A mortar interface coupling-based FEM-SEM hybrid simulation method and system for surface acoustic wave devices

Abstract

The application discloses a kind of Mortar interface coupling-based surface acoustic wave device FEM-SEM hybrid simulation method and system, it is related to surface acoustic wave simulation technical field, solve the problem that traditional low-order FEM needs great grid density when processing large size thick substrate, leading to serious waste of computing resources.The overall geometric model of SAW device is divided into upper layer geometric complex subdomain and lower layer large size propagation subdomain along the depth direction;Traditional low-order FEM with high geometric adaptability is used to discretize the upper layer geometric complex subdomain, and SEM with high precision and low degree of freedom characteristics is used to discretize the lower layer large size propagation subdomain;And Mortar finite element constraint is introduced on the non-conformal interface of the two subdomains to construct the interface coupling matrix, and finally the global system is assembled and solved.It can effectively reduce the total degree of freedom of the system, significantly improve the overall simulation efficiency of surface acoustic wave device, especially the device containing thick substrate structure.

Description

Technical Field

[0001] This invention relates to the field of surface acoustic wave (SAW) simulation technology, specifically to a hybrid FEM-SEM simulation method and system for SAW devices based on Mortar interface coupling. Background Technology

[0002] Surface acoustic wave (SAW) devices play an irreplaceable role in modern radio frequency (RF) communications. As device design evolves towards higher frequencies, wider bandwidths, and miniaturization, extremely high demands are placed on the accurate numerical simulation of device characteristics. The finite element method (FEM), with its strong adaptability to complex geometric boundaries, has become the preferred numerical tool for accurate modeling of SAW devices.

[0003] However, in actual SAW device simulations, traditional FEMs face a severe imbalance in computational resource allocation. Typical SAW devices (such as TF-SAW) consist of complex nanoscale metal electrode structures on the surface and a large-scale support substrate at the micrometer level or even thicker. Traditional FEMs typically use second-order low-order polynomials as unit shape functions. To accurately simulate the propagation and attenuation of sound waves in the thick substrate region, even if the region has regular geometry and relatively smooth physical field changes, traditional FEMs are still forced to divide the data into an extremely large number of fine meshes. This leads to an explosive increase in the overall degrees of freedom (DOF) of the model, consuming massive amounts of computational memory and time.

[0004] The spectral element method (SEM), as a high-order finite element method, employs high-order orthogonal polynomials (such as Lagrange polynomials) and non-uniformly distributed integration nodes (such as Gauss-Lobatto-Legendre, GLL nodes). Its greatest advantage lies in achieving extremely high computational accuracy with only a very small number of large-sized elements. However, SEM has extremely stringent requirements for element shape (usually requiring regular quadrilaterals or hexahedrons), which makes it impossible to fully realize its effects on the arbitrarily shaped and extremely complex interdigitated electrode (IDT) structures on the surface of SAW devices.

[0005] Therefore, existing single numerical methods (pure FEM or pure SEM) cannot achieve a balance between the "surface geometric complexity" and "substrate wave propagation efficiency" of SAW devices. Thus, there is an urgent need for a hybrid simulation framework that retains the geometric flexibility of FEM while leveraging the efficiency of SEM in handling large-area problems. Summary of the Invention

[0006] To address the shortcomings and deficiencies of existing technologies, this invention provides a hybrid FEM-SEM simulation method and system for surface acoustic wave (SAW) devices that combines the geometric flexibility of FEM with the high efficiency and precision of SEM, significantly improving the computation speed and memory utilization of overall finite element simulation of SAW devices.

