CN108595893B - Three-dimensional mechanical modal simulation method based on three-layer preprocessor - Google Patents

Abstract

The invention belongs to the technical field of three-dimensional mechanical vibration analysis numerical solution, and relates to a three-dimensional mechanical modal simulation method based on three layers of pretreatment seeds. The method comprises the steps of firstly carrying out finite element modeling on a target aircraft, introducing a displacement or stress boundary condition to establish a corresponding geometric structure model, adopting tetrahedral mesh subdivision to solve a domain, then selecting a scalar laminated basis function, obtaining a finite element eigenequation of the aircraft structure by using a standard finite element eigen analysis method, finally accelerating the efficiency of solving the equation by an implicit restart Arnoldi iteration method in an ARPACK software package by constructing three layers of preprocessing elements, and finally obtaining characteristic values and characteristic vectors, namely vibration modal frequency and vibration mode, thereby realizing rapid modal analysis.

Description

Three-dimensional mechanical modal simulation method based on three-layer preprocessor

Technical Field

The invention belongs to the technical field of three-dimensional mechanical vibration analysis numerical solution, and relates to a three-dimensional mechanical modal simulation method based on three layers of pretreatment seeds.

Background

The aircraft can vibrate when being subjected to various excitation actions, such as subsonic speed to cause full-aircraft vibration; low supersonic speeds cause rudder and vertical fin vibrations; the whole machine vibrates when the landing gear is put down, and the like. Under the action of the above excitation, the aircraft is subjected to stress unbalance, material failure and even structural disintegration, which is not favorable for the aerodynamic integrated design of the aircraft. The problem of various vibration of the fuselage structure is solved or reduced, which is often not only effective by simply increasing the structural rigidity and strength, but also sometimes can be counterproductive, and because the structural vibration problem is related to the natural frequency and the vibration mode of the fuselage and the local structure thereof, the vibration mode analysis of the aircraft is very necessary. The modal analysis can obtain the vibration characteristics of the aircraft, is an important link of the structural design of the aircraft, and is mainly used for checking whether the structure can avoid resonance when bearing various loads. Therefore, fast and effective modal analysis techniques are critical to aircraft design. The general experimental test method is difficult to apply on a large scale due to high cost and long period, and the numerical simulation method is convenient to apply on a large scale due to low cost and short period.

At present, when the modal simulation analysis is carried out on the aircraft structure by utilizing various mechanical numerical calculation methods, a finite element intrinsic analysis method is adopted, and most of structural analysis commercial software also adopts a finite element method, such as ANSYS, ABAQUS and the like. Finite element analysis generally comprises the steps of modeling, mesh division, unit calculation, matrix integration, boundary and excitation introduction, generalized eigenequation solving, post-processing and the like. In the above process, solving the generalized eigen equation is the most computation time and memory consuming step, so it is critical to find an efficient solution technique.

At present, an implicit restart Arnoldi iteration method is widely adopted for solving, and a linear equation needs to be repeatedly solved in the process. The preprocessing conjugate gradient method is widely applied to solving of linear equations, and for intrinsic problems, due to the fact that a coefficient matrix of the linear equations is unchanged during each intrinsic iteration, preprocessing is only needed once, and therefore calculation efficiency is greatly improved. However, the idea of the preconditioned conjugate gradient method is to transform the original linear equation to a new linear equation that is easy to solve by the preconditioner, which illustrates the importance of a good preconditioner for the entire Arnoldi process. Particularly, when the structure of the aircraft is complex, the grid division is more, the linear equation often reaches ten million orders, the calculation time and the memory consumption of a general preprocessor are often very huge, and the modal analysis of the aircraft structure cannot be completed quickly. Therefore, an efficient preprocessor needs to be constructed to improve the solving efficiency of the preprocessing conjugate gradient method, so that the speed of Arnoldi eigen iteration is increased, and the rapid modal analysis of the aircraft structure is finally realized.

Disclosure of Invention

Aiming at the problems or the defects, the method aims to solve the problem of constructing a high-efficiency pretreatment unit in finite element modal analysis and realize the rapid modal analysis of the aircraft structure; the invention provides a three-dimensional mechanical modal simulation method based on three layers of pretreatment seeds.

