lower sensitivity of the engine to knocking which allows an increase in ignition advance and a reduction in engine pumping losses, the supply of recycled gases allowing an increase in the pressure in the engine's intake manifold for the same air flow rate required to produce torque.

[0004] Generally the quantity of recycled gases is estimated using a differential pressure sensor at a flow control valve in the EGR circuit, called the EGR valve. This sensor makes it possible to estimate the flow rate that has just passed through the valve, called the EGR flow rate. Good control of the EGR flow rate is essential because it determines the EGR rate, that is to say, the ratio between the flow rate of exhaust gases at the intake and the total flow rate of gases admitted into the engine (air and recycled gases), which influences the level of NOx pollutants or fuel consumption. A Poor control of the EGR flow rate can lead to unwanted effects that could compromise vehicle operation. For example, an overestimated EGR flow rate can lead to higher NOx emissions in diesel engines or knocking in spark-ignition engines. This can have drastic consequences in terms of reduced reliability or non-compliance with regulatory pollution control standards.

[0005] The difficulty in estimating the EGR flow rate is due to the fact that a flow sensor is not used at the EGR valve itself, which would be expensive and inaccurate. Instead of such a flow meter, a differential pressure sensor is used which measures the difference between the pressure downstream and upstream of the valve. The flow rate through the valve is estimated based on a model which uses the measured differential pressure and the valve opening level. Even though this model can provide very accurate results, the differential pressure information invariably has a disturbed and very dynamic character because it presents high-frequency oscillations, which are difficult to follow using electronic control units whose computing capabilities are limited.Therefore, it is not practical to make calculations with the information from the differential pressure sensor captured at high frequency, i.e. with a periodicity of 1 millisecond.

[0006] The invention consists of developing a method for calculating at low frequency the estimation of an EGR flow rate, i.e. with a periodicity of approximately 20 milliseconds, while maintaining a level of precision equivalent to a high frequency calculation.

[0007] There are, in the state of the art, different methods for estimating an EGR flow rate.

[0008] The first method consists of producing a model of the total flow rate of the engine intake gases, on the one hand, and determining the air flow rate alone, on the other hand. For this, the total flow rate can be determined by a filling model, from the pressure and temperature in the engine intake manifold, and a filling efficiency value which is itself a function of a set of parameters including at least the speed, the pressure in the intake manifold, and possibly other parameters such as the timing position of the intake and exhaust valves. The fresh air flow rate is determined by a flow meter. The EGR flow rate is then equal to the total flow rate of the intake gases from which the fresh air flow rate is subtracted. This approach is described in particular in document US 2016 / 0069285, or in document FR 2 938 016.

[0009] These approaches are simple, but have several disadvantages.

[0010] For example, for engines that have two EGR circuits, including a low-pressure EGR circuit and a high-pressure EGR circuit, they do not allow a distinction to be made between low-pressure and high-pressure EGR rates, only the EGR rate global that can be estimated using this method.

[0011] Then, the estimation is based on knowledge of the total flow rate of the intake gases, which is classically a volumetric pump model, and the fresh air flow rate, which is directly measured by sensor. The available accuracy of this information is approximately + / -5% for the sensor measurement and approximately + / -3% for the total flow rate, which can generate inconsistent results, particularly on low EGR flow values ​​and transient values.

[0012] A second method consists of using a valve cross-section model, which generally uses the Barré de Saint-Venant equation or "Throttle Equation" in Anglo-Saxon equivalent. The equation gives very precise results, if the position of the valve and the pressure difference that exists at its terminals are controlled.

[0013] On the other hand, it is extremely sensitive to the pressure values ​​at the valve terminals and the angular position of the valve, when the pressure difference is low or when the valve is close to closing.

[0014] In order to improve its accuracy, it is common to add corrective terms taking into account the dispersions, or even to prefer the first method cited above on the problematic operating points. This effective section method is used for example in document EP 1 416 138 with the addition of a correction map taking into account the upstream pressure, the downstream pressure and the effective section of the valve, or in document EP 3 434 888, with the addition of a dynamic correction map depending on the positions of the valve.

