MeshFree无网格技术在航空装备中的应用

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Abstract

The results of the study of stress-strain state of the “shock-absorbing strut” part, which is a load-bearing element of the aircraft landing gear structure, are presented. A monolithic rod structure exposed with complex loading was considered as a mathematical model. Approaches in the study of the stress state of the structure by the finite element method and the finite point method are proposed.Numerical algorithms of approximate solution of differential equations of elasticity theory are implemented in software packages Abaqus and midas MeshFree for modeling the stress-strain state of a shock-absorbing strut. Comparative an alysis and accuracy assessment of the obtained results are given.

Keywords

stresses, deformations, displacements, finite element model, finite difference model

INTRODUCTION

An important task in determining the strength characteristics of any technical structure is the study of the stress-strain state of its elements due to the external influence of volumetric and surface forces,identification of dangerous sections and local zones of stress concentrations, the influence of environmental conditions, physical nonlinearity of materials, etc. The set of influence factors requires the construction of a mathematical model with formalization of its behavior, adequate to the full-scale structure operation process. Currently, this approach is successfully implemented in the development of new aircraft products at the design stage.

Mathematical description of the stress-strain state of a structural element operating under conditions of elasticity, plasticity, and creep under complex loading is carried out using systems of differential, integral,and integral-differential equations, an alytical solutions of which are rarely obtained [1, 2]. One of the main tools for the study of strength characteristics nowadays is detailed mathematical modeling with the use of computer equipment and computer technologies. At the same time, modern software systems implement computational algorithms based on numerical methods of solving systems of equations, for example, the theory of elasticity. The most common method of numerical calculation of strength of structures is the finite element method (FEM), program implementation of which is carried out in modern engineering a nalysis complexes, such as Ansys, Abaqus and others. In many cases, specialists also use the finite difference method (FDM). Many works are devoted to the construction of computational algorithms for solving applied problems of mechanics on the basis of numerical methods and their software implementation, for example, [3–10].

This paper is devoted to the an alysis of computational algorithms and procedures for numerical study of the stress-strain state of a rod structure on the example of the supporting element of the aircraft landing gear, implemented in the software packages Abaqus (FEM) and Midas MeshFree (meshless method).The purpose of this research is to substantiate the possible application of mesh-free methods in problems of deformation of solid bodies.

METHODS AND APPROACHES

The Abaqus software package is widely used by specialists in the field of finite element strength calculations of technical structures. It is based on the finite element method. While, the Midas MeshFree is the technology that allows the accurate an alysis to be carried out without the need to create meshes,developed on the basis of the finite difference method. These complexes implement a matrix formulation of the discrete problem.

When the finite difference method is used, differential equations are approximated using discrete values of quantities given at selected points. In the case of using the finite element method, the structure as a whole (continuous medium) is modeled by dividing it into finite elements (regions), in each of which the behavior of the medium is described using a separate set of selected stress and displacement functions.These sets of functions are defined in such a form as to satisfy the continuity conditions for the characteristics they describe [10]. Both methods require the construction and solution of a system of algebraic equations, which do not always have a simple form.

In FEM, the behavior of an element is described using force and displacement components. To construct the mathematical model, the stress state is represented by generalized forces at the element nodes, and the displacements of the node points are represented by degrees of freedom. The relations between forces and displacements of a finite element are written in one of three basic forms [10]: stiffness equations (1), compliance equations (2), mixed relations between forces and displacements (3):

Here {F}, {Ff}, {Fs} are the vectors of forces and nodal forces, components of nodal forces related to fixing conditions; {Δ}, {Δf}, {Δs} are the vectors of displacements, nodal displacements from nodal forces, reactive displacements; [k], [f], [Ω] are the matrices of stiffness, element (region) compliance,transfer form between forces and displacements.

