CN103246807B - A kind of two-dimensional solid solid photonic crystal hybrid mode band gap optimization method - Google Patents

Abstract

The invention discloses a kind of two-dimensional solid solid photonic crystal hybrid mode band gap optimization method.First it select Two dimensional square lattice type phonon crystal primitive unit cell, and phonon crystal primitive unit cell is divided into M × M square pixel type structure；Then, the Time Migration of Elastic Wave Equation met according to the solid phonon crystal XY of two-dimensional solid and Z mode, the solid phonon crystal XY of two-dimensional solid and the Z mode of developing any materials layout can carry the plane wave expansion method program quickly calculated, calculate its band gap, the most first fix XY mode band gap, again Z mode band gap is compared therewith, so that it is determined that hybrid mode band gap；Finally utilize genetic Optimization Algorithm, according to the requirement to hybrid guided mode band gap, search the material layout that the solid phonon crystal primitive unit cell of two-dimensional solid is optimum.Such an approach achieves the target needing active designs photonic crystal structure according to band gap, obtain the novel photonic crystal structure with optimum band gap properties；Reduce the calculating time simultaneously, improve computational efficiency.

Description

A kind of two-dimensional solid-solid photonic crystal hybrid mode band gap optimization method

Technical field

The invention belongs to vocal cords gap material design field, relate to a kind of vocal cords gap material method of topological optimization design, relate to especially And one is based on genetic algorithm and modified plane wave expansion method (PWE), carries out the distribution of two-dimensional solid-solid phonon crystal primitive unit cell material Topology optimization design, the two-dimensional solid-solid phonon crystal XY required for obtaining and Z hybrid mode band gap characteristic.

Background technology

Phonon crystal refers to by two kinds and two or more material in space by periodic arrangement, has elastic wave band gaps Composite.Under certain conditions, when elastic wave is when by the periodic modulation of material constant, phonon-side bands can be produced, i.e. The propagation of the elastic wave of certain frequency scope is suppressed or forbids.This characteristic of phonon crystal has great theory significance, Before the aspects such as novel acoustic functional material such as passive sound insulation, precision optical machinery platform vibration damping, acoustic filter have a wide range of applications Scape.In above-mentioned application, presence or absence and the band gap size of phonon crystal complete band gap are the most extremely important；And, band gap is the biggest, Its using value is the highest.So, seek the photonic crystal structure with big forbidden band, the always weight of phonon crystal theoretical research Point.

Traditional phonon crystal mentality of designing is usually: at specific lattice types (tetragonal, triangular crystal lattice etc.) bar Under part, use a limited number of kind of symmetric figure (in two-dimensional problems, use cylinder, tetragonal prism etc.) as the scattering object of primitive unit cell, Phonon crystal band gap or improvement is opened by adjusting the geometric parameter of these scattering objects and scattering object and matrix material parameter The characteristic of band gap, and then empirically refine possible design rule, find more preferable phonon crystal the most instructively and newly tie Structure.

But, the phonon crystal primitive cell structure of which kind of form is had desired optimum band gap, according only to experimental Summing up and the prediction of intuitive, be then a problem being difficult to, this has fettered people to a great extent to phonon crystal Application.

Summary of the invention

In order to overcome the deficiency of existing phonon crystal Design and analysis methods, the present invention proposes a kind of two-dimensional solid-solid phonon Crystal hybrid mode band gap optimization method, it is based on genetic algorithm and modified plane wave expansion method, according to two-dimensional solid-solid phonon Crystal XY and the requirement of Z hybrid mode band gap, the phonon crystal material optimal location that Automatic-searching is corresponding, obtain that there is optimum band The novel photonic crystal structure of gap characteristic so that it is reach best technical and economic performance.