[0007] To achieve the above objectives, the present invention provides the following technical solution: The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling, provided by the present invention, includes the following steps: S1. Construct the overall geometric model of the SAW device. Based on the geometric features and physical properties of the SAW device, in space, the SAW device is divided into two non-overlapping subdomains along the depth direction: the upper geometrically complex subdomain containing complex electrode structures and piezoelectric films, and the lower large-size propagation subdomain containing thick dielectric layers and silicon substrates. The two subdomains are separated by a non-conformal interface. S2, Physical Domain Decomposition: The upper-level geometrically complex subdomains are discretized using a highly geometrically adaptable FEM, and low-order shape functions and unstructured meshes are used to extract the complex physical boundaries of the upper-level structure. The lower large-size propagation subdomain is discretized using SEM, which has high precision and low degrees of freedom. Large-size regular spectral units are used, and high-order Lagrange interpolation polynomials and non-uniformly distributed GLL nodes are used inside the units to achieve a high-precision approximation of the wave field in the lower base region with very few degrees of freedom. S3. Constructing the Mortar cross-order projection interface: On the non-conformal interface between the upper FEM subdomain and the lower SEM subdomain, Mortar finite element constraints are introduced. By defining the Lagrange multiplier subspace, the cross-space projection integral between the low-order uniform node shape function and the high-order non-uniform node spectral shape function is calculated to construct the Mortar interface coupling matrix that connects the two different discretization schemes. S4. Global System Solution and Key Parameter Extraction: The local stiffness / mass matrices of the upper FEM subdomain and the lower SEM subdomain are globally assembled with the Mortar interface coupling matrix to construct a saddle point system equation set for the global discrete system. Solving this equation set yields the global displacement field, potential field, and frequency domain admittance characteristics of the SAW device.

[0008] Preferably, in step S2, the unstructured mesh used for the upper geometrically complex subdomain is a triangular or tetrahedral mesh, and the low-order shape function used is the shape function corresponding to the second-order element, specifically the shape function of the second-order nine-node quadrilateral element, which is based on uniformly distributed nodes.

[0009] Preferably, the lower-level large-size propagation subdomain is discretized using the spectral element method (SEM), specifically including: The thick substrate is divided into a small number of large-sized regular units, specifically high-order quadrilateral or hexahedral units. High-order Lagrange orthogonal polynomials are used as basis functions inside the large units, and the order of the high-order Lagrange interpolation polynomials is N≥2. Non-uniformly distributed GLL nodes are used, with nodes dense at the edge of the unit and sparse at the center, so that deep acoustic waves can be accurately simulated with a very small number of nodes.

[0010] Preferably, in step S3, the specific process of constructing the Mortar cross-level projection interface is as follows: S31. Define the FEM side of the interface as the slave surface and the SEM side as the master surface; S32. Define a Lagrange multiplier field on the interface to represent the unknown force or charge flux on the interface. S33. Establish a weakly constrained integral equation. This integral equation requires calculating the inner product between the FEM shape function obtained by low-order polynomial interpolation and the SEM shape function obtained by high-order Lagrange polynomial interpolation. The interface integral calculation adopts the high-order Gauss-Legendre integration rule to ensure the accuracy of the interface coupling matrix calculation. S34. Discretize and numerically integrate the weakly constrained integral equations to transform the continuous constraints into algebraic equations containing cross-order coupling matrices of the FEM and SEM sides.

[0011] Preferably, in step S4, the local stiffness / mass matrices of the upper FEM subdomain and the lower SEM subdomain are globally assembled with the Mortar interface coupling matrix to construct the saddle point system equations of the global discrete system, specifically including: During global assembly, the frequency domain dynamic control equations of the FEM and SEM subdomains are combined with the Mortar interface constraint equations from step S3 to form a global discrete system matrix. The global discrete system matrix includes the frequency domain dynamic stiffness matrix of the FEM subdomain, the frequency domain dynamic stiffness matrix of the SEM subdomain, the Mortar interface coupling matrix, and the Lagrange multiplier correlation matrix. The frequency domain dynamic stiffness matrix includes the stiffness matrix, damping matrix, mass matrix, angular frequency, and imaginary unit.

[0012] Preferably, the local stiffness / mass matrices of the upper geometrically complex subdomain and the lower large-size propagation subdomain are globally assembled with the Mortar interface coupling matrix, and an electric excitation load vector is applied to the global discrete system. The electric excitation load vector is applied only to the electrode region in the upper geometrically complex subdomain, and the excitation source of the lower large-size propagation subdomain is set to 0. The global discrete system is solved using a sparse saddle point system linear algebra solver.

[0013] Preferably, a sparse direct solver is used to perform LU decomposition on the saddle point system and solve it back, or a preconditioned GMRES iterative method is used in conjunction with Schur supplementary preconditioners to accelerate convergence, or a block Gaussian elimination strategy is used to first eliminate the λ variable and then solve the reduced FEM-SEM coupled system to obtain numerical solutions of the global displacement field and potential field, and calculate the port admittance accordingly.

[0014] Preferably, it also includes S5 and performance parameter extraction: Based on the obtained potential and displacement of the electrode region, the frequency domain admittance of the device port is calculated, and then the performance indicators of the device, such as resonant frequency, anti-resonant frequency, effective electromechanical coupling coefficient and quality factor, are extracted.