This approach is also valid for the FDM, where finite differences at the considered points (nodes) are compiled. Turning to the direct method of constructing relations for an element or region, three systems of equations of elasticity theory are considered, namely, equilibrium equations, relations between displacements and deformations, and equations of state. This method is particularly useful in elucidating the fundamental relationships between the finite element or finite difference approximation and the real structure.

Under the assumption of elastic behavior of the element, the relationship between stresses and strains is described by the matrix relation

Here {σ}, {ε} are the vector-columns of stresses and strains; {σ0}, {ε0} are the vector-columns of initial or residual stresses and strains; [D] is the elasticity matrix characterizing material properties.

In many cases, the components of the residual stress-strain state of an element caused, for example, by technological effects are not taken into account in the calculations. Components of stresses and deformations are determined from the known relations of the theory of elasticity. Due to the brevity of presentation, these relations are not given.

The given relations are successfully realized in computational blocks in modern software systems by building a mesh model and loading it in approximation to real acting loads. However, one of the main problems of this method is the procedure of constructing a qualitative mesh, which is largely determined not only by the complexity of the geometric shape of the structure under study, but also by the acquired skills of the performers. That is, it depends on objective and subjective factors. The displacements of any point inside the element e with nodal points i, j, m are given by the column vector

where [N] is the shape function matrix whose components Ni, Nj, Nm are position functions; is the vector-column of nodal displacements of element e; the superscript T is the transpose operation.

To fulfill the conditions of convergence of the computational process, certain requirements are imposed on the shape functions, which cannot always be fulfilled. Difficulties arise in fulfilling the continuity condition when selecting the element displacement functions that ensure continuity along the entire boundary of the element with adjacent elements. Discontinuity of displacements at the node boundaries in mesh construction leads to infinite deformations [11]. In addition, the process of mesh

construction can be more laborious and time consuming than obtaining a numerical solution to the problem under consideration.

It should be noted that at large deformations the distortion of meshes in the process of calculation is inadmissible, since the question arises about the adequacy of the obtained calculation results.

Currently, many specialists refuse to construct mesh computing algorithms, thus developing meshless methods of research and technologies for their realization [12–15]. Such methods have found wide application in solving physical and mechanical problems about the motion of a material continuum, for example, in hydromechanics. Modeling is performed by studying the interactions of conditional particles,for which an integral or other mathematical procedure for restoring the fields of physical parameters of the continuum by the current state of a set of particles is defined.

Let us consider the possibilities of applying the meshless method to the problems of studying the stress-strain state of loaded structures, in which the elastic and plastic deformations can take both small and large values.

BASIC RELATIONS OF THE NUMERICAL SOLUTION OF THE PROBLEM

Mathematical modeling of finite element an alysis of a structure is realized on two approaches. One of them involves solving differential equations describing the behavior of some arbitrary infinitesimal region. The other approach postulates a variational extremum principle that is valid for the entire domain [11]. Although these approaches are equivalent from the mathematical point of view, the ways of obtaining approximate solutions in these approaches differ from each other.

Consider the following differential problem. Let in the region D with boundary G there is a differential problem

with a boundary condition

Here L and l are the differential operators; f and ϕ are the given elements; u is the sought element of some linear normalized spaces F, Φ, and U, respectively.If one of the variables is time t, the most commonly considered regions are of the form

where t is the time; x is the set of spatial coordinates, x= .

The solution is sought in the spatial domain d (x) on the segment [x0, T] with initial conditions t = t0(Cauchy problem), or boundary and mixed problems, in which the boundary conditions are given on the boundary Γ(x) of the domain d (x).

The difference method is often used to solve differential problems. All derivatives included in the equations and boundary conditions are replaced by difference approximations. In the solution of Eq. (2), we define some grid—a finite set of points (nodes)

regional

where h is the grid parameter (the grid spacing).

The approximate solution of the differential problem (2)–(4) is determined by the solution of the difference scheme uh, depending on the parameter h. As a result, the differential operators L and l are replaced by the difference operators Lh and lh:

where u is the desired element of the problem; f and ϕ are the given elements.