The technical solution adopted for the present invention to solve the technical problems is: calculate two with the modified plane wave expansion method improved Dimension solid-solid pixel type phonon crystal XY and the dispersion relation of Z mode, extract corresponding hybrid mode band gap magnitude；Then something lost is applied Pass optimized algorithm, according to hybrid mode band gap target to be reached, search for phonon crystal most optimum materials topological layout.It is characterized in Comprise the following steps:

1) initialize: stochastic generation N_popThe chromosome (individual) of individual binary number representation constitutes initial population, and each two enter Each of number processed is set to 0 or 1 randomly.In general, the two-dimentional sound corresponding to a binary number randomly generated Sub-crystal structure not necessarily has complete forbidden band, the most just because of this, the multiformity of guarantee population, it is easier to converge to the overall situation Optimal solution.

2) calculate ideal adaptation degree: use the modified plane wave expansion method improved, calculate the individual corresponding picture of each heredity Element type phonon crystal primitive unit cell XY mode and the dispersion relation of Z mode.Research finds: solid-solid phonon crystal the first eight energy of XY mode Below Ji, between two adjacent energy levels, not necessarily can form complete band gap, but between every two adjacent energy levels of solid-solid phonon crystal Z mode Band gap can open.So when solid-solid photonic crystal hybrid mode band gap is optimized, first calculating the band of the XY mode specified Gap, then makes comparisons the band gap of Z mode and the band gap of XY mode, so that it is determined that hybrid guided mode band gap.According to the target optimized, use Phonon crystal hybrid guided mode forbidden band structure object function, then measures, with target function value, the fitness that heredity is individual.

3) perform selection, intersection and mutation genetic operation successively and generate population of future generation, make population evolve forward, the most more New population.Wherein, selection is the size according to heredity ideal adaptation angle value, uses elitist selection to combine with roulette mechanism； The operation that intersects uses average cross mode, randomly chooses chromosome and implements row and intersect or row intersect, searches strengthening the overall situation of algorithm Suo Nengli；Individual variation uses position Variation mechanism.Due to " survival of the fittest, survival of the fittest " natural selection and genetic evolution mechanism, The topological structure of fitst water individuality the most increasingly approaches the phonon crystal primitive unit cell optimum topological layout of target call.

4) whether inspection population meets stop condition (such as fixing evolutionary generation or population are the most stable).If met Then export optimal result, exit；Otherwise return step 2).Optimizing Flow is shown in Fig. 1.

Wherein, described step 2) in the individual corresponding phonon crystal primitive unit cell of heredity be tetragonal, lattice paprmeter is a.Sound It is even number that sub-crystal primitive unit cell is divided into M × M(M) square pixel type structure.The bullet that each pixel random arrangement is selected Property material (only consider bi-material, therefore use string of binary characters form to represent).Phonon crystal primitive unit cell is set as about x, y-axis Symmetrical and about 90 ° of rotationally symmetrical models of z-axis, the most only need 1/8 pixel can describe whole primitive unit cell (as shown in Figure 2).Now, sound The Structure Designing Problem of sub-crystal is equivalent to the select permeability of pixel elastomeric material.Obviously, the most permissible when M is sufficiently large Expressing the structure of arbitrary shape, material boundary is the most smooth, but now search scale the most exponentially form increases, and pixel Undersized also brings difficulty to technique preparation.The present invention considers the restriction of operational capability and preparation technology, studies picture The photonic crystal structure topology optimization design problem that element granularity is moderate.

Described step 2) in the wave equation that met of two-dimensional solid-solid pixel type phonon crystal XY mode and Z mode be:

$\left.\begin{array}{c}\frac{{\∂}^2{u}^i}{{\∂t}^2}=\frac{1}{\mathrm{\ρ}}\{\frac{\∂}{{\∂x}_i}\left(\mathrm{\λ}\frac{{\∂u}^l}{{\∂x}_l}\right)+\frac{\∂}{{\∂x}_l}[\mathrm{\μ}(\frac{{\∂u}^i}{{\∂x}_l}+\frac{{\∂u}^l}{{\∂x}_i})\left]\right\}\\ \frac{{\∂}^2{u}^z}{{\∂t}^2}=\frac{1}{\mathrm{\ρ}}\▿\left({\mathrm{\μ}\▿u}^z\right)\end{array}\right\}---\left(1\right)$

Wherein, i, l represent coordinate x, y；uⁱI component for displacement vector u；ρ is density of material；λ and μ is that elastic properties of materials is normal Number (Lame constants).Solving equation group with the modified plane wave expansion method improved, the frequency dispersion that i.e. can get XY and Z mode is closed System.

