Mathematical modelling and design of isolation diaphragms for pressure gauges

The paper discusses some questions of equilibrium, stability and shape optimization of circular membranes with arbitrary profile along the meridian. Such membranes can be used as a flexible barrier to isolate the media on one side of a diaphragm from media on the other side, whilst allowing the transmission of pressure with minimal loss. The main goal of the analysis is to propose techniques for designing such shape of corrugated membranes that satisfy the requirements of so call “slow growing linear or almost linear” behavior of the pressure-volume diagram. The applied Kirchhoff-Love theory for non-shallow shells was used as a base for deriving nonlinear as well as linearized boundary value problems of equilibrium. A genetic algorithm was used to evaluate and optimize the characteristics of corrugated membranes. The presented analysis of the membrane with the maximum linear stroke showed its efficiency and reliability. At the same time, special attention should be paid to the basic parametrization used in modelling, since the mathematically optimal solution is not always suitable from the point of view of membrane production technology.


Introduction
Sensors with the open membrane, including high-temperature sensors, are often required for pressure measurements in the food industry, in the manufacture of plastic articles, building components, for measuring the pressure and the level of liquids. Profiled membranes (also known as corrugated membranes, micro-structured membranes, patterned membranes, membranes with designed topography or notched membranes) are gaining increasing academic and industrial attention and recognition as a viable alternative to flat membranes.
A large number of corrugated membranes are used as elastic elements in the devices of precision mechanics [1,2]. The understanding of deflection, stress and strain of thin diaphragms with clamped edges under various loads is of great importance for designing and fabricating sensors and actuators [3]. A feasible approach to reduce the residual stress is to incorporate a corrugated diaphragm in the microphone structure. This appears to be a more convenient way to increase the mechanical sensitivity of the condenser microphones than the approach of changing the deposition process during the fabrication [4]. The corrugated composite bipolar plate can improve the performance of the vanadium redox flow battery by reducing pumping and ohmic energy losses [5]. The feasible method to compensate for the high-order effect during bunch length compression, thereby enhancing the peak Dynamic Systems and Computer Science: Theory and Applications (DYSC) 2020 Journal of Physics: Conference Series 1847 (2021) 012038 IOP Publishing doi:10.1088/1742-6596/1847/1/012038 2 current of a high-repetition-rate X-ray free-electron laser source is to insert the corrugated structure downstream of the high-order harmonic cavities. This structure functions as a passive linearizer and enhances the longitudinal profile of the electron beam [6]. The profiled ion exchange membranes have shown to significantly improve the performance of reverse electrodialysis by eliminating the spacer shadow effect and by inducing hydrodynamic changes, leading to ion transport rate enhancement [7]. In all the mentioned cases problem of the effective choose of the profile or corrugation is of great importance.
Shape optimization problems are usually solved numerically, by using iterative methods [8]. That is, one starts with an initial guess for a shape, and then gradually evolves it, until it morphs into the optimal shape. Another strategy in the optimization process, based on discretization and linearization techniques, is introduced by reliable finite element formulation and used in [9] and [10] for the membrane form-finding. A survey of shape parametrization techniques for the effective high fidelity shape optimization is presented in [11].
The next alternative in dealing with the shape optimization problems is the genetic algorithm (GA), developed by Holland in 1975 [12], a programming technique that mimics biological evolution as a problem-solving strategy. In [13] it was used to optimize membrane separation modules, while in [14] it was applied to problems of designing axisymmetric shells of minimal weight. A GA optimizer and evolutionary search algorithms have been applied in [15] in order to solve the optimal design of mufflers. A shape optimization system for fluid dynamic problems, which used the genetic code, was presented in [16] and applied to the optimization of bidimensional aerodynamic profiles with geometric constraints.
The present paper starts from the description of the used model of the corrugated circular shell. To describe the large elastic strains of the membrane under hydrostatic pressure two-dimensional nonlinear equations [17,18], based on the Kirchhoff's hypotheses, were used.
Some aspects of the numerical analysis of the equilibrium and stability of such shells are discussed in [19]. The modified genetic algorithm is used to design corrugation providing a sufficiently large value of the linear section length of the membrane loading diagram, i.e. plot of applied pressure versus volume under the membrane surface. Some practical aspects of the parametrizations used in modeling discussed to guarantee that the mathematically optimal solution is suitable for the real-life membrane production technology. In the conclusions, some directions of the future work are presented.

Membranes modelling
We consider the general problem for a circular shell of revolution of arbitrary profile and assume that the profile of such shell having a thickness h and radius ah is given by the function () z f r = in cylindrical coordinates. Shape optimization problems were solved by analysis of multi-parameter expressions of () fr, where all parameters were considered as unknowns. To describe the large elastic strains of the membrane under hydrostatic pressure p we use twodimensional nonlinear equations [17,18], based on the Kirchhoff's hypotheses. In addition to the detailed description of non-axisymmetric equations of the shell equilibrium, the boundary value problem for the ODE system describing the axisymmetric behavior of the shell is provided in [19]. The main method of numerical analysis of this two-point nonlinear boundary value problem was shooting method; the scheme of its application to the nonlinear boundary value problems in consideration was presented in [20]. The essential feature of its realization here is associated with the impossibility to select the one loading parameter for the whole cycle of plotting the loading diagram, i.e. the dependence between force (say, applied pressure p) and geometric (e.g., deflection at the membrane center point, 0 w ) characteristics. Traditionally, either pressure or deflection at the center of the membrane is chosen as such parameter, but in the case of a shape close to a spherical dome one can encounter the situation where there is no functional relationship between these parameters: for non-shallow profiles the loading diagram on the plane 0 pw − is a complex curve with 3 self-intersections. To solve this problem a special algorithm was realized to automatically change and select the loading parameter. This algorithm is in some sense close to that of developed in [21].
The scheme for investigating of stability of the constructed axisymmetric solution is based on the bifurcation approach completely described in [19]. Strictly speaking, such an analysis was not the goal of this work since the characteristic of the optimal membrane should have been linear, and therefore not contains extremum points, indicating a possible loss of stability. However, in the process of finding the optimal form, it was important to ensure such stability. The scheme of introducing restrictions on the range of permissible values of parameters that determine the membrane geometry, which was based upon the information on the stability region of a spherical dome, will be described below.

