CN116205187B - Rapid simulation method, system and related equipment for multi-symmetrical surface acoustic wave device - Google Patents
Rapid simulation method, system and related equipment for multi-symmetrical surface acoustic wave device Download PDFInfo
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Abstract
The application is suitable for the technical field of piezoelectric material power electric coupling, and particularly relates to a rapid simulation method, a rapid simulation system and related equipment of a multi-symmetrical surface acoustic wave device. Aiming at the special structure of the progressive multi-symmetrical surface acoustic wave filter and the mathematical relation of matrix symmetry in the finite element method, the application reduces the construction time of a large number of subunit matrixes and the calculation time for splicing the matrixes by the hierarchical cascading technology by utilizing the inverse and equivalent relation of the matrixes in the simulation process of the surface acoustic wave filter, thereby improving the simulation speed of the progressive multi-symmetrical surface acoustic wave filter and improving the simulation efficiency.
Description
Technical Field
The application is suitable for the technical field of piezoelectric material power electric coupling, and particularly relates to a rapid simulation method, a rapid simulation system and related equipment of a multi-symmetrical surface acoustic wave device.
Background
With the development of wireless technology, the demand for acoustic filter elements in smart communication devices is increasing. To verify the feasibility of the solution, further improving the device performance, designers often use the Finite Element Method (FEM) to accurately simulate acoustic filters. The finite element method is based on a partial differential equation of a physical model, specifically, a continuous area of a known boundary condition is discretely divided into a plurality of grids, and then an integral equation is built by reconnecting grid nodes, so that an accurate numerical solution of the model is obtained by solving. Taking a Surface Acoustic Wave (SAW) resonator as an example, the finite element method can calculate variables such as displacement and potential of any material and shape, and any complex layered structure device. However, the simulation of the full three-dimensional size of the surface acoustic wave resonator needs to split a large number of grids, consumes a large amount of computing resources, and is difficult to achieve in practical situations, and a two-dimensional or quasi-three-dimensional structure is generally adopted under the periodic boundary condition, so that a part of accuracy is sacrificed for time and computing cost.
The proposal of the Hierarchical Cascading Technology (HCT) solves the problem of computational resources in the problem, the method utilizes the periodic structural characteristics of interdigital in the surface acoustic wave resonator, the complete device is divided by taking an electrode corresponding area as a subunit, each subunit eliminates the internal degree of freedom through a Schur (Schur) compensation algorithm, only a small amount of degrees of freedom at the edge and the electrode are left, and the step has no loss of precision; the adjacent edges of the subunits are spliced and the degree of freedom of the contact edges is eliminated, and the matrix of the complete device can be obtained by repeating the above steps, so that the response of the filter can be easily calculated. Because the resource consumption required by simulation is greatly reduced, and the calculation accuracy is reserved to the greatest extent, the hierarchical cascading technology becomes one of the hot methods of finite element simulation of the surface acoustic wave device, and subunits with the same size do not need to repeatedly generate matrixes when the hierarchical cascading technology calculates, because the corresponding matrixes are necessarily the same under the condition of the same structure, and for the traditional surface acoustic wave resonator with strong periodic characteristics, the periodic subunits can be quickly cascaded into a whole in an exponential growth mode of 2 of 1, 2, 4, 8 and the like, which is an important step for realizing acceleration in the hierarchical cascading technology algorithm.
However, with the development of mobile technology, the requirements of the saw filter for low loss and large bandwidth are increasing, so that the resonator fingers with the traditional periodic structure cannot meet the requirements, and thus, a high-performance structure such as a progressive dual-mode coupling saw filter (DMS) appears. The progressive dual-mode coupling surface acoustic wave filter is characterized by comprising a plurality of interdigital transducers (IDTs), wherein the interdigital sizes of the interdigital transducers are not consistent, and the design is beneficial to expanding the bandwidth; several, even tens of, size-changing fingers are arranged between adjacent interdigital transducers for transition so as to reduce acoustic wave propagation loss between the adjacent interdigital transducers. A progressive dual-mode-coupled surface acoustic wave filter may include several tens or even hundreds of interdigital signals with different sizes, which means that a corresponding number of sub-unit matrices need to be constructed before the progressive dual-mode-coupled surface acoustic wave filter is simulated by using a hierarchical cascading technique method, and in the cascading process, the periodic structure is reduced and discontinuous, and the progressive dual-mode-coupled surface acoustic wave filter is difficult to apply in an exponentially growing cascading manner, and can only be cascaded sequentially, so that the calculation time of the hierarchical cascading technique in the simulation process is greatly increased.
Disclosure of Invention
The application provides a rapid simulation method, a rapid simulation system and a rapid simulation related device for a multi-symmetrical surface acoustic wave device, and aims to solve the problem of large calculation amount when the multi-symmetrical surface acoustic wave device is simulated based on a finite element method and a hierarchical cascading technology in the prior art.