In the finite element method, the discretization domain corresponds to the physical domain under study, which is partitioned into a finite number of subareas (finite elements) bounded by some conditions of geometric regularity, such as compliance with the limiting aspect ratio between element dimensions,angles, etc. Solving the spatial problems leads to the need for reliable and efficient mesh generators.

Meshless methods involve discretization of the unknown function and its derivatives, which are determined only by the position of the points located in the an alysis domain and the parameters specified therein [12].

The development of meshless methods has been based on different approaches. For example, in one of them, the domain under study D is extended by obtaining some artificially created computational domain with simpler geometry. The discretization process is carried out using a structured grid Dh, as which in many cases a Cartesian grid is convenient to use, Fig. 1. Here Fig. 1a is the structured grid Dh; Fig. 1b id the Cartesian grid.

The boundary of the physical domain Γ under study divides the extended domain D into an inner structured regular subarea Dh and an outer subarea. In the inner part, the approximate calculation of the linear integral I(f) of the form

is performed using the linear quadrature formulas

where p(x) is the weight function; ci are the quadrature coefficients; xi are the quadrature nodes.

Various mathematical methods can be used to compute integrals in the outer subarea, for example, the method in which the variables are uncorrelated with the nodes (Lagrange’s method of uncertain multipliers) or the method of weighted residuals.

In the method of weighted residuals, the most general procedure for numerical solution of the system of differential equations of problem (2) is the expression in the form [12]

with Neumann and Dirichlet boundary conditions

where the unknown function u is approximated by some trial approximation and equations (5), (6) are replaced by the relation [12]

In expressions (12)–(14) the following notations are adopted: A and B are the differential operators; b and t are the external forces or sources acting in the area Ω along the boundary respectively; is the given value at the boundary are the weight functions.

In order to preserve the local character of the problem leading to a ribbon matrix, the function u should be transformed to the form

Here is the interpolating functions; n(p) is the total number of points in the area under consideration. The interpolating functions should satisfy the conditions [12]

Here is a subarea Ω containing n points, n << n(p); is an approximate value of u at point i such that .

In the method of weighted residuals, the choice of weight functions determines the scheme of obtaining the solution by the following classical methods [11] as collocation at a point, collocation in a subarea, Galerkin method, which constitute the meshless approximation procedures. A detailed consideration of this issue is given in [12]. The authors believe that the advantages of meshless procedures are of the best realized by point collocation.

Finite cell methods can also be used to represent higher order boundaries as well as for numerical integration [15].

Modeling the stress-strain state of a structure of complex geometry using the meshless method allows us to carry out a numerical solution of a system of partial differential equations based on their approximation at arbitrarily placed nodes. One of such methods is the finite point method.The essence of the method is shown in Fig. 1.

For any selected edge point of region D, a local point cloud is built. The approximation is computed as a linear combination of the unknown nodal values of the cloud and certain metric coefficients. The distances between the nodal elements and the scattered points are minimized. Since rm is the function of the coordinates, the value of the function ϕ at a contour point depends on the choice of shape function or it is [11]

In cloud, the problem is solved by applying the least square approximation method.

MODELING OF STRESS-STRAIN STATE OF A STRUCTURE IN ABAQUS AND MIDAS MESHFREE SOFTWARE PACKAGES

We carry out the study of the stress-strain state on the example of the “shock absorber strut” part,the design of which is schematically shown in Fig. 2a. Here 1—the cylinder body; 2—the rod; 3—the crosshead. Figure 2b presents the solid-state finite-element model built in Abaqus software package,Fig. 2c—the solid meshfree model built in the midas MeshFree software environment.