Described step 2) modified plane wave expansion method solving equation group (1) of middle improvement time, group element material constant g(ρ, λ And μ) Fourier transformation g (G-G') most important.And for pixel type phonon crystal, can count according to placement property Calculate Fourier transformation g (G) of the group element material constant of Arbitrary distribution.

If P_aFor filled media g in primitive unit cell_aCollection of pixels.First central point pixel P is calculated₀∈P_a(Fig. 3) material constant g₀Fourier transformation g₀(G) it is:

$g_0\left(G\right)=\left\{\begin{array}{cc}{g}_B+f(g_A-g_B),& G=0\\ f(g_A-g_B)\sin c\left(\frac{G_{x}a}{2N}\right)\sin c\left(\frac{G_{y}a}{2N}\right),& G\≠0\end{array}---\left(2\right)\right.$

Wherein: a is phonon crystal primitive unit cell lattice paprmeter；G is reciprocal lattice vector, G_xAnd G_yIt is respectively x and the y durection component of G；The packing ratio of whole primitive unit cell is accounted for for scattering object.

According to placement property, arbitrary filling g_aPixel P of medium_r∈P_aFourier transformation be:

g_r(G)-g₀(G)e^iG·r (3)

Fourier transformation g (G) that can be calculated the distribution of pixel type phonon crystal material constant by (3) formula is:

$g\left(G\right)=\underset{P_r\∈P_a}{\mathrm{\Σ}}{g}_r\left(G\right)=\underset{r}{\mathrm{\Σ}}{g}_r\left(G\right)\mathrm{\δ}\left(r\right)=g_0\left(G\right)\underset{r}{\mathrm{\Σ}}{e}^{\mathrm{iG}\•r}\mathrm{\δ}\left(r\right)---\left(4\right)$

Wherein:

If $e\left(G\right)=[e^{\mathrm{iG}\·r_1},e^{\mathrm{iG}\·r_2},\·\·\·,e^{\mathrm{iG}\·r_k},\·\·\·,e^{\mathrm{iG}\·r_L}],$ The pixel materials constant distribution of primitive cell structure can represent For δ (r)=[δ (r₁),δ(r₂),…δ(r_k),…,δ(r_L)], r_kRepresenting the center position of kth pixel in primitive unit cell, L represents Pixel count total after the segmentation of phonon crystal primitive unit cell.Then (4) formula is equivalent to

G (G)=g₀(G) e (G). δ (r) (5)

Owing to e (G) can be with calculated off line, (5) formula shows, after the δ (r) of given any primitive cell structure, both can use and add Method calculates g (G).By g (G) calculated off line and store, call for each band gap calculation, thus be substantially reduced operand, improve fortune Calculate speed.

The invention has the beneficial effects as follows: can want according to two-dimensional solid-solid phonon crystal XY and Z hybrid mode band gap Asking, automatic fast searching has the phonon crystal topology of global optimum, obtains the photonic crystal structure of novelty；Break away from biography The Experience Design thinking of system, it is achieved that need the target of active designs photonic crystal structure according to band gap, make phonon crystal can Design becomes strong；Decrease the calculating time simultaneously, improve computational efficiency, and make designed phonon crystal reach best skill Art economic performance.

Accompanying drawing explanation

Fig. 1 is that two-dimensional solid-solid pixel type phonon crystal is tied by the present invention based on genetic algorithm and modified plane wave expansion method Structure XY and Z hybrid mode topology optimization design FB(flow block)；

Fig. 2 is symmetrical about x, y-axis and about z-axis 90 ° rotational symmetric phonon crystal primitive unit cell 10 × 10 picture in the present invention Element structural representation and gene expression figure；

Fig. 3 is the present invention arbitrary filling g_aShift transformation when the material constant Fourier transformation of material pixel cell calculates Figure；

Fig. 4 is present example optimum phonon crystal 3 × 3 primitive unit cell figure；

Fig. 5 is present example optimum phonon crystal energy band diagram；

Fig. 6 is the population average fitness in present example genetic algorithm evolutionary process and the change of optimum individual fitness Figure；

Fig. 7 is the phonon crystal primitive unit cell XY mode that the optimum individual in present example genetic algorithm evolutionary process is corresponding Hybrid mode band gap bound that three-level and four-level are determined and band gap width variation diagram.