In a first aspect, the present application provides a method for rapid simulation of a multiple symmetric surface acoustic wave device, the method comprising the steps of:
s1, acquiring the overall geometric structure of a multi-symmetrical surface acoustic wave device, wherein the multi-symmetrical surface acoustic wave device comprises a plurality of interdigital transducers which are symmetrically arranged according to a periodic rule;
s2, judging whether the multi-symmetrical surface acoustic wave device meets central symmetry or not: if yes, marking the central symmetry position of the surface acoustic wave device, wherein the central pair is called plane axisymmetry;
s3, judging whether the region formed by connecting the interdigital transducers meets central symmetry, if so, marking the region formed by connecting the interdigital transducers meeting central symmetry as a symmetrical interdigital transducer region;
s4, judging whether any two interdigital transducers which do not belong to the symmetrical interdigital transducer area meet central symmetry, and if yes, marking the met interdigital transducers as symmetrical single interdigital transducers;
s5, dividing the surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer, and calculating a basic finite element matrix of the basic subunits meeting preset symmetry and periodicity conditions by using a finite element method;
s6, judging whether the symmetrical single interdigital transducer is marked in the step S4, if so, calculating the basic finite element matrix on the opposite side of central symmetry according to a matrix symmetry rule according to the basic finite element matrix of the symmetrical single interdigital transducer;
s7, judging whether the symmetrical interdigital transducer area is marked in the step S3, if so, splicing the basic finite element matrixes of the interdigital transducers on the same side of the central symmetry in the symmetrical interdigital transducer area according to a hierarchical cascading method to obtain an area matrix, and calculating the area matrix on different sides of the central symmetry according to a matrix symmetry rule;
s8, judging whether the central symmetry position is marked in the step S2, if so, splicing the basic finite element matrix and/or the regional matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure until the central symmetry position is reached to obtain a semi-overall matrix of the multi-symmetrical surface acoustic wave device, and then calculating the overall simulation matrix of the surface acoustic wave device by the semi-overall matrix according to a matrix symmetry rule;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
Still further, the base subunit includes an interdigital subunit and a gap subunit.
Still further, the preset symmetry and periodicity conditions include:
the interdigital subunits that do not satisfy central symmetry;
one of the interdigital subunits satisfying a central symmetry and not belonging to a periodic structure in the multiple symmetric surface acoustic wave device;
one of the interdigital subunits belonging to a periodic structure in the multi-symmetric surface acoustic wave device;
and the gap sub-units with different structures in the multi-symmetrical surface acoustic wave device.
Further, in step S5, the method further includes the steps of:
and eliminating the internal degree of freedom of the basic finite element matrix by using a Shu's complement operation, and reserving the displacement degrees of freedom of the left and right edges of the basic finite element matrix and the internal electrode degrees of freedom.
Still further, the matrix symmetry rule is:
defining the displacement degrees of freedom of the basic finite element matrixes in the x, y and z directions as u, V and w, respectively, and the potential degrees of freedom as V, wherein the displacement degrees of freedom in the x direction between the basic finite element matrixes symmetrical about the center are opposite, and the displacement degrees of freedom in the x and y directions and the potential degrees of freedom V are kept equal correspondingly.
Further, in step S7, the method further includes the steps of:
and eliminating the internal degree of freedom of the area matrix by using a Shull complement operation, and reserving the displacement degrees of freedom of the left and right edges of the area matrix and the internal electrode degrees of freedom.
Further, in step S8, the method further includes the steps of:
and eliminating the internal degrees of freedom of the semi-integral matrix and/or the integral simulation matrix by using a Shuhr complement operation, and reserving the displacement degrees of freedom and the internal electrode degrees of freedom of the left and right edges of the semi-integral matrix and/or the integral simulation matrix.
In a second aspect, the present application also provides a rapid simulation system of a multiple symmetric surface acoustic wave device, the rapid simulation system comprising:
the geometrical acquisition module is used for acquiring the overall geometrical structure of the multi-symmetrical surface acoustic wave device, and the multi-symmetrical surface acoustic wave device comprises a plurality of interdigital transducers which are symmetrically arranged according to a periodic rule;
the overall symmetry judging module is used for judging whether the multi-symmetry surface acoustic wave device meets central symmetry or not: if yes, marking the central symmetry position of the surface acoustic wave device, wherein the central pair is called plane axisymmetry;
the region symmetry judging module is used for judging whether the region formed by connecting the interdigital transducers meets the central symmetry or not, and if so, the region formed by connecting the interdigital transducers meets the central symmetry is marked as a symmetric interdigital transducer region;
the single-finger symmetry judgment module is used for judging whether any two interdigital transducers which do not belong to the symmetrical interdigital transducer area meet central symmetry, and if yes, the met interdigital transducers are marked as symmetrical single-finger interdigital transducers;
the finite element calculation module is used for dividing the surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer and calculating a basic finite element matrix of the basic subunits meeting preset symmetry and periodicity conditions by using a finite element method;
the single-finger symmetry processing module is used for judging whether the single-finger symmetry judging module marks the symmetrical single-finger transducer, if so, calculating the basic finite element matrix on the opposite side of central symmetry according to a matrix symmetry rule according to the basic finite element matrix of the symmetrical single-finger transducer;
the region symmetry processing module is used for judging whether the region symmetry judging module marks the symmetric interdigital transducer region, if so, splicing the basic finite element matrixes of the interdigital transducers on the same side of the central symmetry in the symmetric interdigital transducer region according to a hierarchical cascading method to obtain a region matrix, and calculating the region matrix on the opposite side of the central symmetry according to a matrix symmetry rule;
the overall symmetry processing module is used for judging whether the overall symmetry judging module marks the central symmetry position, if so, the basic finite element matrix and/or the area matrix are spliced according to the hierarchical cascading method according to the sequence of the overall geometric structure until the central symmetry position is reached to obtain a semi-overall matrix of the multi-symmetry surface acoustic wave device, and then the semi-overall matrix is used for calculating an overall simulation matrix of the surface acoustic wave device according to a matrix symmetry rule;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
In a third aspect, the present application also provides a computer device comprising: a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps in the method for fast simulation of a multiple symmetric surface acoustic wave device as in any of the embodiments above when the computer program is executed.
