CN116384138B - Phonon crystal topology optimization method and system containing specific band gap - Google Patents
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Abstract
The invention belongs to the technical field of structural topology optimization, and provides a phonon crystal topology optimization method and a phonon crystal topology optimization system containing specific band gaps, wherein a frequency band constraint expression and an identical side frequency constraint expression are established at first, and the sensitivity of the frequency band constraint expression and the identical side frequency constraint expression is calculated respectively; then integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to sensitivity, establishing a topology optimization model for designing a specific band gap of the photonic crystal, controlling the formation of the elastic band gap by utilizing the same-side frequency constraint auxiliary frequency band constraint together, obtaining the photonic crystal with the specific band gap, and controlling the formation of the elastic band gap by utilizing the same-side frequency constraint auxiliary frequency band constraint together; the sensitivity is integrated into a traditional dynamic topology optimization model by combining two constraint conditions of frequency bands and same-side frequencies, a solving method meeting the universality and robustness required by optimization iteration is established, and the specific band gap design of the phonon crystal in a specified frequency range is realized.
Description
Technical Field
The invention belongs to the technical field of structural topology optimization, and particularly relates to a phonon crystal topology optimization method and system containing specific band gaps.
Background
Phonon crystals block the propagation of elastic waves/sound waves in a specific frequency range through the periodic arrangement of unit cells, and are widely applied to the fields of acoustic function devices, vibration reduction, noise reduction and the like. However, the work about photonic crystal band gap in the prior study is still limited to the optimal model with the largest adjacent two-order band gap, and is difficult to adapt to the actual requirements of engineering structures and devices on the band gap with the specified frequency. Compared with the addition of the structural band constraint, the difficulty of the specific band gap design of the photonic crystal is that the rigidity matrix changes along with the wave vector change, and the problem of typical multi-working-condition characteristic values is solved, and the direct addition of the band constraint often encounters the difficult problems of convergence and the like.
Phonon crystal topology optimization design is always an important research content in the field of novel metamaterials; the inventor finds that in the phonon crystal topology optimization method with the largest adjacent two-order band gaps, which is proposed in the prior art, the structural mode order is required to be predetermined, and an accurate band gap range cannot be given; in addition, at present, a complete scheme is not formed in the field of topology optimization with respect to photonic crystal structure design containing specific band gaps.
Disclosure of Invention
In order to solve the problems, the invention provides a phonon crystal topology optimization method and a system containing a specific band gap, wherein the formation of the elastic wave band gap is controlled jointly by the same-side frequency constraint and the auxiliary band constraint; by combining two constraint conditions of frequency bands and same-side frequencies, integrating the sensitivity of the constraint function into a traditional dynamic topology optimization model through deduction of the sensitivity, a solving method meeting the universality and robustness required by optimization iteration is established, and the specific band gap design of the phonon crystal in a specified frequency range is realized.
In order to achieve the above object, the present invention is realized by the following technical scheme:
In a first aspect, the present invention provides a photonic crystal topology optimization method including a specific band gap, including:
establishing a frequency band constraint expression and a same-side frequency constraint expression, and respectively calculating the sensitivity of the frequency band constraint expression and the sensitivity of the same-side frequency constraint expression;
integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to the calculated two sensitivity, and establishing the topology optimization model;
and (3) carrying out specific band gap design on the photonic crystal through a topological optimization model, and controlling the formation of the elastic wave band gap by utilizing the constraint of the auxiliary frequency band of the same side to obtain the photonic crystal with the specific band gap.
Furthermore, before optimization, a periodic boundary condition is applied to an elastic wave control equation of the phonon crystal, and discretization is carried out by adopting a finite element method.
Further, for different wave vectors, the band constraint expression is improved, and the improved band constraint expression is obtained as follows:
Wherein ω j-max and ω j-min represent the maximum characteristic frequency and the minimum characteristic frequency of all wave vectors of the phonon crystal on the j-th order dispersion curve, respectively; FB m(ωj-max) and FB m(ωj-min) represent a maximum band constraint function and a minimum band constraint function, respectively; j represents the number of structural characteristic frequencies; m represents the number of band gaps.
Further, ω j-max and ω j-min were condensed using KS function:
Wherein, gamma is the coagulation coefficient of KS function; omega j,k(n) represents the structural characteristic frequency corresponding to the nth wave vector point on the j-order dispersion curve; k (n) is the nth wave vector point; n represents the number of wave vector points required.
