CN115563838B - A method for band structure and frequency domain analysis of seismic metamaterials based on COMSOL - Google Patents

Abstract

The invention relates to a COMSOL-based seismic metamaterial energy band structure and a frequency domain analysis method, which comprise the following steps: constructing a geometrical model of the seismic metamaterial and giving material parameters; setting a Floquet period boundary condition; selecting a proper mesh subdivision method to carry out mesh division on the geometric model; performing parameterized scanning and analyzing; the frequency domain of the structure is analyzed. The method can quickly solve the dispersion curve through COMSOL, so that frequency domain analysis is further carried out, the attenuation domain of the limited-row periodic structure is obtained, and the method has important significance for evaluating the damping performance of the seismic metamaterial.

Description

A method for band structure and frequency domain analysis of seismic metamaterials based on COMSOL

technical field

The invention relates to the field of computer simulation of solid mechanics, in particular to a band structure and frequency domain analysis method for solving seismic metamaterials by using COMSOLMultiphysics multiphysics finite element simulation software.

Background technique

Seismic metamaterials are artificial composite periodic structures designed in the sub-wavelength range that can change the local characteristics of the ground. Periodic structures have band gap characteristics, which are usually represented by energy band structures, also known as "dispersion curves", which generally refer to the relationship between frequency and wave number. The energy band structure is usually divided into a forbidden band region or a passband region. The infinite periodic structure is represented by the bandgap energy band structure, and the finite periodic structure is represented by the attenuation domain, that is, elastic waves in the attenuation domain cannot propagate, while elastic waves outside the attenuation domain can propagate in the periodic structure.

The existing theoretical methods for calculating the energy band structure of seismic metamaterials mainly use the finite element method, which has many characteristics such as versatility and effectiveness. In the early days, finite element software such as ANSYS was mainly used to analyze the dispersion curve of periodic structures. Although there are many commercial finite element software, there are few finite element software styles that can handle complex operations.

Contents of the invention

The purpose of the present invention is to provide a COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method, carry out finite element analysis through COMSOL to obtain the energy band structure, and quickly solve the dispersion curve, thereby further carry out frequency domain analysis, and obtain the attenuation domain of the finite row periodic structure.

To achieve the above object, the present invention provides a COMSOL-based seismic metamaterial band structure and frequency domain analysis method, comprising the following steps:

S1. Construct a seismic metamaterial geometric model and assign material parameters;

S2, setting the Floquet periodic boundary condition;

S3. Selecting an appropriate meshing method to mesh the geometric model;

S4, perform parametric scanning, and analyze;

S5, analyzing the frequency domain of the structure.

Preferably, in step S3, the unit cell analysis is taken, the grid is divided, and the algebraic equation is established through the COMSOL standard finite element analysis process:

(K-ω ² M)U=F

In the formula, K is the stiffness matrix, M is the mass matrix, U is the displacement vector of all nodes, F is the load vector of nodes, and ω is the angular frequency.

Preferably, in step S2, the periodic boundary condition is:

u _m (r) = u _m (r+R)e ^-ik·R

In the formula, k is the wave vector, and R is the lattice vector. This formula shows that the displacement u _m (r) is a complex number.

Preferably, in step S5, the displacement in the frequency domain analysis of the structure is expressed as follows:

u(r,t)=u _m (r)e ^-iωt

In the formula, r is the spatial position vector, t is the time, ω is the angular frequency, and i is the complex unit.

Preferably, the boundary corner point displacement relationship and the periodic boundary condition relationship are as follows:

|K _R -ω ² M _R |＝0

In the formula, K _R and _MR are Hermite matrices, and both contain wave loss, and both are complex matrices.

Therefore, the present invention adopts the above-mentioned COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method, and the finite element analysis through COMSOL can quickly obtain the dispersion curve and perform frequency domain analysis. Through the seismic metamaterial energy band structure and frequency domain analysis, the band gap frequency range of the structure under infinite periodic arrangement and the attenuation domain produced by the finite arrangement structure can be obtained, which is of great significance for the evaluation of the shock absorption performance of seismic metamaterials.

The technical solutions of the present invention will be described in further detail below with reference to the accompanying drawings and embodiments.

Description of drawings

Fig. 1 is the flowchart of the embodiment of the present invention;

2 is a schematic diagram of the first Brillouin zone of a two-dimensional periodic structure in an embodiment of the present invention;

Fig. 3 is the schematic diagram of well-shaped structure in the embodiment of the present invention;

Fig. 4 is the energy band structure diagram of the embodiment of the present invention;

Fig. 5 is a frequency domain analysis transmission spectrum of an embodiment of the present invention.

reference sign

1. Steel; 2. Soil.

Detailed ways

The technical solutions of the present invention will be further described below through the accompanying drawings and embodiments.

Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the usual meanings understood by those skilled in the art to which the present invention belongs.