The three-dimensional mechanical modal simulation method based on the three-layer pretreatment son comprises the following steps:

A. finite element modeling is carried out on the target aircraft structure, and a displacement boundary condition or a stress boundary condition is introduced to establish a corresponding geometric structure model;

B. adopting a tetrahedral mesh to subdivide and solve a domain;

C. selecting a second-order scalar laminated basis function, and obtaining a finite element intrinsic equation of the target aircraft structure by using a standard finite element intrinsic analysis method;

D. solving the finite element eigen equation obtained in the step C by adopting an implicit restart Arnoldi iteration method in an ARPACK software package to obtain a characteristic value lambda and a corresponding characteristic vector alpha, namely an amplitude vector, solving a linear equation generated by each step of iteration of the Arnoldi iteration method by adopting a pretreatment conjugate gradient method based on a three-layer pretreatment agent, wherein the specific construction method of the three-layer pretreatment agent is as follows:

the following linear equations need to be solved repeatedly during the process of implicitly restarting Arnoldi iteration

Px＝My_j (1)

Wherein the matrix P is the matrix to be preprocessed, M is the system quality matrix, x is the intermediate vector to be solved, y_jFor iterative vector, a pretreatment conjugate gradient method is adopted to solve the formula (1), and the following pretreatment system needs to be solved in the pretreatment conjugate gradient method

Nz＝r (2)

Wherein N is a preprocessing subunit, z is a preprocessing vector to be solved, and r is a residual vector;

firstly, the matrix P is sorted in blocks according to the three directions of x, y and z to obtain the following block matrix form

Wherein P is_xxIs a matrix in the x direction, P_yyIs a matrix in the y direction, P_zzIs a matrix in the z direction, P_xyAnd P_yxIs a coupling matrix in the x and y directions, P_xzAnd P_zxIs a coupling matrix in the x-and z-directions, P_yzAnd P_zyIs a coupling matrix in the y direction and the z direction, takes the form of a first layer of preprocessors

For block diagonal matrix P in equation (4)_xx，P_yy，P_zzThe second layer preconditioner is constructed with properties based on the stack basis function: first construct P_xxThe second layer of pre-processors of (1) pre-processor, P is processed according to the arrangement sequence of the first order basis function and the second order basis function_xxWritten in block form as follows

Wherein P is₁₁Matrices formed for first-order basis functions, P₂₂Matrices formed for second-order basis functions, P₁₂And P₂₁Taking the second layer of preprocessors as the following form for the coupling matrix between the first order basis function and the second order basis function

Matrix P_yy，P_zzConstruction method of corresponding second-layer preprocessing son and matrix P_xxThe construction methods of the second layer of pretreatment sub are the same, and the names of the second layer of pretreatment sub corresponding to the second layer of pretreatment sub are respectively N_y，N_z。

For the block diagonal matrix P in equation (6)₁₁And P₂₂Constructing a third-layer preprocessor, moment by adopting multi-wavefront incomplete cholesky decompositionArray P₁₁And P₂₂The corresponding third-layer pretreatment seeds are respectively N_lAnd N_h: first, to the matrix P₁₁Performing multi-wavefront incomplete cholesky decomposition to obtain the following form of third-layer pretreatment

N_l＝(D₀P₀)^-1LL^T(P₀TD₀ ^T)^-1. (7)

Wherein D₀As a diagonal matrix, P₀For reordering matrices, L is a matrix P₁₁Cholesky decomposition of (1) lower triangular matrix, N_hMethod of construction of (1) and (N)_lThe same construction method is adopted, and the construction of the three layers of pretreatment seeds is completed;

E. and D, carrying out post-processing on the characteristic value and the corresponding characteristic vector obtained in the step D to obtain the vibration mode frequency and the corresponding vibration mode.

In conclusion, the three-dimensional mechanical modal simulation method of the three-layer pretreatment sub, which is constructed by the invention, can greatly improve the solving speed of the Arnoldi iteration method and realize rapid modal analysis.

Drawings

FIG. 1 is a flow chart of a three-dimensional mechanical modal simulation method based on three layers of pretreatment;

FIG. 2 is a diagram of a finite element model of an embodiment;

fig. 3 is a graph comparing the calculated performance of the examples with ANSYS software.

Detailed Description

The technical solution of the present invention is described in detail below with reference to the accompanying drawings and examples.

Referring to fig. 1, a three-dimensional mechanical modal simulation method based on three layers of pretreatment seeds includes the following steps:

A. finite element modeling is carried out on the target aircraft structure, and a displacement boundary condition or a stress boundary condition is introduced to establish a corresponding geometric structure model.

According to the characteristics of the aircraft, displacement boundary conditions are introduced to establish a corresponding geometric structure model to simulate the free vibration characteristics of the whole structure, as shown in FIG. 2.