Further, the ipsilateral frequency constraint expression is:
IFC＝IFm(ωj-max)-IFm(ωj-min)≤ε,j＝1,...,Jm＝1,...,M
Wherein IFC is ipsilateral frequency constraint; epsilon is a threshold value, IF m(ωj-max) and IF m(ωj-min) are a maximum same-side frequency constraint function and a minimum same-side frequency constraint function respectively, and omega j-max and omega j-min respectively represent the maximum characteristic frequency and the minimum characteristic frequency of all wave vectors of the phonon crystal on a j-th order dispersion curve; j represents the number of structural characteristic frequencies; m represents the number of band gaps.
Further, the ipsilateral frequency constraint is approximated by a modified Heaviside function:
Wherein x is frequency; is the intermediate frequency value of the band gap; m is the number of band gaps; beta 2 is steep.
Furthermore, the design variables in the topological optimization model are iteratively updated, so that the optimization design of the phonon crystal structure is completed.
In a second aspect, the present invention also provides a photonic crystal topology optimization system containing a specific band gap, including:
A computing module configured to: establishing a frequency band constraint expression and a same-side frequency constraint expression, and respectively calculating the sensitivity of the frequency band constraint expression and the sensitivity of the same-side frequency constraint expression;
the topology optimization model building module is configured to: integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to the calculated two sensitivity, and establishing the topology optimization model;
An optimization module configured to: and (3) carrying out specific band gap design on the photonic crystal through a topological optimization model, and controlling the formation of the elastic wave band gap by utilizing the constraint of the auxiliary frequency band of the same side to obtain the photonic crystal with the specific band gap.
In a third aspect, the present invention also provides a computer readable storage medium having stored thereon a computer program which when executed by a processor implements the steps of the photonic crystal topology optimization method of the first aspect with a specific bandgap.
In a fourth aspect, the present invention also provides an electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the photonic crystal topology optimization method with specific band gaps according to the first aspect when executing the program.
Compared with the prior art, the invention has the beneficial effects that:
1. Firstly, establishing a frequency band constraint expression and an ipsilateral frequency constraint expression, and respectively calculating the sensitivity of the frequency band constraint expression and the sensitivity of the ipsilateral frequency constraint expression; then integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to the calculated two sensitivities, establishing the topology optimization model, designing a specific band gap of the photonic crystal through the topology optimization model, and jointly controlling the formation of an elastic wave band gap by utilizing the same-side frequency constraint auxiliary frequency band constraint to obtain the photonic crystal with the specific band gap; by combining two constraint conditions of frequency bands and same-side frequencies, integrating the sensitivity of the constraint function into a traditional dynamic topology optimization model through deduction of the sensitivity, a solving method meeting the universality and robustness required by optimization iteration is established, and the specific band gap design of the phonon crystal in a specified frequency range is realized.
2. The invention aims at different wave vectors, improves the band constraint expression, and simultaneously provides the ipsilateral frequency constraint with continuous micro characteristics, and the auxiliary improved band constraint jointly controls the formation of the elastic band gap.
3. The phonon crystal with the specific band gap obtained by the invention has the natural frequency outside the range of the external excitation elastic wave/sound wave, ensures that the designed phonon crystal blocks the propagation of the elastic wave/sound wave in the corresponding range in the actual working and running process, and realizes the functions of vibration reduction and noise reduction.
Drawings
The accompanying drawings, which are included to provide a further understanding of the embodiments and are incorporated in and constitute a part of this specification, illustrate and explain the embodiments and together with the description serve to explain the embodiments.
FIG. 1 is a flow chart of embodiment 1 of the present invention;
FIG. 2 is an initial single cell diagram of phonon crystal according to example 1 of the present invention;
FIG. 3 is a graph showing initial dispersion of phonon crystals according to example 1 of the present invention;
FIG. 4 is a graph of a phonon crystal topology optimized unit cell according to example 1 of the present invention;
FIG. 5 is a graph of dispersion after phonon crystal topology optimization according to example 1 of the present invention;
FIG. 6 shows phonon crystals spliced from 5X 5 unit cells after topology optimization according to example 1 of the present invention;
FIG. 7 is a graph of a sweep frequency of the photonic crystal depicted in FIG. 6 in accordance with embodiment 1 of the present invention;
Wherein, 1, epoxy resin; 2. lead.
Detailed Description
The invention will be further described with reference to the drawings and examples.
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the application. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
The specific band gap in the present invention is expressed as phonon crystal band gap in the specified frequency range, for example, the specific band gap range can be [6000, 7000] Hz and [9000, 11000] Hz, and can be specified as other ranges.
