Use of Genetic Algorithms for Optimal Design in Mechanics
S.I.Rodin, O.V.Dushko
Department of Material Science and Mechanics

 At well boring in oil and the gas industries for turbo drill driving are used pumps feeding water-based fluid in a borehole. The fluid pressure depends on depth of a borehole and can be as high as 20-30 MPa that constructs rather high offloading on members of the pump. The least resource has the pump piston cup usually made of rubber.

Its resource usually does not exceed several tens of clocks that require labor consuming and often operations on repair. The heightening of a resource of a pump bucket up to a resource of remaining basic elements of system allows constructing the more reliable and cost-effective pump, defining operation of all boring machinery.

Figure 1.
The pump - power stroke.

At driving the cylinder piston 1 (fig.1) under an operation force F piston cup 2 prevents penetration of a fluid 3 in a clearance at an interior surface of the cylinder 4. Thus on piston cup the hydrostatic pressure of a fluid and frictional force on contact piston cup - cylinder operates. For making pre-load and conduct contact stresses in an unyielding state piston cup geometry does not coincide with an interior surface of the cylinder. At an insertion of the cylinder piston in the cylinder the piston cup is deformed.

Piston cup material is homogeneous and isotropic in all area, except for area of contact with the cylinder. Here surface layer 5 (fig.1), for a heightening of a resource, is modified - in rather thin surface layer a module of elasticity is much above than module of elasticity of basic material.

The modification of surface layer raises a resource of a piston cup in some tens of times that is confirmed experimentally. Lowering of frictional force on contact to the cylinder and change of distribution of contact stresses, which appears to be more uniform, that is confirmed by results of analytical study, explain it. The increase of a resource is connected with more uniform distribution of contact stresses on all contact area. It is possible to drive this process by changing initial geometry of a piston cup.

For searching optimum piston cup geometry is used the method of genetic algorithms to solve a delivered problem at presence of many unknown parameters.

The method of genetic algorithms requires solution of several hundred thousands separate tasks with different interior parameters of system for deriving the optimum solution, that requires essential time for computations. For this purpose is designed the effective analytical method of definition of stresses and displacements in a piston cup to make possible for a real-time to receive the required solution of a problem of optimization.

Analytical solution for a piston cup

It's necessary to define the stress-deformed state of a piston cup and contact stresses piston cup-cylinder for a given arbitrary initial geometry of a piston cup and operating pressure of a fluid. The piston cup is symmetric concerning an axis z (fig.1). Thus stresses and displacements vary only along two coordinate axes - z and r.

The strains of a piston cup are small and thus connection between stresses and displacements may be linear (Hooks law). For this problem it is possible to apply linear theory of elasticity.

Figure 2 shows the geometry of a piston cup and loading for power stroke in z-r axes, and figure 3 shows its undeformed geometry.

Figure 2.
Piston cup - loading.

Figure 3.
Piston cup - undeformed geometry.

For definition of stress-deformed state of a piston cup is used the known method of a stress function. At the solution of a problem are used usual methods of theory of elasticity.

Scientific novelty is in accepted stress function for the solution of a problem.

The boundary conditions are satisfied on each contour approximately. Their precise correspondence is fulfilled only in isolated points of a contour, with their arbitrary number. Thus the magnification of these points gives the precision of the solution. These points are shown on fig. 3.

The obtained solution precisely satisfies to all equations of the theory of elasticity and approximately, as much as precise, for boundary conditions.

The solution of a problem is given in the solution of system of the linear algebraic equations. The system order is equal to doubled number of points of a contour for satisfying boundary conditions.

In the field of contact on boundary DA (fig. 4) due to modification the elastic modulus of a material varies along coordinate r on width t from value E1 on a surface up to value E of an elastic body.

Figure 4.
The influence of a modified layer.

The influence of a modified layer on behavior of a whole body is considered approximately by introduction of a complementary borderline field DA* (fig. 4) with a reduced elastic modulus Å*. The tangential stresses (fig. 4) on contact of boundaries DA and DA* cause points displacements of boundary DA and influence stresses and displacements of the whole body.

The contact tangential stresses are represented by an ascending power series through unknown coefficients, which are derived from equilibrium conditions and equality of displacements of boundary DA and modified layer DA* in points of contact.

These equations supplement the basic system for boundary conditions.

The solution of optimization problem

The fitness function defines correspondence of the solution to optimum and is accepted as the weighted combination of the solutions, defining the sum of quadrates of diversions of contact stresses from medial at different stages of erasing of a surface. Thus fitness function enables to define an optimum configuration of a piston cup for as much as possible uniform distributions of contact stresses during operation and erasing of a material. Island ring topology is used to achieve solution.

On figure 5 is shown the initial optimum configuration of a piston cup (curve 3 - full-scale, curve 4 - in an expanded scale) and corresponding contact stresses (curve 1) without the account of erasing of a material.

Figure 5.
Initial optimum configuration - no erasing.

On figure 6 is shown the initial optimum configuration of a piston cup for achieving the greatest possible uniform distribution of contact stresses during operation at its erasing on depth of 0,5 mm (curve 3 - full-scale, curve 4 - in an expanded scale) and corresponding contact stresses (curve 1 - beginning of operation, curve 2 - erased surface).