B. And (5) adopting a tetrahedral mesh to subdivide a solution domain.

The divided solving domain is artificially divided into a plurality of three-dimensional tetrahedral meshes, so that a continuous geometric structure space is converted into a discrete mesh space.

C. And selecting a second-order scalar laminated basis function, and obtaining a finite element intrinsic equation of the target aircraft structure by using a standard finite element intrinsic analysis method.

Final finite element eigen-analysis equation

(-ω²M+K)α＝0 (8)

Wherein M is a mass matrix, K is a stiffness matrix and is collectively called as a characteristic matrix of the system, alpha is a displacement amplitude vector to be solved, and omega is a natural frequency to be solved. The M and K matrices are characteristic matrices of the system, whose matrix elements are related to unit basis functions and unit dispersion, where we use second order stacked basis functions, the form of the basis functions being:

wherein

For a second order basis function space, Dim refers to the spatial dimension, λ₁，λ₂，λ₃，λ₄As a function of the volume coordinate. From the above equation, it can be seen that the second order basis functions comprise first order basis functions.

D. And solving the eigen equation obtained in the step C by adopting an implicit restart Arnoldi iteration method in an ARPACK software package to obtain a characteristic value lambda and a corresponding characteristic vector alpha, namely an amplitude vector, wherein a linear equation generated by each iteration step of the Arnoldi iteration method is solved by adopting a pretreatment conjugate gradient method based on a three-layer pretreatment son.

For the intrinsic problem of the formula (8), the following deformation equation is solved by adopting an implicit restart Arnoldi iteration method

Wherein ω is₀To estimate the lowest modal frequency. The following equations need to be solved repeatedly during the process of implicitly restarting Arnoldi iteration

Px＝My_j (11)

Wherein the matrix

x is the intermediate vector to be solved, y_jIs an iteration vector. The formula (11) is solved by adopting a pretreatment conjugate gradient method, and the following pretreatment system needs to be solved in the pretreatment conjugate gradient method

Nz＝r (12)

Wherein N is a preprocessing sub, z is a preprocessing vector to be solved, and r is a residual vector.

The solution of equation (12) depends on the nature of the preprocessor N, where we propose a three-layer preprocessor approach to construct: firstly, the matrix P is sorted in blocks according to the three directions of x, y and z to obtain the following block matrix form

Wherein P is_xxIs a matrix in the x direction, P_yyIs a matrix in the y direction, P_zzIs a matrix in the z direction, P_xyAnd P_yxIs a coupling matrix in the x and y directions, P_xzAnd P_zxIs a coupling matrix in the x-and z-directions, P_yzAnd P_zyAre coupling matrices in the y-direction and the z-direction. We take the form of a first layer of preconditioners

This is the first layer of pre-processing, for the block diagonal matrix P in equation (14)_xx，P_yy，P_zzWe use the properties based on the stacking basis functions to construct the second layer preconditioner, matrix P_xx，P_yy，P_zzCorresponding second layer of pretreatmentThe construction method is the same, and the names of the second-layer preprocessing sub-elements corresponding to the first-layer preprocessing sub-elements are respectively N_x，N_y，N_zHere in the form of a matrix P_xxFor illustration purposes. Firstly, P is processed according to the arrangement sequence of first-order basis function and second-order basis function_xxWritten in block form as follows

Wherein P is₁₁Matrices formed for first-order basis functions, P₂₂Matrices formed for second-order basis functions, P₁₂And P₂₁Is a coupling matrix between first and second order basis functions. We take the second layer of preconditioners as follows

For the block diagonal matrix P in equation (16)₁₁And P₂₂We use the incomplete cholesky decomposition of the multi-wavefront to construct the third-layer preprocessor, matrix P₁₁And P₂₂The corresponding third-layer pretreatment seeds are respectively N_lAnd N_hTheir construction methods are the same, and we also use P as₁₁For illustration purposes. Our matrix P₁₁After performing the multiwave incomplete cholesky decomposition, the following form of the third layer of preconditioners can be obtained

Wherein D₀As a diagonal matrix, P₀For reordering matrices, L is a matrix P₁₁Decomposes the lower triangular matrix. Thus, the three-layer pretreatment substructure is completed. And applying the preprocessing sub-N to the solving of the formula (12), so that the formula (11) can be quickly solved, the convergence speed of the implicit restart Arnoldi iteration is increased, and the quick modal analysis is realized. Best we will do by implicitly restarting the Arnoldi iteration method in the ARPACK packageFinally, a series of characteristic values lambda are obtained_i(i ═ 1,2, …, n) and corresponding feature vector α_iWhere n is the number of modes of interest, i is 1,2, …, n, i.e., an amplitude vector.