[0015] A computing device, comprising: One or more processors; A storage device for storing one or more programs, which, when executed by one or more processors, enable one or more processors to implement the FEM-SEM hybrid simulation method.

[0016] A computer-readable storage medium storing a program that, when executed by a processor, implements a hybrid FEM-SEM simulation method.

[0017] The FEM-SEM hybrid simulation method and system for surface acoustic wave devices based on Mortar interface coupling proposed in this invention have the following beneficial effects: (1) The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling of the present invention seamlessly mixes FEM and SEM through Mortar cross-order projection interface, combining the geometric flexibility of FEM with the high efficiency and high precision of SEM. At the same time, it uses high-order SEM to replace the massive inefficient mesh of traditional FEM in the thick substrate region, and completes the accurate simulation of the broad substrate region with a very small number of high-order elements and degrees of freedom, which significantly improves the calculation speed and memory utilization of the overall finite element simulation of SAW devices.

[0018] (2) The hybrid simulation framework of the present invention breaks through the application limitations of single numerical methods, does not require major modifications to the existing simulation process, is easy to promote and apply in engineering, and can be widely applied to simulation scenarios of SAW devices with complex surface structures and large-size substrates. It has strong practicality and versatility. Attached Figure Description

[0019] Figure 1 This is a flowchart of the FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling in this invention; Figure 2This is a schematic diagram of the spatial multi-subdomain decomposition and hybrid discretization strategy for SAW devices in this invention; Figure 3 This is a schematic diagram comparing the distribution of FEM and SEM spectral units and nodes in this invention; Figure 4 This is a comparison diagram of the three models in Example 2 of this embodiment; Figure 5 This is a comparison chart of the admittance curves of the three models in the same frequency band in Example 2. Detailed Implementation

[0020] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Example 1

[0021] like Figure 1 As shown, the FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling of the present invention includes the following steps: S1. Construct the overall geometric model of the SAW device. Based on the geometric features and physical properties of the SAW device, spatially divide the SAW device along the depth direction into two non-overlapping subdomains: an upper geometrically complex subdomain containing complex electrode structures and piezoelectric thin films. And a lower-level large-size propagation subdomain comprising a thick dielectric layer and a silicon substrate. The two subdomains are physically separated by a non-conformal interface. .

[0022] like Figure 2 As shown, the metal interdigitated electrodes and piezoelectric thin film region at the top of the device have complex structures and severe acoustic-electric field gradients, while the underlying temperature compensation layer (such as SiO2) and deep substrate (such as Si) regions have regular geometry and relatively smooth acoustic field attenuation. Based on the geometric characteristics and physical properties of the SAW device, the SAW device model is divided into two non-overlapping subdomains along the depth direction.

[0023] Among them, the upper-level geometrically complex subdomain This refers to the region covering the area from the top of the interdigitated metal electrode to part of the bottom of the piezoelectric film in a direction perpendicular to the device surface. Its geometric boundaries exhibit high irregularity, typically including acute angles at the edges of the interdigitated electrodes, local undulations at the interfaces of multilayer stacks, and micro-nano scale features at the contact surfaces between the piezoelectric film and the electrodes.

[0024] Lower-level large-size propagation subdomain This refers to the region extending downwards from the bottom boundary of the upper subdomain to the bottom of the silicon substrate, including the absorbing boundary layer. Its geometric contour is macroscopically regular, primarily planar or gently curving surfaces. The physical field within this region changes smoothly and exhibits strong wave propagation characteristics. Non-conformal interface. This is the theoretical spatial separation surface between the upper and lower subdomains. It is independently discretized on the FEM and SEM sides, with inconsistent node positions, numbers, and distribution patterns on both sides, failing to satisfy a one-to-one node correspondence. Through this spatial decomposition, the modeling task of the overall geometric model of the SAW device is decoupled into two sub-problems: geometry-driven and wave-driven, laying the foundation for subsequent differentiated discretization.

[0025] S2. Physical Domain Decomposition: The upper geometrically complex subdomain is discretized using a highly geometrically adaptable FEM, employing low-order shape functions and unstructured meshes to extract the complex physical boundaries of the upper structure; the lower large-size propagation subdomain is discretized using a SEM with high accuracy and low degrees of freedom, employing large-size regular spectral elements, with high-order Lagrange interpolation polynomials and non-uniformly distributed GLL nodes within the elements, to achieve a high-accuracy approximation of the wave field in the lower base region with very few degrees of freedom.