In a fourth aspect, an embodiment of the present application further provides a computer readable storage medium, where a computer program is stored, where the computer program when executed by a processor implements the steps in the method for fast simulation of a multiple symmetric surface acoustic wave device according to any of the above embodiments.
The application has the beneficial effects that aiming at the special structure of the progressive multi-symmetrical surface acoustic wave filter and the mathematical relation of matrix symmetry in the finite element method, the inversion and peer relation of the matrix is utilized in the simulation process of the surface acoustic wave filter, so that the construction time of a large number of subunit matrixes and the calculation time of splicing the matrixes by a hierarchical cascading technology are reduced, the simulation speed of the progressive multi-symmetrical surface acoustic wave filter is improved, and the simulation efficiency is improved.
Drawings
Fig. 1 is a schematic step flow diagram of a method for fast simulation of a multiple symmetric surface acoustic wave device according to an embodiment of the present application;
FIG. 2 is a schematic diagram of a structure of a progressive multi-symmetric dual-mode coupled SAW filter;
FIG. 3 is a schematic diagram of a multi-symmetrical dual-mode coupling SAW filter according to an embodiment of the present application;
FIG. 4 is a schematic diagram of a subunit provided in an embodiment of the application;
FIG. 5 is a schematic diagram of symmetry of a basic finite element matrix provided by an embodiment of the present application;
FIG. 6 is a schematic diagram of symmetry of a region matrix provided by an embodiment of the present application;
fig. 7 is a schematic structural diagram of a rapid simulation system 200 of a multiple symmetric surface acoustic wave device according to an embodiment of the present application;
fig. 8 is a schematic structural diagram of a computer device according to an embodiment of the present application.
Detailed Description
The present application will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present application more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the application.
Referring to fig. 1, fig. 1 is a schematic step flow diagram of a rapid simulation method of a multiple symmetric surface acoustic wave device according to an embodiment of the present application, where the rapid simulation method includes the following steps:
s1, acquiring the overall geometric structure of a multi-symmetrical surface acoustic wave device, wherein the multi-symmetrical surface acoustic wave device comprises a plurality of interdigital transducers which are symmetrically arranged according to a periodic rule.
Referring to fig. 2, fig. 2 is a schematic structural diagram of a progressive multi-symmetrical dual-mode coupling surface acoustic wave filter, where the overall geometry of the embodiment of the present application includes electrode thickness, metallization rate, finger period (pitch), interdigital number, etc. of the filter, and reference numerals 1, 2, and 3 in fig. 2 are respectively interdigital transducers (IDTs) with different center-to-center distances, and two sides of the interdigital transducers are reflective gratings with other finger periods. The interdigital transducer and the edge of the reflecting grating have different finger periods with the center, and generally have a regular arrangement mode.
S2, judging whether the multi-symmetrical surface acoustic wave device meets central symmetry or not: if yes, marking the central symmetry position of the surface acoustic wave device, wherein the central pair is called plane axisymmetry.
In the prior art, the progressive multi-symmetrical dual-mode coupling surface acoustic wave filter is of a central symmetrical structure, because the filter is designed to be symmetrical, and three transit echoes can be obviously restrained. In the embodiment of the application, if the surface acoustic wave filter meets the central symmetry, the central symmetry position of the surface acoustic wave filter is marked so as to construct an integral matrix according to the matrix symmetry at the central symmetry position in the subsequent simulation calculation. The central symmetry in the embodiments of the present application is with respect to the overall geometry of the dual mode coupled surface acoustic wave filter, unless specifically stated. Exemplary embodiments. As shown in fig. 3, fig. 3 is a schematic diagram of symmetry lines of a multi-symmetrical dual-mode coupling surface acoustic wave filter according to an embodiment of the present application, where a symmetry line 2 in fig. 3 corresponds to the central symmetry position.
And S3, judging whether the region formed by connecting the interdigital transducers meets the central symmetry, and if so, marking the region formed by connecting the interdigital transducers meeting the central symmetry as a symmetrical interdigital transducer region.
As shown in fig. 3, the symmetry line 1 is the center of symmetry of the symmetric interdigital transducer region. When the area formed by the connection of the interdigital transducers is not limited to a boundary specifically, if the central symmetrical position is marked in step S2, the area formed by the connection of the interdigital transducers on one side of the symmetry is sequentially determined, that is, a plurality of mutually non-overlapping areas of the symmetrical interdigital transducers may be obtained in the whole dual-mode coupling surface acoustic wave filter.
S4, judging whether any two interdigital transducers which do not belong to the symmetrical interdigital transducer area meet central symmetry, and if yes, marking the met interdigital transducers as symmetrical single interdigital transducers.
