CN115563838B - COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method - Google Patents
COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method Download PDFInfo
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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
Technical Field
The invention relates to the field of computer simulation solid mechanics, in particular to an energy band structure and frequency domain analysis method for solving a seismic metamaterial by adopting COMSOL Multiphysics multi-physical-field finite element simulation software.
Background
The seismic metamaterial is an artificial composite periodic structure designed in a sub-wavelength range and capable of changing the local characteristics of the ground, wherein the periodic structure has band gap characteristics and is generally represented by an energy band structure, and the periodic structure is also called a 'dispersion curve', and generally refers to the relationship between frequency and wave number. The band structure is generally divided into a forbidden band region or a passband region, the infinite periodic structure is represented by the band gap band structure, and the finite periodic structure is represented by an attenuation domain, i.e., an elastic wave in the attenuation domain cannot propagate, while an elastic wave outside the attenuation domain can propagate in the periodic structure.
The existing method for calculating the energy band structure theory of the seismic metamaterial mainly adopts a finite element method, has the characteristics of universality, effectiveness and the like, and mainly adopts finite element software such as ANSYS and the like to analyze the dispersion curve of a periodic structure in the early stage, however, the periodic structure is different from the natural vibration analysis of a general structure in the application of periodic boundary conditions, the periodic boundary conditions relate to phase factors, and the periodic boundary conditions contain a plurality of phase factors, so that great inconvenience is brought to analysis. Although there are many commercial finite element software, there are few finite element software versions capable of handling complex operations.
Disclosure of Invention
The invention aims to provide a COMSOL-based seismic metamaterial energy band structure and a frequency domain analysis method, which are used for obtaining an energy band structure through COMSOL finite element analysis and rapidly solving a dispersion curve, so that frequency domain analysis is further carried out, and an attenuation domain of a finite-row periodic structure is obtained.
In order to achieve the above purpose, the invention provides a COMSOL-based seismic metamaterial energy band structure and a frequency domain analysis method, which comprises the following steps:
s1, constructing a geometrical model of the seismic metamaterial and giving material parameters;
s2, setting a Floquet period boundary condition;
s3, selecting a proper mesh subdivision method to carry out mesh subdivision on the geometric model;
s4, performing parameterized scanning and analyzing;
s5, analyzing the frequency domain of the structure.
Preferably, in step S3, unit cell analysis is performed to perform grid division, and an algebraic equation is established through a COMSOL standard finite element analysis process:
(K-ω 2 M)U＝F
wherein K is a rigidity matrix, M is a mass matrix, U is all node displacement vectors, F is a node load vector, and ω represents angular frequency.
Preferably, in step S2, the cycle boundary condition is:
u m (r)＝u m (r+R)e -ik·R
wherein k is a wave vector and R is a lattice vector, the formula describes 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
where r is a spatial position vector, t is time, ω is angular frequency, and i is a complex unit.
Preferably, the relationship between the displacement of the boundary corner and the periodic boundary condition is as follows:
|K R -ω 2 M R |＝0
wherein K is R And M is as follows R Is a Hermite matrix, contains wave loss, and is a complex matrix.
Therefore, by adopting the COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method, the frequency dispersion curve can be rapidly obtained through COMSOL finite element analysis and the frequency domain analysis can be carried out, the band gap frequency range of the structure under infinite periodic arrangement and the attenuation domain generated by the structure in finite rows can be obtained through the seismic metamaterial energy band structure and the frequency domain analysis, and the method has important significance for evaluating the damping performance of the seismic metamaterial.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is a schematic diagram of a first Brillouin zone of a two-dimensional periodic structure according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a well-shaped structure according to an embodiment of the present invention;
FIG. 4 is a diagram of an energy band structure of an embodiment of the present invention;
FIG. 5 is a frequency domain analysis transmission spectrum of an embodiment of the present invention.
Reference numerals
1. Steel; 2. soil.
Detailed Description
The technical scheme of the invention is further described below through the attached drawings and the embodiments.
Unless defined otherwise, technical or scientific terms used herein should be given the ordinary meaning as understood by one of ordinary skill in the art to which this invention belongs.
Example 1
A COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method comprises the following steps:
s1, constructing a geometrical model of the seismic metamaterial and giving material parameters;
and carrying out modeling analysis calculation by using a groined structure, and carrying out energy band structure and frequency domain analysis. The groined structural unit is formed by welding four steel plates with equal size through a process, the length of each steel plate is 1.6m, the thickness of the first steel plate is 0.1m, and the height of the first steel plate is 10m; the soil is arranged below, the interface between the upper well-shaped combined steel and the soil at the lower part is flush with the ground, the intersection of the steel plates makes the steel plates trisected, the well-shaped structure formed by the steel plates is highly symmetrical, and the centroid is positioned in the center of the structure.
The seismic metamaterial is composed of two materials, namely steel and soil, and the material parameters are shown in table 1.