[0015] The cross-section method has the disadvantage of being highly non-linear in very closed or very open positions of the valve. On the other hand, pressure variations are very rapid dynamic phenomena, of the order of a millisecond, which are generally filtered for use in the management of the EGR system, the order of magnitude of the time constant of which is around 10 to 20 milliseconds. However, given the highly non-linear nature of the equation in certain positions, significant errors are introduced by using filtered pressure values. Statement of the invention

[0016] In view of the above, the aim of the invention is to improve the accuracy of the estimation of an EGR flow rate, in particular for use in the management of the EGR system at low frequency.

[0017] The subject of the invention is a method for calculating the flow rate of recirculation of exhaust gases at the intake of an internal combustion engine allowing the control of said engine.

[0018] According to the method:

[0019] - The minimum nmin and maximum Ilmax values ​​of an expansion ratio n are measured, defined as the ratio between the pressure measured upstream and the pressure measured downstream of the exhaust gas recirculation valve, - The average value of the expansion ratio nmoy is measured, - The Barré de Saint-Venant function applied to the average value of the expansion ratio nmoy is calculated, - We calculate the second derivative of the Barré de Saint-Venant function applied to the average value of the expansion rate nmoy, - The exhaust gas recirculation flow rate is calculated.

[0020] The method for calculating the EGR flow rate thus makes it possible to calculate the EGR flow rate precisely using the Barré de Saint-Venant function and the second derivative of this function, applied to the average value of the expansion rate nmoy.

[0021] Advantageously, the second derivative of the Barré de Saint-Venant function applied to the average value of the expansion rate Ilmoy is calculated using three terms obtained by applying the Barré de Saint-Venant function to the envelope values ​​nmin and nmax and to the average value nmoy.

[0022] Advantageously, the exhaust gas recirculation flow rate is calculated by summing a first term directly proportional to the Barré de Saint-Venant function applied to the average value of the expansion rate nmoy and a second term directly proportional to the second derivative of the Barré de Saint-Venant function applied to the average value of the expansion rate Ilmoy.

[0023] For example, the minimum nmin and maximum nmax values ​​can be measured at high frequency.

[0024] For example, the minimum nmin and maximum nmax values ​​can be measured at low frequency.

[0025] For example, the average value nmoy can be measured at low frequency.

[0026] Advantageously, the internal combustion engine is equipped with at least one high-pressure partial exhaust gas recirculation circuit and at least one low-pressure partial exhaust gas recirculation circuit.

[0027] The invention also relates to a system for calculating the flow rate of recirculation of exhaust gases at the intake of an internal combustion engine, allowing the control of said engine.

[0028] The flow rate calculation system comprises means for measuring the minimum Ilmin and maximum nmax values ​​of an expansion ratio n, defined as the ratio between the pressure measured upstream and the pressure measured downstream of the exhaust gas recirculation valve, means for measuring the average value of the expansion ratio nmoy, means for calculating the Barré de Saint-Venant function applied to the average value of the expansion ratio Ilmoy, means for calculating the derivative second of the Barré de Saint-Venant function applied to the average value of the expansion rate nmoy, and means of calculating the exhaust gas recirculation flow rate. Brief description of the drawings

[0029] Other aims, characteristics and advantages of the invention will appear on reading the following description, given solely by way of non-limiting example, and made with reference to the appended drawings in which:

[0030] [Fig-1] illustrates, schematically, the structure of a combustion engine internal of a motor vehicle equipped with a high-pressure partial exhaust gas recirculation circuit, a low-pressure partial exhaust gas recirculation circuit and an EGR flow rate calculation system according to the invention;

[0031] [Fig.2] illustrates a flow diagram of the EGR flow calculation method, implemented implemented by the calculation system, according to an implementation mode of the invention;

[0032] [Fig.3] illustrates schematically the information accessible at the level of the EGR valves.

[0033] [Fig.4] is a flowchart used for the classic estimation of EGR flow.