Example 1:
at present, in the existing phonon crystal topology optimization method with the maximum adjacent two-order band gaps, the structural mode order to be considered needs to be predetermined, and an accurate band gap range cannot be given. Compared with the addition of the structural band constraint, the difficulty of the specific band gap design of the photonic crystal is that the rigidity matrix changes along with the wave vector change, and the problem of typical multi-working-condition characteristic values is solved, and the direct addition of the band constraint often encounters the difficult problems of convergence and the like.
In view of the above problems, the present embodiment provides a photonic crystal topology optimization method including a specific band gap, which establishes a band constraint expression and an ipsilateral frequency constraint expression, and calculates sensitivity of the band constraint expression and sensitivity of the ipsilateral frequency constraint expression, respectively; then integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to sensitivity, establishing the topology optimization model, designing a specific band gap of the photonic crystal through the topology optimization model, controlling the formation of the elastic band gap by utilizing the same-side frequency constraint auxiliary frequency band constraint to obtain the photonic crystal containing the specific band gap, and controlling the formation of the elastic band gap by utilizing the same-side frequency constraint auxiliary frequency band constraint; by combining two constraint conditions of frequency bands and same-side frequencies, integrating the sensitivity of the constraint function into a traditional dynamic topology optimization model through deduction of the sensitivity, a solving method meeting the universality and robustness required by optimization iteration is established, and the specific band gap design of the phonon crystal in a specified frequency range is realized.
Optionally, before optimization, applying periodic boundary conditions to an elastic wave control equation of the phonon crystal according to the Floque-Bloch theorem, and performing discretization processing by adopting a finite element method to obtain a finite element equation related to a wave vector:
(K(k)-ω2M)u＝0 (1)
Wherein K and M respectively represent an overall stiffness matrix and an overall mass matrix of the structure; k is a wave vector; ω and u are the feature frequency and feature vector of the structure, respectively.
As shown in fig. 2, the initial unit cell configuration of the photonic crystal is determined, the size can be set to 0.1m×0.1m, the material is composed of lead 2 (elastic modulus e=40.8 Gpa) and epoxy 1 (elastic modulus e=4.35 Gpa), the initial dispersion curve of the photonic crystal is shown in fig. 3, and no band gap is found in [0, 13000] hz; the unit cells are meshed according to the actual conditions and boundary conditions are added, and the unit cell density ρ e (e=1,., 6400) in the design domain is used as a design variable. Calculating a cell stiffness matrix K e (e=1,..6400) and a cell mass matrix M e (e=1,..6400), and establishing an overall stiffness matrix K and an overall mass matrix M;
and (3) applying frequency band constraint and ipsilateral frequency constraint, establishing a topological optimization model, and carrying out bi-material specific band gap design on the photonic crystal through topological optimization, wherein the specific band gap range can be as follows: [6000, 7000] Hz and [9000, 11000] Hz, and obtaining the phonon crystal meeting the requirements.
Specifically, in order to explain the technical solution of the present embodiment, specific steps are described.
S1, applying a periodic boundary condition:
Wherein r is a position vector;
S2, band constraint is applied:
Wherein ω j-max and ω j-min represent maximum and minimum characteristic frequencies of all wave vectors of the phonon crystal on the jth order dispersion curve, FB m(ωj-max) and FB m(ωj-min) represent maximum and minimum frequency band constraint functions, respectively; j represents the number of structural characteristic frequencies, M represents the number of band gaps; optionally, the number of structural characteristic frequencies j=6 and the number of band gaps m=2;
s3, applying same-side frequency constraint:
IFC＝IFm(ωj-max)-IFm(ωj-min)≤ε,j＝1,...,J m＝1,...,M (4)
Wherein IFC is the same-side frequency constraint, IF m(ωj-max) and IF m(ωj-min) are maximum and minimum same-side frequency constraint functions, respectively; epsilon is a threshold and the initial value of epsilon can be set to 1, which is every 30 steps epsilon=max { epsilon-0.1, 0.3} with the number of iterations.
The functional expression of the band constraint function FB m in step S2 may be:
wherein, Is the intermediate frequency value of the band gap,/>Β determines the steepness of the function, which is half the band gap width; alternatively,/> The initial value of β may be 1, which every 50 steps β=min {1+β,16} with the number of iterations.
S5, omega j-max and omega j-min in the step S2 can be condensed by using KS function:
Wherein ω j,k(n) represents the structural characteristic frequency corresponding to the nth wave vector point on the j-order dispersion curve; gamma > 0 is the coacervation coefficient of the KS function, and the initial value of gamma may be 5, which is y=min { γ+1,40} every 15 steps with the number of iterations;
The ipsilateral frequency constraint in step S3 is approximated by a modified Heaviside function:
Where β 2 determines the steepness of the function, the initial value of β 2 may be 10, which is every 30 steps of β 2＝min{1+β2, 40 with the number of iterations.