Embodiment one

A COMSOL-based seismic metamaterial band structure and frequency domain analysis method, the steps are as follows:

S1. Construct a seismic metamaterial geometric model and assign material parameters;

A well-shaped structure is used as a modeling analysis calculation, and energy band structure and frequency domain analysis are performed. The unit cell of the well-shaped structure is welded by four equal-sized steel plates. The length of each steel plate is 1.6m, the thickness of steel plate one is 0.1m, and the height of the steel plate is 10m; the bottom is set as soil, and the interface between the upper well-shaped combined steel and the lower soil is set flush with the ground. The intersection of the steel plate and the steel plate divides the steel plate into three equal parts.

The seismic metamaterial is composed of steel and soil, and the material parameters are listed in Table 1.

Table 1 Material parameters

Material Density(kg/m ³ ) Elastic modulus (Pa) Poisson's ratio steel 7784 2.07*10 ¹¹ 0.3 soil 1300 3*10 ⁷ 0.3

Select "3D Modeling" in the COMSOL model, then select the "Eigenfrequency" module in the "Solid Mechanics" field, and perform modeling in the "Geometry" option of the model tree.

First, build the lower soil, build a soil column with a side length of 2m and a height of 40m; then build the upper well-shaped structure, and the two parts are rigidly connected; finally, delete and merge the redundant surfaces of the upper well-shaped structure to form a union with the lower soil column.

S2, setting the Floquet periodic boundary condition;

The geometric model built in the above steps is a unit cell, or a typical unit, and the structure composed of infinitely periodic unit cells becomes a seismic metamaterial structure, and the root cause of the band gap in the seismic metamaterial structure is periodicity.

After right-clicking under the "Solid Mechanics" option, find "Periodic Boundary Conditions" and select Floquet period under "Periodic Type"; Floquet period k is the wave vector, set to kx and ky;

The application of periodic boundary conditions needs to keep the opposite sides consistent. The model studied in this example consists of two sets of opposite sides, and two sets of periodic boundary conditions need to be set. In addition, fixed constraints are set at the bottom of the soil column.

S3. Selecting an appropriate meshing method to mesh the geometric model;

The finite element grid divides several models into a certain number of domains. Each geometric domain is called a unit. The physical parameters and boundary conditions are applied to each unit according to the settings, and the numerical solution is performed; therefore, grid division is required before calculating the energy band structure. Selecting an appropriate unit type and structure size grid can greatly improve the solution efficiency.

This embodiment adopts free tetrahedral meshing, which can adjust the mesh of a certain area or the entire geometric area independently. The default "regular" size is selected for the meshing size. Here, the thickness of the meshing can be changed as needed.

It is undeniable that the finer the grid division, the more units corresponding to one side, the more grids, the higher the calculation accuracy, but at the same time, the longer the calculation time, the higher the performance requirements of the computer.

S4, perform parametric scanning, and analyze;

The parametric sweep is mainly for the wave vector k; therefore, before the parametric sweep, define the parameter k under the "Global Parameters" tab.

The principle of parametric scanning is to perform parametric scanning along the Γ-Х-M-Γ direction in the first irreducible Brillouin zone through the Bloch wave vector k, so as to calculate the required energy band.

Define the value of k from 0-3 to correspond to the Γ-Х-M-Γ direction, where 0-1 means along the horizontal Γ-Х direction, 1-2 means along the vertical Х-M direction, and 2-3 means along the diagonal M-Γ direction; during the scanning process of the wave vector k, it is set as kx and ky components which are perpendicular to each other; a,(3-k)*pi/a)); Similarly, the expression of ky along the Γ-Х-M-Γ direction is defined as if(k<1,0,if(k<2,(k-1)*pi/a,(3-k)*pi/a)).

In the parametric scan, enter range(0,3/36,3) as the parameter value, which means that k starts from 0 to 3, and the step size is 3/36; then set the required setting frequency to 18, which means that 18 energy band curves are selected.

The energy band structure obtained after the parametric analysis can be viewed directly in COMSOL, and the data generated after the parametric scanning calculation can be viewed in the "results". In the "results", add the "one-dimensional drawing group" and add the "global" definition option. The data set comes from the "parametric solution" obtained above. The y-axis data is expressed as frequency (Hz), and the x-axis source data is selected as "external solution"; that is, the dispersion curve is drawn in the right graph column.

The energy band structure needs to add the sound cone curve. The sound cone curve is the first dispersion curve for calculating the energy band structure when the structure (pure soil) is not included. After adding the sound cone curve, the final energy band structure diagram is obtained.

The energy band structure diagram of the embodiment of the present invention shows that there are three band gaps in the structure, the first band gap is 0.37-0.50 Hz, the second band gap is 5.96-6.77 Hz, and the third band gap is 9.43-16.66 Hz, indicating that seismic waves in these frequency ranges cannot propagate to rear buildings through the seismic metamaterial structure.