[0034] [Fig.5] is a flowchart used in the invention for estimating the EGR flow rate. Detailed description of at least one embodiment

[0035] In the example illustrated in [Fig.l], the internal combustion engine 10 comprises, in a non-limiting manner, four cylinders 12 in line, a fresh air intake manifold 14, an exhaust manifold 16, a turbocharging system 18, a high-pressure partial exhaust gas recirculation circuit (“high-pressure EGR circuit”) and a low-pressure partial exhaust gas recirculation circuit (“low-pressure EGR circuit”).

[0036] The cylinders 12 are supplied with air via the intake manifold 14, or intake distributor, itself supplied by a pipe 20 provided with an air filter 22 and the compressor 18b of the turbocharger 18 of the engine 10.

[0037] The turbocharger 18 essentially comprises a turbine 18a driven by the exhaust gases and a compressor 18b mounted on the same shaft as the turbine 18a and providing compression of the air distributed by the air filter 22, with the aim of increasing the quantity (mass flow rate) of air admitted into the cylinders 12 of the engine 10.

[0038] A heat exchanger 24 is placed after the outlet of the compressor 18b equipping the supply line 14a of the intake manifold 14 with fresh air.

[0039] The internal combustion engine 10 comprises an intake circuit Ca and an exhaust circuit Ce.

[0040] The intake circuit Ca comprises, from upstream to downstream in the direction of circulation of the air:

[0041] - the air filter 22 or air box;

[0042] - a flow meter 26 arranged in the intake pipe 20 downstream of the air filter 22 to measure the actual value of the air flow entering the engine 10;

[0043] - an air intake valve 28;

[0044] - the compressor 18b of the turbocharger 18;

[0045] - a throttle body 30 or a gas intake valve in the engine;

[0046] - a heat exchanger 32 for cooling the intake gases corresponding to a mixture of fresh air and exhaust gases recirculated after their compression in the compressor 18b; and

[0047] - the intake manifold 14.

[0048] The compressor is associated with a bypass circuit with an inlet discharge valve 55 which opens in the event of sudden closure of the throttle body 30, to prevent the compressed air, located between the compressor 18b and the throttle body 30, from passing through the compressor 18b and damaging it, when, for example, the driver of the vehicle suddenly lifts his foot off the accelerator pedal.

[0049] The exhaust circuit Ce comprises, from upstream to downstream in the direction of circulation of the burnt gases:

[0050] - the exhaust manifold 16;

[0051] - the turbine 18a of the turbocharger 18; and

[0052] - a system 40 for depolluting the combustion gases of the engine.

[0053] As regards the exhaust manifold 16, the latter recovers the exhaust gases resulting from the combustion and evacuates them to the outside, via a gas exhaust duct 34 opening onto the turbine 18a of the turbocharger 18 and via an exhaust line 36 mounted downstream of said turbine 18a.

[0054] The engine 10 here comprises two circuits for partial recirculation of the exhaust gases at the intake, called “EGR” circuits (“exhaust gas recirculation” in English terms), namely a high-pressure EGR circuit 15 and a low-pressure EGR circuit 38.

[0055] The circuit 38, here a low-pressure exhaust gas recirculation circuit, called “EGR BP”, originates at a point on the exhaust line 36, downstream of said turbine 18a, and in particular downstream of the gas depollution system 40, and returns the exhaust gases to a point on the fresh air supply pipe 20, upstream of the compressor 18b of the turbocharger 18, in particular downstream of the flow meter 26. The flow meter 26 only measures the flow of fresh air alone.

[0056] As illustrated, this recirculation circuit 38 comprises, in the direction of circulation of the recycled gases, a cooler 38a, a filter 38b, and a valve 38c intended to regulate the flow of low-pressure exhaust gases. Valve 38c is arranged downstream of cooler 38a and upstream of compressor 18b.

[0057] It will be noted that the air intake valve 28 can also be used to force the circulation of a flow of low-pressure exhaust gases in the BP EGR circuit in the case where the vacuum between the exhaust circuit and the intake circuit is insufficient. In this case, closing the valve 28 makes it possible to create a vacuum downstream thereof, capable of sucking gases from the BP EGR circuit.