The optimization task
The main goal of the research was elaboration the technique of the designing the shape of the corrugated membrane that satisfies the requirements of so call "slow growing linear or almost linear" behavior of the pressure-volume diagram. More precisely these requirements can be formulated by means of the figure 1 and a set of notations presented below. (2) The condition of linearity means, firstly, that the nonlinearity parameter  should be small enough and, secondly, that pV − curve, or the characteristic curve of the corrugated diaphragm, can intersect the straight line (its linear approximation) no more than at the three points.
One extra requirement is connected with the boundedness of the mechanical stresses: within the region min max p p p  all mechanical stresses should be limited by the elastic limit of the membrane material.
One can see that conditions (1)-(3) can be easily checked by numerical analysis of the loading diagram (i.e. plot of pV − dependence) though the construction of this diagram is not so easy. The calculations were based upon the nonlinear equilibrium equations for the thin membrane of arbitrary shape within the limits of axisymmetric deformations mentioned in the previous section. This process requires the solution of a series of nonlinearly boundary value problems.

The optimization technique
The optimization process was divided into two steps. At the first one some shape of the membrane profile was chosen and N-parametrical description of this shape was constructed. Possible example of such shape and its description by the set of 25 n + parameters ( ) is presented at the figure 2. Actually some of them are more important while others are not so significant so usually from 5 to 10 parameters were used for the optimization purposes.

Figure 2. Example of the shape parametrization
The optimization step, or the process of finding the optimal values of the chosen set of parameters, was performed by means of the genetic algorithm (GA). This algorithm has been inspired by evolutionary biology and incorporates techniques such as inheritance, mutation, inverse, selection and crossover to find a better solution. The used algorithm can be called modified as it compares different techniques, such as three-stepwise single-point crossover, elitism strategy and truncation method in order to achieve certain results in an acceptable time. The detailed description of the used GA was presented in [22].
Very important part of this algoritm is the chosing of the fitness function F , i.e. the function we want to minimize. In our case it was chosen to reflect requirements (1) and (3) so roughly speaking it could be written as 12 3 10 The role of coefficients i K in (4) was to reflect the importance of linearity (so we set 21 KK ) or the importance of the slow growth (so we set 12 KK ). These coefficients were also used as penalty functions depending upon the volume V with quick growth in the area req VV  to guarantee condition (2) fulfilment. To narrow the search range for the optimal values of the geometry parameters, the following approach was used, based on the requirement of the membrane operation without loss of stability. Such check for a given membrane shape is rather time-consuming, so it is inexpediently to use it every time when calculating the objective function F. Therefore, for each checking geometry, before calculating the objective function, a rough empirical assessment of possible problems with stability was performed. To do this, in the vicinity of the membrane surface, two spherical domes were builtthe inner and the outer (see figure 3). The membrane was considered suitable for consideration if both were stable in the required range of applied pressure max pp  . The classical expression from [23] was used as a criterion for the stability of the dome. Stability tests were realized regardless of the GA program by analyzing only optimal forms. , and the profile itself is built according to a given set based on the spline interpolation. In the process of optimization, a set of parameters constituting the membrane profile with the maximum linear stroke was found. The constructed membrane differs very little from a circular plate; however the difference in loading diagrams between the corrugated membrane and the plate is not only significant, but also qualitative. The example in [24] demonstrates the good mathematical possibilities of the method, however, the chosen formal parametrization led to an unsatisfactory result in terms of technology: the production of a thin biconvex membrane (the obtained values for 1 2 5 , ,..., z z z are positive though 6 z is negative) is very laborious, which immediately reduces the efficiency of the resulting solutions.
Therefore, as a numerical example, consider a more realistic problem. As a parametrization, we will use the scheme shown in figure 2, where the number of rollers is three, all of which have the same radii. The idea to use the flat area in the center of membrane was borrowed from [25]. We assume that this flat area has a radius of 5 mm, and the horizontal zone of fixing the edge of the membrane has a width of 2 mm. Thus, we obtain seven parameters for optimization:  It turned out that the characteristics (nonlinearity and stiffness) of this membrane are slightly (5-10%) worse than those obtained earlier in [24], but the technology of its manufacturing is not so difficult.
To check whether the designed diaphragm is stable under the applied pressure the bifurcation equations were analyzed. In particular it has been shown that for pressure within limits from zero to 200 MPa no bifurcations with mode from 0 to 10 can happen. This proves the fact of mechanical stability of the proposed construction.
To guarantee that proposed design can be realized technologically it is necessary to show that it is stable with respect to possible technological imperfections, that is, to check how minor changes in the shape will influence upon the performance of the membrane. To check this necessary condition a series of numerical experiments were performed and a lot of data on the dependence of the "pressurevolume" graph upon the parameters of the geometry was obtained. Analysis of these distortions of the membrane shape shows that in some cases they can slightly change the shape of the plot but it still will satisfy the requirements (1)-(3).

Conclusions
A genetic algorithm was used to evaluate and optimize the characteristics of corrugated membranes. The presented analysis of the membrane with the maximum linear stroke showed its efficiency and reliability. The future research in this field will be related with increasing the speed of the proposed a) b)