TABLE 1 Material parameters
| Material | Density (kg/m) 3 ) | Elastic modulus (Pa) | Poisson's ratio |
| Steel and method for producing same | 7784 | 2.07*10 11 | 0.3 |
| Soil and method for producing soil | 1300 | 3*10 7 | 0.3 |
The "three-dimensional modeling" in the COMSOL model is selected, then the "characteristic frequency" module in the "solid mechanics" field is selected, and modeling is performed on the "geometric" option of the model tree.
Firstly, establishing lower soil, and establishing a soil column with the side length of 2m and the height of 40 m; then establishing an upper groined structure, wherein the two parts are rigidly connected; and finally, deleting and merging redundant surfaces of the upper well-shaped structure, and forming a combination body with the lower soil column.
S2, setting a Floquet period boundary condition;
the geometric model built in the above step is a unit cell or a typical unit, the structure formed by infinite unit cells arranged periodically becomes a seismic metamaterial structure, and the root cause of band gap generation of the seismic metamaterial structure is periodicity.
After right clicking under the option of 'solid mechanics', finding 'periodic boundary conditions', and selecting a Floquet period under 'periodic type'; the Floquet period k is a wave vector and is set to kx and ky;
the application of periodic boundary conditions requires that opposite sides be kept consistent, and the model studied in this embodiment consists of two sets of opposite sides, two sets of periodic boundary conditions are required to be set, and in addition, fixed constraints are set at the bottom of the soil column.
S3, selecting a proper mesh subdivision method to carry out mesh subdivision on the geometric model;
dividing a plurality of models into a plurality of domains by a finite element grid, wherein each geometric domain is called a unit, applying physical parameters and boundary conditions into each unit according to the setting, and carrying out numerical value solving; therefore, grid division is needed before the energy band structure is calculated, and the selection of a grid with proper unit type and structure size can greatly improve the solving efficiency.
The embodiment adopts free tetrahedral meshing, which can independently adjust meshing of a certain area or the whole geometric area, the meshing size selects default conventional size, and the thickness degree of meshing can be changed according to the requirement.
The finer the undeniable grid division is, the more the number of units corresponding to the single side is, the more the number of grids is, the higher the calculation accuracy is, but the longer the calculation time is, the higher the performance requirement on the computer is.
S4, performing parameterized scanning and analyzing;
the parameterized scanning is mainly aimed at wave vector k; thus, parameter k is defined under the "global parameters" tab prior to parameterized scanning.
The principle of parametric scanning is that the required energy band is calculated by parametric scanning of the bloch wave vector k along the Γ -mj- Γ direction in the first irreducible brillouin zone.
Defining a value of k from 0 to 3 to correspond to the gamma-gamma direction, wherein 0 to 1 represents a horizontal gamma-gamma direction, 1 to 2 represents a vertical gamma-gamma direction, and 2 to 3 represents a diagonal gamma-gamma direction; in the scanning process of the wave vector k, the wave vector k is set to be perpendicular to kx and ky components; the expression kx along Γ -b-M- Γ may be preset in a "global parameter" and defined as if (k <1, pi/a x k, if (k <2, pi/a, (3-k) x pi/a)); the expression ky along Γ -t-M- Γ is defined as if (k <1,0, if (k <2, (k-1) pi/a, (3-k) pi/a)).
In parametric scanning, a parameter value is input into range (0, 3/36,3), wherein k starts from 0 to 3, and the step length is 3/36; setting the required set frequency number to 18, which means that 18 energy band curves are obtained through selection.
The energy band structure obtained after parameterization analysis is directly checked in COMSOL, data generation after parameterization scanning calculation is checked in a result, a one-dimensional drawing group is added in the result, a global definition option is added, a data set is derived from the parameterized solution, y-axis data is expressed as frequency (Hz), and x-axis source data is selected as an external solution; i.e. a dispersion curve is drawn in the right hand graphic column.
The energy band structure needs to be added with an acoustic cone curve, the acoustic cone curve is a first dispersion curve of the energy band structure calculated when the energy band structure does not contain a structure (pure soil), and a final energy band structure diagram is obtained after the acoustic cone curve is added.
The energy band structure diagram of the embodiment of the invention shows that the structure has three band gaps, the first band gap is 0.37-0.50Hz, the second band gap is 5.96-6.77Hz, and the third band gap is 9.43-16.66Hz, which indicates that the earthquake waves in the frequency ranges can not be transmitted to the rear building through the earthquake metamaterial structure.
S5, analyzing a frequency domain of the structure;
the frequency domain analysis of the structure is also called transmission spectrum simulation, firstly geometric modeling is carried out, a metamaterial structure consisting of 7 rows of single cells is arranged through an array, and secondly a low reflection boundary or Perfect Matching Layer (PML) is arranged at the soil boundary.