[0026] Among them, the highly geometrically adaptable FEM is a numerical method based on variational principles, which locally approximates field variables through element shape functions. It can flexibly adapt to geometries with arbitrary topologies and boundaries. Low-order shape functions refer to interpolation basis functions with lower polynomial orders, first or second order. They have local support and continuity within the element and are suitable for describing the physical response of geometrically complex regions with strong field gradients. Unstructured meshes refer to mesh systems composed of irregular elements such as triangles or tetrahedrons. Their node distribution is not constrained by the global coordinate system and can automatically fit arbitrary geometric features such as the edges of interdigitated electrodes and the curvature of multi-layer interfaces. This discretization method can easily adapt to arbitrary geometric irregularities at the electrode edges. Let the degree-of-freedom vector after FEM discretization be... .

[0027] The SEM, characterized by high accuracy and low degrees of freedom, is based on a high-order finite element variant. It employs high-order Lagrange orthogonal polynomials as basis functions within regular elements, combined with non-uniform integral GLL nodes, achieving exponential convergence accuracy for smooth wave fields while maintaining a minimal number of elements. Large-size regular spectral elements refer to discrete elements with side lengths greater than the upper-layer FEM elements and rectangular or hexahedral geometric shapes. High-order Lagrange interpolation polynomials refer to Lagrange-type interpolation basis functions of order N≥2, exhibiting global continuity and high smoothness within the element. Non-uniformly distributed GLL nodes refer to a node distribution that is dense at element boundaries and sparse in the central region. This distribution effectively suppresses Runge oscillations caused by high-order interpolation and improves the numerical fidelity for the dispersion characteristics of the wave equation. Let the degree-of-freedom vector after SEM discretization be... .

[0028] The differences between FEM and SEM spectral units and node distributions are as follows: Figure 3 As shown in the image. At this point, on the interface... Above, there are dense low-order FEM nodes, and below, there are sparse and non-uniformly distributed high-order SEM nodes. There is a serious mismatch between the two in terms of geometric size, node position and shape function order.

[0029] S3. Constructing the Mortar cross-order projection interface: On the non-conformal interface between the upper FEM subdomain and the lower SEM subdomain, Mortar finite element constraints are introduced. By defining the Lagrange multiplier subspace, the cross-space projection integral between the low-order uniform node shape function and the high-order non-uniform node spectral shape function is calculated to construct the Mortar interface coupling matrix connecting the two different discretization schemes. The specific process is as follows: S31, Define the interface The FEM side is from the surface SEM side is the main plane ; S32. Define a Lagrange multiplier field on the surface. , used to represent unknown force or charge flux at an interface; S33. Establish a weakly constrained integral equation. This integral equation requires calculating the inner product between the FEM shape function obtained by low-order polynomial interpolation and the SEM shape function obtained by high-order Lagrange polynomial interpolation. The interface integral calculation adopts the high-order Gauss-Legendre integration rule to ensure the accuracy of the interface coupling matrix calculation. The specific calculation is as follows: ; in, Obtained by low-order polynomial interpolation, It is obtained by interpolation using higher-order Lagrange polynomials. Since the integrand contains higher-order polynomials, this embodiment must employ a sufficiently high-order numerical integration scheme (such as a higher-order Gauss-Legendre integration rule) to ensure the accuracy of the interface coupling matrix calculation during interface integral calculation.

[0030] S34. Discretize and numerically integrate the weakly constrained integral equations to transform the continuous constraints into algebraic equations containing cross-order coupling matrices from the FEM and SEM sides: ; in, and It is a cross-order coupling matrix.

[0031] Mortar finite element constraints are a weak formal constraint technique for coupling non-conformal mesh interfaces. They introduce Lagrange multipliers to satisfy the continuity conditions of displacement and potential physical fields on both sides of the interface. The Lagrange multiplier space refers to an independent function space defined on the master surface, typically with a lower dimension than the shape function space of the master surface, used to characterize unknown generalized forces or fluxes at the interface. Low-order uniform node shape functions refer to low-order polynomial interpolation functions constructed based on uniformly distributed nodes used in the upper-level FEM subdomain, such as the biquadratic shape function corresponding to a second-order nine-node quadrilateral element. High-order non-uniform node spectral shape functions refer to high-order Lagrange interpolation functions constructed based on GLL non-uniform nodes used in the lower-level SEM subdomain, such as the biquartic shape function corresponding to a fourth-order sixteen-node quadrilateral element. The cross-spatial projection integral can refer to the integral applied at the interface... The inner product operation is performed on the shape functions of two different function spaces. The result of this integration constitutes the core element of the Mortar interface coupling matrix, which is used to establish a linear mapping relationship between the FEM degrees of freedom and the SEM degrees of freedom.