[0058] The high pressure EGR circuit 15, called “HP EGR”, originates at a point in the exhaust circuit Ce, upstream of said turbine 18a and returns the exhaust gases to a point in the intake circuit Ca, downstream of the heat exchanger 32.

[0059] As illustrated, this recirculation circuit 15 comprises a valve 15a configured to regulate the flow rate of high pressure exhaust gases.

[0060] The engine is associated with a fuel circuit comprising, for example, fuel injectors (not referenced) injecting gasoline directly into each cylinder from a fuel tank (not shown).

[0061] The engine comprises an electronic control unit 70 configured to control the various elements of the internal combustion engine from data collected by sensors at different locations in the engine.

[0062] The electronic control unit 70 comprises a calculation module 72, a measurement module 73 and a control module 74.

[0063] As illustrated in [Fig.3], several pieces of information are accessible at the EGR valves 15a and 38c.

[0064] For example, the gas pressure downstream of the EGR valve called Pavai is measured using a relative pressure sensor. The differential pressure AP is measured using a differential pressure sensor. The opening angle of the Oegr valve is measured using a position sensor placed on the electric motor that drives the valve.

[0065] It is generally accepted that the flow rate through the Qegr valve can be described according to the following equation: 100661 = V e e „)BSv[^ ) (I) A7 r upstream \ r mw! /

[0067] With:

[0068] Qe^n mass flow rate, in kg / s

[0069] Se, the effective section of the valve, in mm2

[0070] BSV, the Barré de Saint-Venant function calculated according to equation (2), below and expressed in kgxK°5 sxPaxmrn?

[0071] Tamont, the temperature upstream of the valve, in K

[0072] Pamont, the pressure upstream of the valve, expressed in Pa.

[0073] The Barré de Saint-Venant function is expressed when the flow is non-sonic, that is, for all the calculations that interest us, by the following expression:

[0074] bsv ( n ) = (2)

[0075] With:

[0076] n, the relaxation rate is the ratio between Pamont and Pavai, dimensionless

[0077] y, the adiabatic index of gases, dimensionless

[0078] r, the ratio between the ideal gas constant divided by the molar mass of the gas in question, expressed in J-kg'-Kr1

[0079] By multiplying equation (1) by II on each side, we obtain the following equation: 100801 6,. / 7= = S,(fts,-)B5V2()<3) s 2 aval \ 2 avili /

[0081] With:

[0082] BSV f \ = LLBSV ( ) <4)

[0083] As previously indicated, the pressure information is extremely dynamic due to the acyclism of the behavior of an internal combustion engine. This acyclism creates high volatilities of the differential pressure signal and thereby implies strong variations in the expansion rate n.

[0084] In order to limit the noise on the total flow rate, it is necessary to filter this information. However, filtering the pressure information has a direct impact on the accuracy of the flow rate recalculated by equation (1) or (3). Indeed, these equations use the Barré de Saint-Venant (BSV) function which is non-linear.

[0085] It is therefore not equivalent to calculate the average of the function over all the pressures encountered, which corresponds to a true average of the flow rate, than to calculate the average of the pressures and apply the Barré de Saint-Venant function to it. The difference between these two calculations can be quite significant, sometimes greater than 20% according to certain calculations, which risks leading to an overestimation of the flow rate interpreted by the rest of the engine control.

[0086] It is thus proposed to get as close as possible to the real value of the EGR flow rate while carrying out all the calculations at a lower frequency, i.e. every 10 milliseconds for example.