A specified displacement is set as the seismic excitation point at a distance from the soil boundary 3a (a is a unit cell constant, a is 2m in the present embodiment): specifying u in x and z directions ox ＝0.1，u oz Displacement excitation of =0.1 (m), the "frequency domain" analysis was chosen in the "study", range (0.1,0.2,25) was set, indicating excitation frequencies from 0.1-25Hz, with a step size of 0.2.
Setting a three-dimensional intercept point at the rear 2a of the seismic metamaterial structure, namely responding to an acquisition point, and processing acquired data to obtain a transmission spectrum coefficient, wherein the definition of the transmission spectrum coefficient is FRF=20log 10 (A 1 /A 0 ) Wherein A is 1 Expressed as acceleration response in the presence of a seismic metamaterial structure, A 0 Expressed as the acceleration response without the seismic metamaterial structure.
The acceleration response obtained in this example is the root mean square of the acceleration magnitude, and when FRF < 0 indicates the presence of an attenuation domain, as shown by the transmission spectrum of the frequency domain analysis in this example, it can be seen that the range of the attenuation domain is enlarged compared to the band gap, probably due to the fact that the soil itself stimulates a certain attenuation effect for displacement in the x-z direction.
The band gap frequency range of the structure under infinite periodic arrangement and the attenuation domain generated by the structure in finite arrangement can be obtained through the energy band structure and the frequency domain analysis of the seismic metamaterial.
Therefore, by adopting the COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method, the frequency dispersion curve can be rapidly obtained through COMSOL finite element analysis and the frequency domain analysis can be carried out, the band gap frequency range of the structure under infinite periodic arrangement and the attenuation domain generated by the structure in finite rows can be obtained through the seismic metamaterial energy band structure and the frequency domain analysis, and the method has important significance for evaluating the damping performance of the seismic metamaterial.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention and not for limiting it, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that: the technical scheme of the invention can be modified or replaced by the same, and the modified technical scheme cannot deviate from the spirit and scope of the technical scheme of the invention.
Claims (3)
1. A COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method comprises the following steps:
s1, constructing a geometrical model of the seismic metamaterial and giving material parameters;
the seismic metamaterial is composed of steel and soil; the method comprises the steps of constructing a geometrical model of the seismic metamaterial, namely selecting a three-dimensional modeling in a COMSOL model, then selecting a frequency characteristic module in the field of solid mechanics, and modeling in geometrical options of a model tree;
s2, setting a Floquet period boundary condition;
s3, selecting a proper mesh subdivision method to carry out mesh subdivision on the geometric model;
s4, performing parameterized scanning and analyzing;
s5, analyzing a frequency domain of the structure;
firstly, geometric modeling is carried out, a metamaterial structure composed of single cells is arranged through an array, and then a low reflection boundary or a perfect matching layer is arranged on the soil boundary; setting a seismic excitation point at a distance 3a, designating u in the x and z directions ox ＝0.1，u oz Displacement excitation of =0.1 (m), range is set for frequency domain analysis, range= 0.1,0.2,25, indicating excitation frequency from 0.1-25Hz, step size of 0.2, a is a unit cell constant;
a response acquisition point is located at the rear 2a of the seismic metamaterial structure,the acquired data are processed to obtain a transmission spectrum coefficient, and the transmission spectrum coefficient is FRF=20log 10 (A 1 /A 0 )，
Wherein A is 1 Expressed as acceleration response in the presence of a seismic metamaterial structure, A 0 Acceleration response expressed as a non-seismic metamaterial structure;
in the step S3, unit cell analysis is adopted, grid division is carried out, and an algebraic equation is established through a COMSOL standard finite element analysis process:
(K-ω 2 M)U＝F
wherein K is a rigidity matrix, M is a mass matrix, U is all node displacement vectors, F is a node load vector, and omega represents angular frequency;
in step S5, the displacement in the frequency domain analysis of the structure is represented as follows:
u(r，t)＝u m (r)e -iωt
where r is a spatial position vector, t is time, ω is angular frequency, and i is a complex unit.
2. The COMSOL-based seismic metamaterial energy band structure and frequency domain analysis method according to claim 1, wherein in step S2, the periodic boundary conditions are:
u m (r)＝u m (r+R)e -ik·R
wherein k is a wave vector and R is a lattice vector, the formula describes the displacement u m (r) is a complex number.
3. The method for analyzing the energy band structure and the frequency domain of the seismic metamaterial based on the COMSOL according to claim 1, wherein the relationship between the displacement of boundary corner points and the relationship between the periodic boundary conditions are as follows:
|KR-ω 2 M R |＝0
wherein K is R And M is as follows R Is a Hermite matrix, contains wave loss, and is a complex matrix.
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| CN110543669A (en) * | 2019-07-24 | 2019-12-06 | 华南理工大学 | Sound insulation simulation calculation method of acoustic metamaterial plate |
| CN114818292A (en) * | 2022-04-15 | 2022-07-29 | 中国核动力研究设计院 | Energy band structure analysis method for phononic crystal current-carrying pipeline |
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