S7, respectively solving the first derivative of the design variable according to the function in the step S2-the step S6, carrying out iterative updating on the design variable in the topology optimization model by adopting an MMA algorithm, and completing the optimization design of the phonon crystal structure, wherein the calculation steps are as follows:
Sensitivity of band constraints:
according to equation (3), the derivative of the band-constraint function with respect to the design variable can be expressed as:
wherein, And/>Can be obtained by differentiating the formula (5):
According to formula (6), the derivative of omega j-max and omega j-min characteristic frequency omega j,k is obtained:
Sensitivity of ipsilateral frequency constraint:
the derivative of the ipsilateral frequency constraint function with respect to the design variable can be expressed as follows:
wherein, And/>Can be obtained by differentiating the formula (7):
Putting constraint function sensitivity obtained by solving into an optimization list, defining a non-target optimization problem meeting frequency band constraint and ipsilateral frequency constraint in a corresponding topology model, and establishing the topology optimization model as follows:
where λ and ρ e are design variables, superscript represents the indices d, i and e, The cell densities of the diffusion field, the intermediate field, and the corrosion field, respectively.
And (3) carrying out iterative updating on the design variables by adopting an MMA algorithm to finish the optimal design of the phonon crystal. The obtained single cell diagram after phonon crystal topology optimization is shown in figure 4; the dispersion plot after phonon crystal topology optimization is shown in fig. 5, and it can be seen from fig. 5 that elastic wave/acoustic wave specific band gaps are generated in the applied constraints [6000, 7000] hz and [9000, 11000] hz; the phonon crystal graph spliced by 5×5 single cells after topology optimization is shown in fig. 6; the sweep analysis of the photonic crystal described in fig. 6 is performed, and the resulting sweep graph is shown in fig. 7. It can be seen from fig. 7 that specific band gaps of elastic wave/sound wave are generated in the range of the applied constraints of [6000, 7000] hz and [9000, 11000] hz, and the accuracy of the topology optimization method of the photonic crystal with specific band gaps is verified.
Aiming at different wave vectors, the embodiment improves the band constraint expression in the prior art, and simultaneously proposes another ipsilateral frequency constraint with continuous micro characteristics, and the improved band constraint is assisted to jointly control the formation of the elastic band gap; the constraint conditions of frequency band constraint and same-side frequency constraint are combined, sensitivity of a constraint function is deduced and integrated into a traditional dynamic topology optimization model, a solving method meeting the universality and robustness required by optimization iteration is established, and the specific band gap design of the phonon crystal in a specified frequency range is realized.
Example 2:
the embodiment provides a photonic crystal topology optimization system with specific band gaps, which comprises the following components:
A computing module configured to: establishing a frequency band constraint expression and a same-side frequency constraint expression, and respectively calculating the sensitivity of the frequency band constraint expression and the sensitivity of the same-side frequency constraint expression;
the topology optimization model building module is configured to: integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to the calculated two sensitivity, and establishing the topology optimization model;
An optimization module configured to: and (3) carrying out specific band gap design on the photonic crystal through a topological optimization model, and controlling the formation of the elastic wave band gap by utilizing the constraint of the auxiliary frequency band of the same side to obtain the photonic crystal with the specific band gap.
The working method of the system is the same as that of the photonic crystal topology optimization method containing the specific band gap in embodiment 1, and is not repeated here.
Example 3:
The present embodiment provides a computer-readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of the photonic crystal topology optimization method with specific band gaps described in embodiment 1.
Example 4:
The present embodiment provides an electronic device including a memory, a processor, and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the photonic crystal topology optimization method including a specific bandgap of embodiment 1 when executing the program.
The above description is only a preferred embodiment of the present embodiment, and is not intended to limit the present embodiment, and various modifications and variations can be made to the present embodiment by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present embodiment should be included in the protection scope of the present embodiment.