As shown in fig. 3, the region where the symmetry line 3 is located is the symmetry center of the symmetrical single interdigital transducer. The symmetrical single interdigital transducer is also not limited by a boundary, and when the central symmetrical position is marked in the step S2, judgment can be sequentially performed between any two interdigital transducers which do not belong to the symmetrical interdigital transducer area on one side of the symmetry.
S5, dividing the surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer, and calculating a basic finite element matrix of the basic subunits meeting preset symmetry and periodicity conditions by using a finite element method.
Still further, the base subunit includes an interdigital subunit and a gap subunit. FIG. 4 is a schematic view of a subunit of an embodiment of the present application, wherein the embodiment of the present application divides a piezoelectric layer within a range of a distance between an electrode and a contact with the electrode into a base subunit; and the area without electrode and with only the piezoelectric layer is divided into a spacer unit.
Specifically, the finite element method is a method of discretely dividing a calculation domain into a finite number of units which are not overlapped with each other and are connected with each other, selecting a basis function in each unit, and approximating a true solution in the unit by using a linear combination of the basis functions. Wherein, unlike the prior art that calculates matrices for all basic subunits, the present application performs finite element matrix calculation only for the basic subunits satisfying the preset symmetry and periodicity conditions including:
the interdigital subunits that do not satisfy central symmetry;
one of the interdigital subunits satisfying a central symmetry and not belonging to a periodic structure in the multiple symmetric surface acoustic wave device;
one of the interdigital subunits belonging to a periodic structure in the multi-symmetric surface acoustic wave device;
and the gap sub-units with different structures in the multi-symmetrical surface acoustic wave device.
If the base subunits are symmetrical about a center, the application can calculate the matrix of one side by the matrix of the other side through the subsequent steps so as to reduce the finite element calculation; the periodic structure refers to a part with the same design in the multi-symmetrical surface acoustic wave device, and if the multi-symmetrical surface acoustic wave device has the periodic structure, a matrix between the interdigital subunits in the periodic structure also has the periodic characteristic; the gap sub-unit is a region without electrodes and only with piezoelectric layers in the filter, and the gap sub-unit is only structurally distinguished without calculating opposite displacement degrees of freedom according to a symmetrical mode, so that the gap sub-unit only needs to calculate a single matrix according to different structures and then uses the single matrix for calculating other positions. The embodiment of the application reduces the time for generating and calculating the needed matrix by at least half according to the conditions.
Further, in step S5, the method further includes the steps of:
and eliminating the internal degree of freedom of the basic finite element matrix by using a Shu's complement operation, and reserving the displacement degrees of freedom of the left and right edges of the basic finite element matrix and the internal electrode degrees of freedom.
S6, judging whether the symmetrical single interdigital transducer is marked in the step S4, and if so, calculating the base finite element matrix on the opposite side of central symmetry according to a matrix symmetry rule according to the base finite element matrix of the symmetrical single interdigital transducer.
Still further, the matrix symmetry rule is:
defining the displacement degrees of freedom of the basic finite element matrixes in the x, y and z directions as u, V and w, respectively, and the potential degrees of freedom as V, wherein the displacement degrees of freedom in the x direction between the basic finite element matrixes symmetrical about the center are opposite, and the displacement degrees of freedom in the x and y directions and the potential degrees of freedom V are kept equal correspondingly.
Specifically, the dual-mode coupled surface acoustic wave filter works based on the piezoelectric effect, and there is a coupling of force and electricity, so that the displacement degrees of freedom u, V, w and the potential degrees of freedom V in three different directions need to be calculated in the finite element simulation calculation thereof.
Exemplary, referring to FIG. 5, FIG. 5 is a schematic diagram illustrating the symmetry of a basic finite element matrix with left and right edges thereof retaining a displacement u for the basic finite element matrix with internal degrees of freedom eliminated according to an embodiment of the present application b 、v b 、w b And degree of freedom of electric potential V b The interface between the electrode and the substrate maintains the potential degree of freedom V m 。
Since the displacement in the x-direction is symmetrical with respect to the x-direction, the displacement in the x-direction should take the opposite number (l, left; r, right), i.e.:
；
while the rest of displacement and potential degrees of freedom are independent of the x-direction, so the left and right one-to-one and equal:
；
further, defining the known first underlying finite element matrix 1 on the left as K, then there is a relationship:
；
wherein Q is m1 The amount of charge at the interface of the electrode of the first basic finite element matrix 1 and the substrate.
The second basic finite element matrix 2 is symmetrical to the first basic finite element matrix 1, so that similar relations exist, the second basic finite element matrix 2 is K', and the charge quantity is Q m2 。
Further, according to the degree of freedom relationship between the first base finite element matrix 1 and the second base finite element matrix 2, the matrix K' of the second base finite element matrix 2 satisfies:
。
therefore, the embodiment of the application obtains the mutually symmetrical basic finite element matrixes through the matrix relation, thereby correspondingly reducing the number of times of performing the Shu's complement on half of the basic finite element matrixes and saving the calculation time.