S5, analyzing the frequency domain of the structure;

The frequency domain analysis of the structure is also called the transmission spectrum simulation. Firstly, the geometric modeling is carried out, and the metamaterial structure composed of 7 rows of unit cells is set through the "array", and then the "low reflection boundary" or "perfectly matched layer (PML)" is set at the soil boundary.

Set the specified displacement as the seismic excitation point at a distance of 3a from the soil boundary (a is a unit cell constant, and a in this embodiment is 2m): specify the displacement excitation of u _ox =0.1, u _oz =0.1 (m) in the x and z directions, select "frequency domain" analysis in "research", and set range (0.1,0.2,25), indicating that the excitation frequency is from 0.1-25Hz and the step size is 0.2.

A three-dimensional intercept point is set at 2a behind the seismic metamaterial structure, that is, the response collection point. The acquired data is processed to obtain the transmission spectral coefficient. The definition of the transmission spectral coefficient is FRF=20log ₁₀ (A ₁ /A ₀ ), where A ₁ represents the acceleration response when there is a seismic metamaterial structure, and A ₀ represents the acceleration response when there is no seismic metamaterial structure.

The acceleration response obtained in this embodiment is the root mean square of the acceleration magnitude. When FRF<0, it means that there is an attenuation domain. As shown in the transmission spectrum of the frequency domain analysis in this embodiment, it can be seen that compared with the band gap, the range of the attenuation domain is expanded. This may be due to the soil itself having a certain attenuation effect on the displacement excitation in the x-z direction.

Through seismic metamaterial band structure and frequency domain analysis, the bandgap frequency range of the structure in the infinite periodic arrangement and the attenuation domain in the finite arrangement structure can be obtained.

Therefore, the present invention adopts the above-mentioned COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method, and the finite element analysis through COMSOL can quickly obtain the dispersion curve and perform frequency domain analysis. Through the seismic metamaterial energy band structure and frequency domain analysis, the band gap frequency range of the structure under infinite periodic arrangement and the attenuation domain produced by the finite arrangement structure can be obtained, which is of great significance for the evaluation of the shock absorption performance of seismic metamaterials.

Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention rather than limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that: it can still modify or equivalently replace the technical solution of the present invention, and these modifications or equivalent replacement cannot make the modified technical solution deviate from the spirit and scope of the technical solution of the present invention.

Claims (3)

1. A method for analyzing the energy band structure and frequency domain of seismic metamaterials based on COMSOL, comprising the following steps: S1. Construct a seismic metamaterial geometric model and assign material parameters; The seismic metamaterial is composed of steel and soil; the construction of the seismic metamaterial geometric model is to select the three-dimensional modeling in the COMSOL model, then select the frequency feature module in the field of solid mechanics, and model in the geometry option of the model tree; S2, setting the Floquet periodic boundary condition; S3. Selecting an appropriate meshing method to mesh the geometric model; S4, perform parametric scanning, and analyze; S5, analyzing the frequency domain of the structure; Firstly, geometric modeling is carried out, and the metamaterial structure composed of unit cells is set through an array, and then a low-reflection boundary or a perfect matching layer is set at the soil boundary; the seismic excitation point is set at a distance of 3a, and the displacement excitation of u _ox = 0.1, u _oz = 0.1 (m) is specified in the x and z directions, and the range is set for frequency domain analysis. Range = 0.1, 0.2, 25, indicating that the excitation frequency is from 0.1-25Hz, the step size is 0.2, and a is the unit cell constant; The response acquisition point is set at 2a behind the seismic metamaterial structure, and the acquired data is processed to obtain the transmission spectral coefficient, the transmission spectral coefficient FRF=20log ₁₀ (A ₁ /A ₀ ), where A ₁ represents the acceleration response when there is a seismic metamaterial structure, and A ₀ represents the acceleration response when there is no seismic metamaterial structure; In step S3, the unit cell analysis is taken, the grid is divided, and the algebraic equation is established through the COMSOL standard finite element analysis process: (K-ω ² M)U=F In the formula, K is the stiffness matrix, M is the mass matrix, U is the displacement vector of all nodes, F is the load vector of nodes, and ω is the angular frequency; In step S5, the displacement in the frequency domain analysis of the structure is expressed as follows: u(r,t)=u _m (r)e ^-iωt In the formula, r is the spatial position vector, t is the time, ω is the angular frequency, and i is the complex unit. 2. a kind of seismic metamaterial energy band structure and frequency domain analysis method based on COMSOL according to claim 1, is characterized in that, in step S2, described periodic boundary condition is: u _m (r) = u _m (r+R)e ^-ik·R In the formula, k is the wave vector, and R is the lattice vector. This formula shows that the displacement u _m (r) is a complex number. 3. according to claim 1, a kind of seismic metamaterial energy band structure and frequency domain analysis method based on COMSOL, it is characterized in that, boundary corner point displacement relation and periodic boundary condition relation are as follows: |KR-ω ² M _R |＝0 In the formula, K _R and _MR are Hermite matrices, and both contain wave loss, and both are complex matrices.