[0087] To achieve this, the variations in the relaxation rate II presented above are assimilated to a sinusoid which is added to a mean signal varying as a function of time, the sinusoid having an unknown amplitude A and an unknown pulsation co as described in the following equation:

[0088]

[0089]

[0090]

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[0095]

[0096]

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[0100]

[0101]

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[0108] n(t) = (t) + Asin(early)(5) To calculate the EGR flow rate from equation (3), it is necessary to correctly estimate the following BSV-, ( ü(t) ) function: BSV2 (n(t)) = BSV2 (n»wy+Asin( early)) (6) For this purpose, we use Taylor developments in their approximate forms, namely: Which we apply here with a = n^y and h = Asin( wt) ■ We then obtain the following second-order approximation: BSV2 (n(t) )~BSV2 + Asin(cot)BSV2 '(rUy) + ^^bsV2 "(ILnoy) (8) The classical estimation method, which consists, as explained above, first of filtering or averaging the expansion rate and then applying the Barré de Saint-Venant function, would lead to the result in the following form: 5W(n(t))~BW(iM (9) Which corresponds only to the first term of approximation (8). The present invention corresponds to a method which consists of using the following terms of the Taylor development in order to supplement the available information without excessive increase in the computational load. To calculate the average flow rate over a given period T, we integrate equation (3) over this period: '■ '■ y * If we consider that the calculation period is quite short, for example of the order of a pressure oscillation, we can admit that the position of the valve and T upstream is a constant. We can also admit that the pressure Pavai remains constant and that most of the oscillations occur on Pamont- We can ask: Upstream And rewrite equation (10) as: Using equations (12) and (8) we then obtain: + AsÙl(œt)BSV 2 '(H^y) + ^^1BSV2 "(aWy) dt (13)

[0109] Which can be rewritten as:

[0110] = (n^.) +f ABSV2 "(n^OfoSin^w / )^ (14)

[0111] Over a period T large compared to the pulsation w, we obtain

[0112] f7 . ( _netf7 . 2 / , . . / _ J 0 sin ( œt ) dt^QJ 0 sm ( cot )dt^ 1 / 2

[0113] By choosing a filter with an adequate time constant we then obtain: 101141 a bsv2 (¼)+^bsv2 "(ru,,) <*»

[0115] To calculate the value of the average EGR flow rate, it is necessary to calculate the second derivative of the BSV2 function.

[0116] Using a numerical scheme of the centered difference type, we can then evaluate the second derivative of the BSV2 function as follows:

[0117] "Zjj .455^2(^ 2 V avg} , 2 s 1 Lmix 1 LminJ bsv2 ( n,„aj ) -2 bsv2 ( ru,, ) +bsv2( nmin )

[0118] We then obtain the following expression for the average EGR flow rate: [01191 Qlgr„ = a bsv2 ( il”, ) + ^ (BSv2(n,„„) -2 bsv2( n,„„) + bsv2(ru) ) (17)

[0120] This amounts to considering the classic estimate a BSV2 ( n7MOV ) to which we add a correction linked to the maximum flow rate and minimum flow rate estimates. We note that the correction is not very demanding in computing resources because it consists of carrying out weightings for the following three pieces of information only: for the maximum flow rate, / 2 for the average flow rate and for the minimum flow rate.

[0121] It can also be noted that all odd derivatives of the sine functions result in a zero flow rate average because r7 . oz+i / -, , , Expression (17) is thus an ap- J 0 sin \ <ùt)dt~^q, pour tout k naturel third-order approximation, which ensures acceptable accuracy even when the differences between nm(„ and become significant.

[0122] Figure 4 illustrates the procedure used in the conventional estimation, in order to better appreciate the contribution of the invention, the procedure of which is presented in Figure 5. Conventionally, the information on the expansion rate is captured at high frequency, i.e. every millisecond. The information is filtered by taking the average nwoy at low frequency, i.e. every 10 milliseconds. The BSV2 function is then applied to the average value and multiplied by the coefficient a to obtain the estimate of the EGR flow rate.

[0123] With reference to Figure 5, the information is captured at high frequency, i.e. all milliseconds. We then retrieve the maximum, minimum n / fM„ and average nOTOy values ​​at low frequency, i.e. every 10 milliseconds. We then apply the BSV2 function to these three maximum II„My, minimum ripl / „ and average nraOy values ​​with a weighting of % for the maximum value, / 2 for the average value and % for the minimum value and we multiply by the coefficient a to obtain the estimate of the EGR flow rate.