And S7, judging whether the symmetrical interdigital transducer area is marked in the step S3, if so, splicing the basic finite element matrixes of the interdigital transducers on the same side of the central symmetry in the symmetrical interdigital transducer area according to a hierarchical cascading method to obtain an area matrix, and calculating the area matrix on the different side of the central symmetry according to a matrix symmetry rule.
Further, in step S7, the method further includes the steps of:
and eliminating the internal degree of freedom of the area matrix by using a Shull complement operation, and reserving the displacement degrees of freedom of the left and right edges of the area matrix and the internal electrode degrees of freedom.
Specifically, referring to fig. 6, fig. 6 is a schematic diagram illustrating symmetry of a region matrix provided by an embodiment of the present application, where a first region matrix and a second region matrix are symmetrical to each other, and the left-side basic finite element matrices 11 and 12 are cascaded according to the manner of the step S6, and the internal degrees of freedom are eliminated by compensation, so as to preserve displacement and potential degrees of freedom on the left and right sides, and potential degrees of freedom of two electrodes; similarly, the area matrices of the base finite element matrices 21 and 22 on the right are calculated by matrix symmetric transformation.
In contrast, if n interdigital transducers are arranged on the left side of the symmetrical interdigital transducer region, 2n times of Shu-Er operation of subunits are needed when cascade splicing is performed in the prior art, and 2n-1 times of Shu-Er operation is needed when subunits are spliced, but in the embodiment of the application, only n times of Shu-Er operation of subunits and the Shu-Er operation when subunits are spliced are needed, and half of calculation time is saved.
S8, judging whether the central symmetry position is marked in the step S2, if so, splicing the basic finite element matrix and/or the regional matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure until the central symmetry position is reached to obtain a semi-overall matrix of the multi-symmetrical surface acoustic wave device, and then calculating the overall simulation matrix of the surface acoustic wave device by the semi-overall matrix according to a matrix symmetry rule;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
Further, in step S8, the method further includes the steps of:
and eliminating the internal degrees of freedom of the semi-integral matrix and/or the integral simulation matrix by using a Shuhr complement operation, and reserving the displacement degrees of freedom and the internal electrode degrees of freedom of the left and right edges of the semi-integral matrix and/or the integral simulation matrix.
After step S8 is completed, a simulation result of the surface acoustic wave filter can be obtained, and frequency response, such as admittance, can be calculated based on the overall simulation matrix.
The application has the beneficial effects that aiming at the special structure of the progressive multi-symmetrical surface acoustic wave filter and the mathematical relation of matrix symmetry in the finite element method, the inversion and peer relation of the matrix are utilized in the simulation process of the surface acoustic wave filter, so that the construction time of a large number of subunit matrixes and the calculation time for splicing the matrixes by the hierarchical cascading technology are reduced, the simulation speed of the progressive multi-symmetrical surface acoustic wave filter is improved, and the simulation efficiency is improved.
Referring to fig. 7, fig. 7 is a schematic structural diagram of a rapid simulation system 200 of a multiple symmetric surface acoustic wave device according to an embodiment of the present application, which includes:
a geometry acquisition module 201, configured to acquire an overall geometry of a multiple symmetric surface acoustic wave device, where the multiple symmetric surface acoustic wave device includes a plurality of interdigital transducers symmetrically arranged according to a periodic rule;
the overall symmetry determination module 202 is configured to determine whether the multiple symmetric surface acoustic wave device satisfies central symmetry: if yes, marking the central symmetry position of the surface acoustic wave device, wherein the central pair is called plane axisymmetry;
the region symmetry judging module 203 is configured to judge whether a region formed by connecting the interdigital transducers meets central symmetry, and if yes, mark the region formed by connecting the interdigital transducers meeting central symmetry as a symmetric interdigital transducer region;
the single-finger symmetry judgment module 204 is configured to judge whether any two of the interdigital transducers that do not belong to the symmetric interdigital transducer region satisfy central symmetry, and if yes, mark the interdigital transducer that satisfies as a symmetric single-finger interdigital transducer;
a finite element calculation module 205, configured to divide the surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer, and calculate a basic finite element matrix of the basic subunits that satisfies preset symmetry and periodicity conditions using a finite element method;
a single-finger symmetry processing module 206, configured to determine whether the single-finger symmetry determination module 204 marks the symmetric single-finger transducer, and if yes, calculate, according to a matrix symmetry rule, the base finite element matrix on the opposite side of central symmetry according to the base finite element matrix of the symmetric single-finger transducer;
the region symmetry processing module 207 is configured to determine whether the region symmetry determination module 203 marks the symmetric interdigital transducer region, if so, splice the basic finite element matrices of the interdigital transducers on the same side of the central symmetry in the symmetric interdigital transducer region according to a hierarchical cascading method to obtain a region matrix, and calculate the region matrix on the different side of the central symmetry according to a matrix symmetry rule;
the overall symmetry processing module 208 is configured to determine whether the overall symmetry determination module 202 marks the central symmetry position, if yes, splice the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the order of the overall geometry until the central symmetry position to obtain a semi-overall matrix of the multiple-symmetry surface acoustic wave device, and then calculate the semi-overall matrix according to a matrix symmetry rule to obtain an overall simulation matrix of the surface acoustic wave device;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
The rapid simulation system 200 of the multiple symmetric surface acoustic wave device can implement the steps in the rapid simulation method of the multiple symmetric surface acoustic wave device in the above embodiment, and can implement the same technical effects, and the description in the above embodiment is omitted herein.
