# Method of mathematical and computer modelling

FIELD: measuring instrumentation.

SUBSTANCE: invention refers to modelling and can be applied for behaviour modelling for aviation structures and products with uncertain input parameters. Method involves behaviour modelling for aviation structures and products with uncertain input parameters at two levels: at macro level by method of end element modelling, and at micro level by quantum and molecular mechanics methods; first, microscopic samples of the model similar geometrically to standard samples used in mechanical tests are examined and tested by molecular dynamics methods, and obtained mechanical parameters of microscopic samples are used as missing macroscopic parameters in material models for end element modelling; transition from micro to macro level and back is made with the use of scale invariance of mechanical parameters and laws.

EFFECT: increased accuracy of mechanical and operation property tests of developed and reconstructed units and components.

4 dwg

The invention relates to methods of mathematical and computer modeling to create a model of the behavior of structures and components of aircraft in conditions of uncertainty of input parameters.

The prior art method of mathematical modeling [1], which are used and compared methods finite element and molecular dynamics simulation. A significant disadvantage of the proposed method is that it addresses only limited modeling fracture and are not predicted properties of materials. In addition, this method does not allow to consider the issues of modeling with the uncertainty of resource product.

Known {2] the method of the hierarchical structural modeling, including the use of basic modeling elements and additional connecting elements, geometrically similar simulated, thus splitting at the elementary elements is used to simplify the computational model and is not associated with the uncertainty of the design parameters. The disadvantage of this method is the inability to obtain the unknown parameters in the simulation of aircraft, since the elements of the partition are assumed to be given and has known mechanical properties

Well-known computer mod is whether the behavior of complex technical systems, implemented on different hierarchical levels [3]. The closest analogue, i.e. a prototype. According to the method for modeling the active zone of the nuclear reactor is introduced cellular structure of several hierarchical levels, the simulated parameters are considered at the level of the minimum cell and subsequently Srednyaya and used to estimate the behavior of a design at a higher scale level. The disadvantage of this method is that the use of the averaging procedure leads to the neglect of important, albeit single elements, for example, inclusions of foreign substances in the active zone. The presence of this kind of single inclusions can impair the functioning of the structure, to cause catastrophic consequences. In addition, the prototype does not take into account effects due to uncertainty of input parameters.

The essence of the problem is the modeling of the aircraft associated with the presence of two types of uncertainties. The first type is incomplete information on the properties of the materials used. Some of these data can be obtained from manufacturers, but the accuracy of the information questionable, which is unacceptable for the design of critical parts. These data can be obtained by conducting a bench test, but it is quite expensive, and Ino is even possible. The second type of uncertainty is associated with the real stresses on the aircraft during operation. The study of the specific loading conditions and the resulting force factors in the design generally requires expensive experiments in wind tunnels or in test flights.

Problem which is solved by this invention is a method of multilevel computer simulations that allow a high degree of accuracy to virtually test the mechanical and operational properties of the developed and rebuilt units and parts at different scale levels, allowing the use of model materials and products that meet real.

This task is solved by the method of mathematical and computer modeling involves creating a hierarchical model. The difference is that modeling involves creating a model of the behavior of structures and components of aircraft in conditions of uncertainty of input parameters at two levels: macroscopic method, finite element modeling and microscopic methods of quantum mechanics and molecular dynamics, where the first considers the microscopic samples representing the model is geometrically similar to the standard samples used for m is a mechanical test, which virtual testing methods, molecular dynamics, and the resulting mechanical properties of microscopic samples is used as the missing macroscopic parameters in models of materials for finite element modeling, and the transition from microscopic to macroscopic level modeling and back using the scale invariance of the mechanical parameters and laws.

In our method is not a simple hierarchy of scales, but a more complex hierarchy of laws, this means that to obtain the mechanical parameters of the upper hierarchical level parameters are used, derived from other than mechanical, patterns, approaches (based on megalomania quantum-mechanical interaction), which allow a high degree of accuracy to virtually test. In addition, the uncertainty of the parameters in our case due to the empirical law behavior of materials (material model for finite element analysis.

The invention is illustrated by drawings, where

figure 1 presents the original sample for the MD simulation,

figure 2 - deformed sample,

figure 3 - molecular-dynamic model of titanium VT1-0) to conduct virtual experiments

figure 4 - finite element m is del to conduct virtual experiments and comparison with the results of molecular dynamic calculation.