Claims (6)
1. A method for optimizing photonic crystal topology including a specific band gap, comprising:
Establishing a frequency band constraint expression and a same-side frequency constraint expression, and respectively calculating the sensitivity of the frequency band constraint expression and the sensitivity of the same-side frequency constraint expression; aiming at different wave vectors, the frequency band constraint expression is improved, and the improved frequency band constraint expression is:
Wherein ω j-max and ω j-min represent the maximum characteristic frequency and the minimum characteristic frequency of all wave vectors of the phonon crystal on the j-th order dispersion curve, respectively; FB m(ωj-max) and FB m(ωj-min) represent a maximum band constraint function and a minimum band constraint function, respectively; j represents the number of structural characteristic frequencies; m represents the number of band gaps; omega j-max and omega j-min were condensed with the KS function:
Wherein, gamma is the coagulation coefficient of KS function; omega j,k(n) represents the structural characteristic frequency corresponding to the nth wave vector point on the j-order dispersion curve; k (n) is the nth wave vector point; n represents the number of wave vector points required;
The ipsilateral frequency constraint expression is:
IFC＝IFm(ωj-max)-IFm(ωj-min)≤ε,j＝1,...,J m＝1,...,M
Wherein IFC is ipsilateral frequency constraint; epsilon is a threshold value, IF m(ωj-max) and IF m(ωj-min) are a maximum same-side frequency constraint function and a minimum same-side frequency constraint function respectively, and omega j-max and omega j-min respectively represent the maximum characteristic frequency and the minimum characteristic frequency of all wave vectors of the phonon crystal on a j-th order dispersion curve; j represents the number of structural characteristic frequencies; m represents the number of band gaps; the ipsilateral frequency constraint is approximated by a modified Heaviside function:
Wherein x is frequency; is the intermediate frequency value of the band gap; m is the number of band gaps; beta 2 is the degree of steepness;
integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to the calculated two sensitivity, and establishing the topology optimization model;
and (3) carrying out specific band gap design on the photonic crystal through a topological optimization model, and controlling the formation of the elastic wave band gap by utilizing the constraint of the auxiliary frequency band of the same side to obtain the photonic crystal with the specific band gap.
2. A photonic crystal topology optimization method of defined in claim 1, wherein prior to optimization, periodic boundary conditions are applied to the elastic wave control equation of the photonic crystal and discretization is performed using a finite element method.
3. The method for optimizing photonic crystal topology including a specific band gap as recited in claim 1, wherein the design variables in the topology optimization model are iteratively updated to complete the optimization design of the photonic crystal structure.
4. A photonic crystal topology optimization system having a specific band gap, comprising:
a computing module configured to: establishing a frequency band constraint expression and a same-side frequency constraint expression, and respectively calculating the sensitivity of the frequency band constraint expression and the sensitivity of the same-side frequency constraint expression; aiming at different wave vectors, the frequency band constraint expression is improved, and the improved frequency band constraint expression is:
Wherein ω j-max and ω j-min represent the maximum characteristic frequency and the minimum characteristic frequency of all wave vectors of the phonon crystal on the j-th order dispersion curve, respectively; FB m(ωj-max) and FB m(ωj-min) represent a maximum band constraint function and a minimum band constraint function, respectively; j represents the number of structural characteristic frequencies; m represents the number of band gaps; omega j-max and omega j-min were condensed with the KS function:
Wherein, gamma is the coagulation coefficient of KS function; omega j,k(n) represents the structural characteristic frequency corresponding to the nth wave vector point on the j-order dispersion curve; k (n) is the nth wave vector point; n represents the number of wave vector points required;
The ipsilateral frequency constraint expression is:
IFC＝IFm(ωj-max)-IFm(ωj-min)≤ε,j＝1,...,J m＝1,...,M
Wherein IFC is ipsilateral frequency constraint; epsilon is a threshold value, IF m(ωj-max) and IF m(ωj-min) are a maximum same-side frequency constraint function and a minimum same-side frequency constraint function respectively, and omega j-max and omega j-min respectively represent the maximum characteristic frequency and the minimum characteristic frequency of all wave vectors of the phonon crystal on a j-th order dispersion curve; j represents the number of structural characteristic frequencies; m represents the number of band gaps; the ipsilateral frequency constraint is approximated by a modified Heaviside function:
Wherein x is frequency; is the intermediate frequency value of the band gap; m is the number of band gaps; beta 2 is the degree of steepness;
the topology optimization model building module is configured to: integrating the frequency band constraint expression and the same-side frequency constraint expression into a dynamic topology optimization model according to the calculated two sensitivity, and establishing the topology optimization model;
An optimization module configured to: and (3) carrying out specific band gap design on the photonic crystal through a topological optimization model, and controlling the formation of the elastic wave band gap by utilizing the constraint of the auxiliary frequency band of the same side to obtain the photonic crystal with the specific band gap.
5. A computer-readable storage medium, on which a computer program is stored, characterized in that the program, when being executed by a processor, implements the steps of the photonic crystal topology optimization method comprising a specific band gap as claimed in any one of claims 1-3.
6. An electronic device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the processor implements the steps of the photonic crystal topology optimization method of any one of claims 1-3 that includes a specific bandgap when the program is executed.
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