Referring to fig. 8, fig. 8 is a schematic structural diagram of a computer device according to an embodiment of the present application, where the computer device 300 includes: a memory 302, a processor 301 and a computer program stored on the memory 302 and executable on the processor 301.
The processor 301 invokes the computer program stored in the memory 302 to execute the steps in the method for fast simulation of a multiple symmetric surface acoustic wave device according to the embodiment of the present application, please refer to fig. 1, specifically including the following steps:
s1, acquiring the overall geometric structure of a multi-symmetrical surface acoustic wave device, wherein the multi-symmetrical surface acoustic wave device comprises a plurality of interdigital transducers which are symmetrically arranged according to a periodic rule;
s2, judging whether the multi-symmetrical surface acoustic wave device meets central symmetry or not: if yes, marking the central symmetry position of the surface acoustic wave device, wherein the central pair is called plane axisymmetry;
s3, judging whether the region formed by connecting the interdigital transducers meets central symmetry, if so, marking the region formed by connecting the interdigital transducers meeting central symmetry as a symmetrical interdigital transducer region;
s4, judging whether any two interdigital transducers which do not belong to the symmetrical interdigital transducer area meet central symmetry, and if yes, marking the met interdigital transducers as symmetrical single interdigital transducers;
s5, dividing the surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer, and calculating a basic finite element matrix of the basic subunits meeting preset symmetry and periodicity conditions by using a finite element method;
s6, judging whether the symmetrical single interdigital transducer is marked in the step S4, if so, calculating the basic finite element matrix on the opposite side of central symmetry according to a matrix symmetry rule according to the basic finite element matrix of the symmetrical single interdigital transducer;
s7, judging whether the symmetrical interdigital transducer area is marked in the step S3, if so, splicing the basic finite element matrixes of the interdigital transducers on the same side of the central symmetry in the symmetrical interdigital transducer area according to a hierarchical cascading method to obtain an area matrix, and calculating the area matrix on different sides of the central symmetry according to a matrix symmetry rule;
s8, judging whether the central symmetry position is marked in the step S2, if so, splicing the basic finite element matrix and/or the regional matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure until the central symmetry position is reached to obtain a semi-overall matrix of the multi-symmetrical surface acoustic wave device, and then calculating the overall simulation matrix of the surface acoustic wave device by the semi-overall matrix according to a matrix symmetry rule;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
Still further, the base subunit includes an interdigital subunit and a gap subunit.
Still further, the preset symmetry and periodicity conditions include:
the interdigital subunits that do not satisfy central symmetry;
one of the interdigital subunits satisfying a central symmetry and not belonging to a periodic structure in the multiple symmetric surface acoustic wave device;
one of the interdigital subunits belonging to a periodic structure in the multi-symmetric surface acoustic wave device;
and the gap sub-units with different structures in the multi-symmetrical surface acoustic wave device.
Further, in step S5, the method further includes the steps of:
and eliminating the internal degree of freedom of the basic finite element matrix by using a Shu's complement operation, and reserving the displacement degrees of freedom of the left and right edges of the basic finite element matrix and the internal electrode degrees of freedom.
Still further, the matrix symmetry rule is:
defining the displacement degrees of freedom of the basic finite element matrixes in the x, y and z directions as u, V and w, respectively, and the potential degrees of freedom as V, wherein the displacement degrees of freedom in the x direction between the basic finite element matrixes symmetrical about the center are opposite, and the displacement degrees of freedom in the x and y directions and the potential degrees of freedom V are kept equal correspondingly.
Further, in step S7, the method further includes the steps of:
and eliminating the internal degree of freedom of the area matrix by using a Shull complement operation, and reserving the displacement degrees of freedom of the left and right edges of the area matrix and the internal electrode degrees of freedom.
Further, in step S8, the method further includes the steps of:
and eliminating the internal degrees of freedom of the semi-integral matrix and/or the integral simulation matrix by using a Shuhr complement operation, and reserving the displacement degrees of freedom and the internal electrode degrees of freedom of the left and right edges of the semi-integral matrix and/or the integral simulation matrix.
The computer device 300 provided in the embodiment of the present application can implement the steps in the method for fast simulating the multiple symmetric surface acoustic wave device in the above embodiment, and can implement the same technical effects, and is not described herein again with reference to the description in the above embodiment.
The embodiment of the application also provides a computer readable storage medium, on which a computer program is stored, which when executed by a processor, implements each process and step in the rapid simulation method of the multiple symmetric surface acoustic wave device provided by the embodiment of the application, and can implement the same technical effects, so that repetition is avoided and redundant description is omitted.
Those skilled in the art will appreciate that implementing all or part of the above-described methods in accordance with the embodiments may be accomplished by way of a computer program stored on a computer readable storage medium, which when executed may comprise the steps of the embodiments of the methods described above. The storage medium may be a magnetic disk, an optical disk, a Read-Only Memory (ROM), a random access Memory (Random Access Memory, RAM) or the like.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one … …" does not exclude the presence of other like elements in a process, method, article, or apparatus that comprises the element.