The method of mathematical and computer modeling, including the creation of a hierarchical model, consider the example of modeling elements of aircraft structures made of α-titanium alloys VT1, VT1-0), as titanium is a classic of aviation material.

Consider as an example the possibility of obtaining modules of elasticity (required parameters for finite element modeling) from molecular dynamics simulation, i.e. simulation at the microscopic level - the methods of quantum mechanics and molecular dynamics, where the first considers the microscopic samples representing the model geometrically similar to the standard samples used for mechanical testing, which virtual testing methods finite element simulation.

Create (generate by computer) sample material in accordance with the experimental values of lattice parameters, the direction of the crystallographic axes, the parameters used interatomic potential (figure 1). This procedure can be performed by any standard method known free packet molecular dynamics simulation (for example, Lampps, NWChem). Then deformiruem. sample programmatically according to specified the program loading (figure 2).
In this case, were stretching along the Z-axis at a given speed. Performed procedure corresponds to the standard macroscopic procedure of the study sample in the gap defined by the existing engineering practices and standards. According to the results of a computer experiment-defined values_{α}- radius-vector defining the position of a particle with number α relative initial particle in the reference configuration, which are parameters in the formula

where c_{ijkl}- modulus of rigidity; Φ is used in the model potential; V_{∗}- the volume of the elementary crystal cell of the lattice; and_{α}is the radius-vector that defines the position frequent the hospitals with the number of α relative initial particle in the reference configuration.
In the formula the distance between the atoms is expressed in the form of_{kl}- Lagrangian deformation volume of the material.

The obtained values are used as input parameters for finite element modeling. It is known that finite element modeling, for example in packages LSDYNA, ANSYS, the material behavior is governed by a set of parameters from which the tensor of elastic modulus (inverse of stiffness) is the main mechanical characteristics.

From the results of a computer experiment (figure 2) also defines a module Poisson (similar to the standard macroscopic experimental determination of this module).

In using scale invariance is the fact that the equations of mechanics do not change their form depending on the chosen system of units and scale length. This allows you to distribute the settings and dependencies obtained for the mechanical system of some size, other sizes.

On microscopica the com layer model body,
geometrically similar to the standard samples used for the macroscopic mechanical tests (figure 3). The number of atoms in a microscopic model of the system on the left coincides with the number of nodes in a macroscopic system (figure 4). Necessary for microscopic simulation parameters (type and lattice parameters, the parameters of the interatomic potential) must coincide with the experimentally defined for this type of material parameters. Microscopic virtual samples are tested by the methods of molecular dynamics, for example made an impact on the sample at a specified rate to a specified value of deformation. From the resulting computer model known from the physics and mechanics of solids dependencies calculated the necessary parameters, for example the tensor modules of elasticity/stiffness of

For example, if the MD simulation is used atomic system of units (AMU / Angstrom / PS), to obtain the macroscopic values of the modulus of elasticity (dimension^{7},

where [E] is the dimension of the modulus of elasticity

[F] is the dimension of force,

[S] is the dimension of the square.

Besides finding the missing mechanical properties of materials MT-model allows to investigate the dynamics of the processes of deformation and fracture - see figures show a fairly good agreement of results when predicting the propagation of cracks, and other processes.

Sources of information

1. N.V.R. Coffman, J.P.Sethna, Heber G, M.Liu, A.Ingraffea, P.Bailey and E.I.Barker / A comparison of finite element and atomistic modeling of fracture // ModellingSimul. Mater. Sci. Eng. 16 (2008) 065008 (15pp)).

2. Patent of Russia №2376563, publ. 20.97.2008,

3. Patent of Russia №2315375, publ. 20.01.2008 - prototype.

The method of mathematical and computer modeling, including the creation of a hierarchical model, wherein the simulation includes creating a model of the behavior of structures and components of aircraft in conditions of uncertainty of input parameters at two levels: macroscopic method, finite element modeling and microscopic methods of quantum mechanics and molecular dynamics, where the first deals with microscopy the mini samples representing the model geometrically similar to the standard samples used for mechanical testing, which virtual testing methods, molecular dynamics, and the resulting mechanical properties of microscopic samples is used as the missing macroscopic parameters in models of materials for finite element modeling, and the transition from microscopic to macroscopic level modeling and back using the scale invariance of the mechanical parameters and laws.

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