Detailed description of the invention

Below in conjunction with Figure of description, the specific embodiment of the present invention is illustrated:

The present embodiment studies the Two dimensional square lattice solid-solid phonon crystal being made up of bi-material, lattice paprmeter a=1m.Material Material A is steel, its density p_A=7780kg/m³, material Lame constants be respectively λ_A=122GPa、μ_A=81GPa；Material B is asphalt mixtures modified by epoxy resin Fat, its density p_B=1180kg/m³, material Lame constants be respectively λ_B=4.43GPa、μ_B=1.59GPa。

The present embodiment comprises the following steps:

1) initialize: the chromosome (individual) of stochastic generation binary number representation constitutes initial population, initial population scale N_pop=60。

2) ideal adaptation degree is calculated: phonon crystal primitive unit cell is divided into 20 × 20 square pixel type structures.By each picture The material (steel or epoxy resin) that element random arrangement is selected, uses string of binary characters form to represent, 1 and 0 represents material respectively Material steel and material epoxy resin.The present invention considers the restriction of operational capability and preparation technology, and research phonon crystal primitive unit cell closes Under the conditions of symmetrical in x, y-axis and rotationally symmetrical about z-axis 90 °, photonic crystal structure topology optimization design problem.Now, only need 1/8 pixel can describe whole primitive unit cell, the individual string of binary characters a length of 55 of heredity.

Use the modified plane wave expansion method improved, calculate the individual corresponding pixel type phonon crystal primitive unit cell XY of each heredity Mode and the dispersion relation of Z mode, obtain the forbidden band numerical value of correspondence.Research finds: before steel-epoxy resin phonon crystal XY mode In eight energy levels, the band gap of the third and fourth energy level and the 6th and the 7th energy inter-stage is easier to open, and every phase of Z mode The band gap of adjacent energy inter-stage is all easily opened.So when steel-epoxy resin photonic crystal hybrid mode band gap is optimized, first fixing The band gap of XY mode, then makes comparisons the band gap of Z mode and the band gap of XY mode, so that it is determined that hybrid guided mode band gap.This example The target of topological optimization is set to the mixing that the band gap of steel-epoxy resin phonon crystal XY mode the third and fourth energy inter-stage is determined Modular belt gap relative value is maximum, it may be assumed that

max: $\frac{\mathrm{\Δ}{\mathrm{\ω}}_3}{\max {\mathrm{\ω}}_3\left(k\right)}=\frac{\min {\mathrm{\ω}}_4\left(k\right)-\max {\mathrm{\ω}}_3\left(k\right)}{\max {\mathrm{\ω}}_3\left(k\right)}$

maxω₃(k) and min ω₄K () is respectively steel-epoxy resin phonon crystal XY mode three-level and four-level The maximum of the hybrid guided mode band gap lower limb eigenfrequency determined and the minima of top edge eigenfrequency.Then with this target Functional value measures the fitness that heredity is individual.

3) perform selection, intersection and mutation genetic operation successively and generate population of future generation, make population evolve forward, the most more New population.Crossing-over rate is 0.5, aberration rate is 0.1～0.8.

4) whether inspection population meets stop condition (population is the most stable).If meeting, exporting optimal result, exiting； Otherwise return step 2).