From the above description of the embodiments, it will be clear to those skilled in the art that the above-described embodiment method may be implemented by means of software plus a necessary general hardware platform, but of course may also be implemented by means of hardware, but in many cases the former is a preferred embodiment. Based on such understanding, the technical solution of the present application may be embodied essentially or in a part contributing to the prior art in the form of a software product stored in a storage medium (e.g. ROM/RAM, magnetic disk, optical disk) comprising instructions for causing a terminal (which may be a mobile phone, a computer, a server, an air conditioner, or a network device, etc.) to perform the method according to the embodiments of the present application.
While the embodiments of the present application have been illustrated and described in connection with the drawings, what is presently considered to be the most practical and preferred embodiments of the application, it is to be understood that the application is not limited to the disclosed embodiments, but on the contrary, is intended to cover various equivalent modifications and equivalent arrangements included within the spirit and scope of the appended claims.
Claims (10)
1. A rapid simulation method of a multiple symmetric surface acoustic wave device, the rapid simulation method comprising the steps of:
s1, acquiring the overall geometric structure of a multi-symmetrical surface acoustic wave device, wherein the multi-symmetrical surface acoustic wave device comprises a plurality of interdigital transducers which are symmetrically arranged according to a periodic rule;
s2, judging whether the multi-symmetrical surface acoustic wave device meets central symmetry or not: if yes, marking the central symmetry position of the multi-symmetry surface acoustic wave device, wherein the central pair is called plane axisymmetry;
s3, judging whether the region formed by connecting the interdigital transducers meets central symmetry, if so, marking the region formed by connecting the interdigital transducers meeting central symmetry as a symmetrical interdigital transducer region;
s4, judging whether any two interdigital transducers which do not belong to the symmetrical interdigital transducer area meet central symmetry, and if yes, marking the met interdigital transducers as symmetrical single interdigital transducers;
s5, dividing the multi-symmetrical surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer, and calculating a basic finite element matrix of the basic subunits meeting preset symmetry and periodicity conditions by using a finite element method;
s6, judging whether the symmetrical single interdigital transducer is marked in the step S4, if so, calculating the basic finite element matrix on the opposite side of central symmetry according to a matrix symmetry rule according to the basic finite element matrix of the symmetrical single interdigital transducer;
s7, judging whether the symmetrical interdigital transducer area is marked in the step S3, if so, splicing the basic finite element matrixes of the interdigital transducers on the same side of the central symmetry in the symmetrical interdigital transducer area according to a hierarchical cascading method to obtain an area matrix, and calculating the area matrix on different sides of the central symmetry according to a matrix symmetry rule;
s8, judging whether the central symmetry position is marked in the step S2, if so, splicing the basic finite element matrix and/or the regional matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure until the central symmetry position is reached to obtain a semi-overall matrix of the multi-symmetrical surface acoustic wave device, and then calculating the overall simulation matrix of the multi-symmetrical surface acoustic wave device by the semi-overall matrix according to a matrix symmetry rule;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
2. The method of fast simulation of a multiple symmetric surface acoustic wave device of claim 1, wherein the base subunit comprises an interdigital subunit and a gap subunit.
3. The method of rapid simulation of a multiple symmetric surface acoustic wave device of claim 2, wherein the preset symmetry and periodicity conditions include:
the interdigital subunits that do not satisfy central symmetry;
one of the interdigital subunits satisfying a central symmetry and not belonging to a periodic structure in the multiple symmetric surface acoustic wave device;
one of the interdigital subunits belonging to a periodic structure in the multi-symmetric surface acoustic wave device;
and the gap sub-units with different structures in the multi-symmetrical surface acoustic wave device.
4. The method for rapid simulation of a multiple symmetric surface acoustic wave device according to claim 1, further comprising the step of, in step S5:
and eliminating the internal degree of freedom of the basic finite element matrix by using a Shu's complement operation, and reserving the displacement degrees of freedom of the left and right edges of the basic finite element matrix and the internal electrode degrees of freedom.
5. The method for rapid simulation of a multiple symmetric surface acoustic wave device of claim 1, wherein the matrix symmetry rule is:
defining the displacement degrees of freedom of the basic finite element matrixes in the x, y and z directions as u, V and w, respectively, and the potential degrees of freedom as V, wherein the displacement degrees of freedom in the x direction between the basic finite element matrixes symmetrical about the center are opposite, and the displacement degrees of freedom in the x and y directions and the potential degrees of freedom V are kept equal correspondingly.
6. The method for rapid simulation of a multiple symmetric surface acoustic wave device according to claim 1, further comprising the step of, in step S7:
and eliminating the internal degree of freedom of the area matrix by using a Shull complement operation, and reserving the displacement degrees of freedom of the left and right edges of the area matrix and the internal electrode degrees of freedom.
7. The method for rapid simulation of a multiple symmetric surface acoustic wave device according to claim 1, further comprising the step of, in step S8:
and eliminating the internal degrees of freedom of the semi-integral matrix and/or the integral simulation matrix by using a Shuhr complement operation, and reserving the displacement degrees of freedom and the internal electrode degrees of freedom of the left and right edges of the semi-integral matrix and/or the integral simulation matrix.