The investigated part is a load-bearing structure of the aircraft main landing gear of pneumohydraulic type with a single-chamber pneumohydraulic shock absorber, has a tubular branched structure and consists of a cylinder body and a crosshead. The cylinder body has lugs for attaching components of the main chassis support (attachment of slotted joints, slopes, etc.). The structure is made of separate metal blocks joined by welding. The shock absorber strut is designed to absorb impact energy during landing, absorb load during run-up, run-through, taxiing and towing of the aircraft on the airfield. At landing, the energy of landing impact of the aircraft is transferred to the work of joint compression of shock absorbers and wheel pneumatics, determined taking into account the arrangement of supports and landing gear loading conditions [17]. Requirements for landing gear strength are defined through operational loads and regulated by aviation regulations [18]. Calculated cases of loading of landing gear of different schemes are given in the section “Strength” [18].

As a calculation model, we take a rod system taking into account the location of the internal cavity.During loading, the supporting rod experiences the complex loading, namely, compression, torsion, and bending. The crosshead axis is subjected to joint bending in two perpendicular directions.

The study of the stress-strain state of the strut component is carried out using the Abaqus (FEM) and midas MeshFree (meshless method) software packages.

Finite Element Model in Abaqus Software Package

The constructed finite element model of the “strut” part includes 43269 finite elements (FE) and 31557 nodes. The number of quadrilateral 2D FEs is 33591 and the number of triangular 2D FEs is 2492.

The ratio of triangular to quadrangular FEs is less than 0.08%.

The finite element model and mesh convergence plot are shown in Fig. 2b and Fig. 3, respectively.

Model in Midas MeshFree Software Package

Model construction in the mesh domain was performed on the original geometry (see Fig. 2c).The distribution of computational nodes is characterized by the mesh density and is shown in Fig. 4.

The degree of mesh density is medium to high. The change of mesh density is dictated by the need to obtain more accurate data on the stress-strain state of the structure in dangerous sections or the most loaded segments. Construction of a qualitative mesh is not required in this method, which leads to a significant reduction in the labor intensity of modeling the problem. The method does not require verification of mesh convergence. In addition, the meshless method yields a more sparse coupling matrix compared to the FEM method, for which, the stiffness matrix has a ribbon structure, which is more efficient, both in terms of memory utilization and problem solution time.

The models were loaded absolutely identical according to the calculation scheme. The values of forces and moments were taken from published open sources [19–22]. For example, in the traverse, the force was applied to the lugs along the two axes of the global coordinate system. The boundary conditions were set by constraining all degrees of freedom of the surfaces of the upper part of the crosshead. The physical and mechanical properties of the material corresponded to the parameters of medium carbon alloy steel.

Let us show the results of the von Mises stresses study for finite element model in Abaqus—Fig. 5a and the midas MeshFree model—Fig. 5b.

Figures 6 and 7 show the results of modeling the tangential stresses and displacements, respectively.Here Figs. 6a and 7a—the finite element model Abaqus; Figs. 6b and 7b—the Midas MeshFree model

The results were compared based on visual a nalysis of the color distribution zones for two models and quantitative an alysis of the point magnitudes of the calculated values. Five magnitudes of these values were taken at selected points with the same coordinates in different diametric directions.

The results of the quantitative an alysis are summarized in Table 1.

The performed quantitative an alysis showed that the convergence of the results is quite close, except for the rather large von Mises stress deviations at points “2” and “3”, equal to 0.1241 and 0.591. Such errors may be related to the distortion of the mesh element in the zone of complex geometry, which requires additional refinement.

CONCLUSIONS

The obtained results of the study allow us to conclude that meshless methods of numerical investigation of physical processes are quite applicable for structural a nalysis of machine parts and aircraft structures in estimation of strength and reliability.

With further study of meshless methods and their programming, new problems can be realized not only for modeling the motion of particles in a continuous medium, but also problems of plasticity and creep. These methods allow studying the statics and dynamics of loaded solid bodies with acceptable accuracy for engineering practice.

The main advantage of meshless methods compared to the finite element method is the reduction of labor intensity in the creation of a computational model, which does not require the construction of a high-quality finite element mesh, as well as the absence of the problem of unrealistic parameters obtained in complex areas of the geometry of the object under study.

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