Through 100 loop iterations, obtain the optimum topological structure that two dimensional steel-epoxy resin XY and Z hybrid mode is corresponding, Its 3 × 3 primitive unit cell is shown in Fig. 4；By Fig. 4 it is found that the hybrid guided mode band gap phase determined by XY mode three-level and four-level To value maximum corresponding to optimum topological structure in, steel (big material constant) periodic arrangement but and be not attached to, be scattering object (figure Black part in 4), matrix material is epoxy resin (white portion in Fig. 4)；Scattering object is approximating square, filling rate It is 50%.The energy band diagram of optimum phonon crystal is shown in Fig. 5, and in figure, transverse axis coordinate is wave vector, and ordinate of orthogonal axes is frequency (unit: kHz)； Owing to using 1/8 model, ask two dimensional steel-epoxy resin phonon crystal XY and Z mode dispersion relation with modified plane wave expansion method Time, wave vector scans along all limits of Γ XM, irreducible Brillouin zone (i.e. Γ → X → M → Γ direction)；Steel-epoxy resin phonon crystal Second energy level that top is Z mode of the hybrid guided mode band gap that XY mode three-level and four-level are determined, and it is XY below The three-level of mode, i.e. this hybrid guided mode band gap are together decided on by XY mode and Z mode；Hybrid guided mode band gap relative width Reaching maximum, the top edge minima of band gap is 1872Hz, and lower limb maximum is 837.4Hz, and band gap width is 1034.6Hz. The average fitness of population and the fitness of optimum individual change (Fig. 6) with optimization process, and in figure, transverse axis coordinate is that heredity is excellent Changing evolution number of times (unit: secondary), ordinate of orthogonal axes is the fitness that XY with Z hybrid mode represents relative to band gap magnitude；Population average Fitness is the meansigma methods of population each ideal adaptation degree, the optimum individual that individuality is this population that wherein fitness is maximum；By Fig. 6 it is found that genetic algorithm optimization evolution early stage, tempo of evolution quickly, and in the later stage, time particularly close to optimal solution, speed Degree is very slow, and the ideal adaptation degree corresponding to optimal solution is 1.235.Steel-epoxy resin phonon crystal XY mode in evolutionary process Fig. 7 is shown in hybrid guided mode band gap bound that three-level and four-level are determined and band gap width change, by Fig. 7 it is found that with And optimize the carrying out evolved, the top edge value minima of band gap is becoming big, and lower limb maximum is diminishing, and the width of band gap is becoming Greatly.

Being shown by result: along with the carrying out of iteration, population is evolved forward, has finally obtained the two dimensional steel-asphalt mixtures modified by epoxy resin of optimum Fat phonon crystal, its hybrid mode band gap fully opens, and band gap magnitude reaches maximum relatively, it is achieved that the target of optimization；This Global Topological optimization method has surmounted the limitation of Experience Design, it is achieved that the active designs of phonon crystal.

Last it is noted that above example only in order to the present invention is described and and unrestricted technology described in the invention Scheme；Therefore, although this specification with reference to each above-mentioned embodiment to present invention has been detailed description, but, this Field it is to be appreciated by one skilled in the art that still the present invention can be modified or equivalent；And all are without departing from sending out The technical scheme of bright spirit and scope and improvement thereof, it all should be contained in the middle of scope of the presently claimed invention.

Claims (5)

1. a two-dimensional solid-solid photonic crystal hybrid mode band gap optimization method, calculates with the modified plane wave expansion method improved Two-dimensional pixel type solid-solid phonon crystal XY and the dispersion relation of Z mode, obtain corresponding band gap magnitude, determine its hybrid mode band Gap；Then Genetic Optimization Algorithm, according to hybrid mode band gap target to be reached, search phonon crystal most optimum materials topology Layout；It is characterized in that: comprise the following steps:

1) initialize: stochastic generation N_popThe chromosome of individual binary number representation, constitutes initial population；

2) ideal adaptation degree is calculated: the wave equation that two-dimensional solid-solid pixel type phonon crystal XY mode and Z mode are met is:

$\left.\begin{array}{c}\frac{{\∂}^2{u}^i}{\∂t^2}=\frac{1}{\mathrm{\ρ}}\{\frac{\∂}{\∂x_i}\left(\mathrm{\λ}\frac{\∂u^l}{\∂x_l}\right)+\frac{\∂}{\∂x_l}\[\mathrm{\μ}(\frac{\∂u^i}{\∂x_l}+\frac{\∂u^l}{\∂x_i})\]\}\\ \frac{{\∂}^2{u}^2}{\∂t^2}=\frac{1}{\mathrm{\ρ}}\▿(\mathrm{\μ}\▿u^z)\end{array}\right\}---\left(1\right)$