8. A rapid simulation system for a multiple symmetric surface acoustic wave device, the rapid simulation system comprising:
the geometrical acquisition module is used for acquiring the overall geometrical structure of the multi-symmetrical surface acoustic wave device, and the multi-symmetrical surface acoustic wave device comprises a plurality of interdigital transducers which are symmetrically arranged according to a periodic rule;
the overall symmetry judging module is used for judging whether the multi-symmetry surface acoustic wave device meets central symmetry or not: if yes, marking the central symmetry position of the multi-symmetry surface acoustic wave device, wherein the central pair is called plane axisymmetry;
the region symmetry judging module is used for judging whether the region formed by connecting the interdigital transducers meets the central symmetry or not, and if so, the region formed by connecting the interdigital transducers meets the central symmetry is marked as a symmetric interdigital transducer region;
the single-finger symmetry judgment module is used for judging whether any two interdigital transducers which do not belong to the symmetrical interdigital transducer area meet central symmetry, and if yes, the met interdigital transducers are marked as symmetrical single-finger interdigital transducers;
the finite element calculation module is used for dividing the multi-symmetrical surface acoustic wave device into a plurality of basic subunits according to the interdigital transducer and calculating a basic finite element matrix of the basic subunits meeting preset symmetry and periodicity conditions by using a finite element method;
the single-finger symmetry processing module is used for judging whether the single-finger symmetry judging module marks the symmetrical single-finger transducer, if so, calculating the basic finite element matrix on the opposite side of central symmetry according to a matrix symmetry rule according to the basic finite element matrix of the symmetrical single-finger transducer;
the region symmetry processing module is used for judging whether the region symmetry judging module marks the symmetric interdigital transducer region, if so, splicing the basic finite element matrixes of the interdigital transducers on the same side of the central symmetry in the symmetric interdigital transducer region according to a hierarchical cascading method to obtain a region matrix, and calculating the region matrix on the opposite side of the central symmetry according to a matrix symmetry rule;
the overall symmetry processing module is used for judging whether the overall symmetry judging module marks the central symmetry position, if so, the basic finite element matrix and/or the area matrix are spliced according to the hierarchical cascading method according to the sequence of the overall geometric structure until the central symmetry position is reached to obtain a semi-overall matrix of the multi-symmetrical surface acoustic wave device, and then the semi-overall matrix is used for calculating an overall simulation matrix of the multi-symmetrical surface acoustic wave device according to a matrix symmetry rule;
and if not, splicing the basic finite element matrix and/or the area matrix according to the hierarchical cascading method according to the sequence of the overall geometric structure to obtain the overall simulation matrix of the multi-symmetrical surface acoustic wave device.
9. A computer device, comprising: a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps in the method for fast simulation of a multiple symmetric surface acoustic wave device as claimed in any one of claims 1 to 7 when the computer program is executed.
10. A computer readable storage medium, characterized in that the computer readable storage medium has stored thereon a computer program which, when executed by a processor, implements the steps of the method for fast simulation of a multiple symmetric surface acoustic wave device according to any of claims 1-7.
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Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP1104952A1 (en) * | 1999-05-14 | 2001-06-06 | Kabushiki Kaisha Toshiba | Surface acoustic wave device |
| CN109716336A (en) * | 2016-08-29 | 2019-05-03 | 谐振公司 | The layering cascade of acoustic wave filter device two-dimensional finite element method simulation |
| CN113544687A (en) * | 2018-12-11 | 2021-10-22 | 谐振公司 | Fast and high precision full finite element surface acoustic wave simulation |
| CN115470735A (en) * | 2022-09-09 | 2022-12-13 | 深圳飞骧科技股份有限公司 | Method, system and related device for SAW physical simulation |
| CN115577604A (en) * | 2022-12-06 | 2023-01-06 | 深圳飞骧科技股份有限公司 | Simulation method and system for quickly constructing overall matrix and related equipment |
Family Cites Families (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2014504827A (en) * | 2011-01-24 | 2014-02-24 | エプコス アクチエンゲゼルシャフト | Surface acoustic wave filter |
| US11182522B2 (en) * | 2016-08-29 | 2021-11-23 | Resonant, Inc. | Hierarchical cascading in FEM simulations of SAW devices |
-
2023
- 2023-04-28 CN CN202310476962.3A patent/CN116205187B/en active Active
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP1104952A1 (en) * | 1999-05-14 | 2001-06-06 | Kabushiki Kaisha Toshiba | Surface acoustic wave device |
| CN109716336A (en) * | 2016-08-29 | 2019-05-03 | 谐振公司 | The layering cascade of acoustic wave filter device two-dimensional finite element method simulation |
| CN113544687A (en) * | 2018-12-11 | 2021-10-22 | 谐振公司 | Fast and high precision full finite element surface acoustic wave simulation |
| CN115470735A (en) * | 2022-09-09 | 2022-12-13 | 深圳飞骧科技股份有限公司 | Method, system and related device for SAW physical simulation |
| CN115577604A (en) * | 2022-12-06 | 2023-01-06 | 深圳飞骧科技股份有限公司 | Simulation method and system for quickly constructing overall matrix and related equipment |
Non-Patent Citations (1)
| Title |
|---|
| 单端SAW器件设计及有限元仿真;谢易微;许绍俊;;科技风(23);第97-98页 * |
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