E. And D, carrying out post-processing on the characteristic value and the corresponding characteristic vector obtained in the step D to obtain the vibration mode frequency and the corresponding vibration mode.

Obtaining a characteristic value lambda for the step D_iProcessing the vibration mode frequency of

According to the corresponding feature vector alpha_iAnd performing finite element post-processing by combining the basis function properties to obtain the displacement distribution in the solution domain, namely the vibration mode corresponding to the vibration mode frequency.

FIG. 2 is a diagram of a finite element model according to an embodiment; fig. 3 shows a comparison of the computational performance of an embodiment of the present invention and ANSYS software, and it can be seen that the computation speed of the particular embodiment is 2 times faster than that of ANSYS software, while the memory consumption is only 76% of that of ANSYS software.

Claims (1)

1. A three-dimensional mechanical modal simulation method based on three-layer pretreatment seeds comprises the following steps:

A. finite element modeling is carried out on the target aircraft structure, and a displacement boundary condition or a stress boundary condition is introduced to establish a corresponding geometric structure model;

B. adopting a tetrahedral mesh to subdivide and solve a domain;

C. selecting a second-order scalar laminated basis function, and obtaining a finite element intrinsic equation of the target aircraft structure by using a standard finite element intrinsic analysis method;

D. solving the finite element eigen equation obtained in the step C by adopting an implicit restart Arnoldi iteration method in an ARPACK software package to obtain a characteristic value lambda and a corresponding characteristic vector alpha, namely an amplitude vector, and solving a linear equation generated by each iteration step of the Arnoldi iteration method by adopting a pretreatment conjugate gradient method based on three layers of pretreatment elements;

the specific construction method of the three-layer pretreatment unit is as follows:

the following linear equations need to be solved repeatedly during the process of implicitly restarting Arnoldi iteration

Px＝My_j (1)

Wherein the matrix P is the matrix to be preprocessed, M is the system quality matrix, x is the intermediate vector to be solved, y_jFor iterative vector, a pretreatment conjugate gradient method is adopted to solve the formula (1), and the following pretreatment system needs to be solved in the pretreatment conjugate gradient method

Nz＝r (2)

Wherein N is a preprocessing subunit, z is a preprocessing vector to be solved, and r is a residual vector; firstly, the matrix P is sorted in blocks according to the three directions of x, y and z to obtain the following block matrix form

Wherein P is_xxIs a matrix in the x direction, P_yyIs a matrix in the y direction, P_zzIs a matrix in the z direction, P_xyAnd P_yxIs a coupling matrix in the x and y directions, P_xzAnd P_zxIs a coupling matrix in the x-and z-directions, P_yzAnd P_zyIs a coupling matrix in the y direction and the z direction, takes the form of a first layer of preprocessors

For block diagonal matrix P in equation (4)_xx，P_yy，P_zzConstructing a second layer of preconditioners with properties based on the stack basis functions; first construct P_xxThe second layer of pre-processors of (1) pre-processor, P is processed according to the arrangement sequence of the first order basis function and the second order basis function_xxWritten in block form as follows

Wherein P is₁₁Matrices formed for first-order basis functions, P₂₂Matrices formed for second-order basis functions, P₁₂And P₂₁Taking the second layer of preprocessors as the following form for the coupling matrix between the first order basis function and the second order basis function

Matrix P_yy，P_zzConstruction method of corresponding second-layer preprocessing son and matrix P_xxThe second-layer preprocessing units have the same construction method, and the names N are respectively given to the corresponding second-layer preprocessing units_y，N_z；

For the block diagonal matrix P in equation (6)₁₁And P₂₂Constructing a third-layer preprocessor, matrix P, by using multi-wavefront incomplete cholesky decomposition₁₁And P₂₂The corresponding third-layer pretreatment seeds are respectively N_lAnd N_hFirst, to the matrix P₁₁Performing multi-wavefront incomplete cholesky decomposition to obtain the following form of third-layer pretreatment

Wherein D₀As a diagonal matrix, P₀For reordering matrices, L is a matrix P₁₁Cholesky decomposition of (1) lower triangular matrix, N_hMethod of construction of (1) and (N)_lThe same construction method is adopted, and the construction of the three layers of pretreatment seeds is completed;

E. and D, carrying out post-processing on the characteristic value and the corresponding characteristic vector obtained in the step D to obtain the vibration mode frequency and the corresponding vibration mode.

Images