Wherein, i, l represent coordinate x, y；uⁱI component for displacement vector u；ρ is density of material；λ and μ is material Lame constants；With The modified plane wave expansion method improved solves equation group, i.e. can get the dispersion relation of XY and Z mode, and then determines XY and Z Hybrid guided mode band gap, the most first fixes the band gap of XY mode, then the band gap of Z mode and the band gap of XY mode is made comparisons, thus really Determine hybrid guided mode band gap；According to the target optimized, with phonon crystal hybrid guided mode band gap structure object function, then with target function value Measure the fitness that heredity is individual；

3) perform selection, intersection and mutation genetic operation successively and generate population of future generation, make population evolve forward；

4) whether inspection population meets stop condition, satisfied then export optimal result, exits；Otherwise return step 3).

A kind of two-dimensional solid-solid photonic crystal hybrid mode band gap optimization method the most according to claim 1, its feature exists In: the individual corresponding phonon crystal primitive unit cell of heredity is tetragonal；Phonon crystal primitive unit cell is divided into M × M square pixel type Structure, wherein M is even number；By any one in two kinds of elastomeric materials of each pixel random arrangement, therefore use binary-coded character String form represents；Phonon crystal primitive unit cell is set as symmetrical about x, y-axis and about 90 ° of rotationally symmetrical models of z-axis.

A kind of two-dimensional solid-solid photonic crystal hybrid mode band gap optimization method the most according to claim 1, its feature exists In: solve two-dimensional solid-solid pixel type phonon crystal XY with the modified plane wave expansion method improved and fluctuation side that Z mode is met During journey group, Fourier transformation g (G) of the group element material constant g of Arbitrary distribution, can calculate according to placement property, wherein group Unit material constant g includes ρ, λ and μ；If P_aFor filled media g in primitive unit cell_aCollection of pixels, first calculate central point pixel P₀∈ P_aMaterial constant g₀Fourier transformation g₀(G)；According to placement property, arbitrary filled media g_aPixel P_r∈P_aMaterial constant Fourier transformation is:

g_r(G)=g₀(G)e^iG·r

Then Fourier transformation g (G) of pixel type phonon crystal material constant distribution is:

Wherein:

IfThe pixel materials constant distribution of primitive cell structure is represented by δ (r)=[δ (r₁),δ(r₂),…δ(r_k),…,δ(r_L)], then:

G (G)=g₀(G)e(G)·δ(r)

Wherein, G is reciprocal lattice vector, r_kRepresenting the center position of kth pixel in primitive unit cell, L is total after representing the segmentation of phonon crystal primitive unit cell Pixel count；

By g (G) offline storage, call for each band gap calculation.

A kind of two-dimensional solid-solid photonic crystal hybrid mode band gap optimization method the most according to claim 1, its feature exists In: the target of hybrid guided mode band gap optimization be set to XY mode third and fourth can the hybrid guided mode band gap that determined of the band gap of inter-stage relative Value maximum, it may be assumed that

$\begin{array}{cc}max:& \frac{{\mathrm{\Δ\ω}}_3}{{\mathrm{max\ω}}_3\left(k\right)}=\frac{{\mathrm{min\ω}}_4\left(k\right)-{\mathrm{max\ω}}_3\left(k\right)}{{\mathrm{max\ω}}_3\left(k\right)}\end{array}$

maxω₃(k) and min ω₄K () is respectively steel-epoxy resin phonon crystal XY mode three-level and four-level is determined The maximum of fixed hybrid guided mode band gap lower limb eigenfrequency and the minima of top edge eigenfrequency.

A kind of two-dimensional solid-solid photonic crystal hybrid mode band gap optimization method the most according to claim 1, its feature exists In: wherein, described step 3) in, selection is the size according to heredity ideal adaptation angle value, uses elitist selection and roulette phase Binding mechanism；Intersect to operate and use average cross mode, randomly choose chromosome and implement to go to intersect or arrange to intersect, to strengthen algorithm Ability of searching optimum；Individual variation uses position Variation mechanism.