[0124] The flowchart shown in [Fig.2] illustrates the method of calculating the EGR flow rate, implemented by the calculation system 70.

[0125] During a first step 61, the maximum and minimum values ​​of the expansion rate are measured, then, during the following step 62, the average value of the expansion rate is measured.

[0126] The method 60 further comprises a step 63 of calculating the BSV2 function applied to the average expansion rate, according to equations (2) and (4), and a step 64 of calculating the second derivative of the BSV2 function applied to the average expansion rate, according to equation (16).

[0127] The method 60 continues with a step 65 of calculating the average flow rate, according to equation (17).

[0128] The method 60 finally comprises a step 66 of engine control, through an EGR flow rate instruction Q.

[0129] Thus the invention proposes a method for estimating the EGR flow rate used for engine control, which requires little computing resources and which can use the pressure values ​​filtered at low frequency, with sufficient precision of the results.

Claims

Claims

1. Method for calculating the exhaust gas recirculation flow rate at the intake of an internal combustion engine (10) allowing the control of said engine (10), characterized in that: a. Minimum (Elmin) and maximum (Ilmax) values ​​of an expansion ratio n are measured, defined as the ratio between a pressure measured upstream and a pressure measured downstream of an exhaust gas recirculation valve (15a and / or 38c), b. An average value of the expansion ratio (Ilmoy) is measured, c. A Barré de Saint-Venant function is calculated applied to the average measured value of the expansion ratio (Ilmoy), d. A second derivative of the Barré de Saint-Venant function is calculated applied to the average measured value of the expansion ratio (Ilmoy), e. The exhaust gas recirculation flow rate (Q) is calculated from the calculated Barré de Saint-Venant function and the second derivative of the Barré de Saint-Venant function.

2. The method of claim 1, wherein the second derivative of the Barré de Saint-Venant function applied to the measured mean value of the expansion rate (Ilmoy) is calculated using three terms obtained by applying the Barré de Saint-Venant function to the envelope values ​​(Ilmin) and (Ilmax) and to the mean value (Ilmoy).

3. A method according to claim 1, wherein the exhaust gas recirculation flow rate is calculated by summing a first term directly proportional to the Barré de Saint-Venant function applied to the measured mean value of the expansion ratio (Ilmoy) and a second term directly proportional to the second derivative of the Barré de Saint-Venant function applied to the measured mean value of the expansion ratio (Ilmoy).

4. Method according to any one of claims 1, 2 or 3, in which the minimum (Ilmin) and maximum (Ilmax) values ​​are measured at high frequency.

5. A method according to any one of claims 1, 2 or 3, in

6.

7.

8. in which the minimum (Ilmin) and maximum (Ilmax) values ​​are measured at low frequency. Method according to any one of claims 1, 2 or 3, in which the average value (Ilmoy) is measured at low frequency. Method according to any one of the preceding claims, in which the internal combustion engine (10) is equipped with at least one high-pressure exhaust gas recirculation circuit (15) and at least one low-pressure exhaust gas recirculation circuit (38). System for calculating the flow rate of recirculation of exhaust gases at the intake of an internal combustion engine (10) allowing the control of said engine (10), characterized in that it comprises: a. Means for measuring the minimum (Ilmin) and maximum (Ilmax) values ​​of an expansion ratio II, defined as the ratio between a pressure measured upstream and a pressure measured downstream of an exhaust gas recirculation valve (15a and / or 38c), b. Means of measuring an average value of the relaxation rate (Ilmoy), c. Means of calculating a Barré de Saint-Venant function applied to the measured average value of the expansion rate (Ilmoy), d. Means of calculating a second derivative of the Barré de Saint-Venant function applied to the measured mean value of the expansion rate (Ilmoy), e. Means of calculating the exhaust gas recirculation flow rate (Q) from the calculated Barré de Saint-Venant function and the second derivative of the Barré de Saint-Venant function.

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