[0032] S4. Global System Solution and Key Parameter Extraction: The local stiffness / mass matrices of the upper FEM subdomain and the lower SEM subdomain are globally assembled with the Mortar interface coupling matrix to construct a saddle point system equation set for the global discrete system. Solving this equation set yields the global displacement field, potential field, and frequency domain admittance characteristics of the SAW device.

[0033] The local stiffness / mass matrix refers to the system matrix obtained by discretizing the frequency domain dynamic control equations of the upper FEM subdomain and the lower SEM subdomain, respectively. The Mortar interface coupling matrix is ​​a cross-order projection matrix constructed to connect the degrees of freedom of the FEM and SEM subdomains. The matrix form of the saddle point system equations of the constructed global discrete system is as follows: ;

[0034] in, and These are the frequency domain dynamic stiffness matrices for the low-order FEM and high-order SEM subdomains, respectively. Their specific structures are as follows: ; In the formula, the subscript i represents FEM or SEM; Here is the stiffness matrix. Here is the damping matrix. For the quality matrix, Angular frequency, The imaginary unit, This represents the electrical excitation load vector applied to the electrode region in the geometrically complex subdomain above. Since the bottom layer typically lacks an excitation source, Set to 0. The displacement field and electric potential field refer to the mechanical displacement components and electric potential values ​​at each node obtained by solving, which together constitute the complete steady-state response solution of the SAW device; the frequency domain admittance characteristic can refer to the complex admittance Y calculated by the relationship between port voltage and current, whose amplitude and phase change with frequency to form an admittance curve, which is the core indicator for evaluating the resonant performance of the device.

[0035] Due to the significant reduction in order of the SEM subdomain, the matrix dimension of the global system is significantly reduced compared to the traditional FEM model. By using the MUMPS linear algebra solver to solve the equations of this sparse saddle point system, the steady-state response of each node in the entire multilayer SAW device can be efficiently obtained, including the global displacement field, potential field, and frequency domain admittance characteristics of the device.

[0036] S5. Performance Parameter Extraction: Based on the obtained potential and displacement of the electrode region, the frequency domain admittance Y parameter of the device port is calculated, and then the resonant frequency, anti-resonant frequency, effective electromechanical coupling coefficient and quality factor Q value of the device are extracted to complete the device simulation evaluation.

[0037] This invention divides a SAW device along its depth into an upper geometrically complex subdomain and a lower large-size propagation subdomain. It leverages the high geometric adaptability of FEM (Flexible Electron Modeling) to accurately model the electrode-piezoelectric interface in the former, while utilizing SEM (Structured Electron Modeling) to efficiently simulate the high-precision, low-degree-of-freedom characteristics of the latter, enabling efficient simulation of acoustic wave propagation in thick substrates. Furthermore, by constructing a Mortar cross-order projection interface on a non-conformal interface, and utilizing Lagrange multiplier space and cross-space projection integrals, it bridges the function space between low-order uniform node shape functions and high-order non-uniform node spectral shape functions. Finally, by globally assembling and solving a saddle-point system, it simultaneously obtains the displacement field, potential field, and frequency domain admittance characteristics. This collaborative mechanism fundamentally avoids the degree-of-freedom explosion problem caused by global low-order FEM and avoids the modeling failure risk caused by geometric constraints in global SEM. Without sacrificing the fidelity of physical modeling, it significantly improves the computational efficiency and memory utilization of SAW device simulation. Example 2

[0038] To verify the performance of the FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling in Example 1, this invention uses a two-dimensional single-cycle surface acoustic wave (SAW) resonator as the verification model, and its design parameters are shown in Table 1.

[0039]

[0040] The specific simulation process is as follows: S1. Taking a two-dimensional TF-SAW resonator as the object, the upper geometrically complex subdomain includes electrodes and thin film layers, and the lower large-size propagation subdomain includes dielectric layers, polycrystalline silicon layers, silicon substrates and PML layers. The interface between the two subdomains is located in the middle of the piezoelectric thin film, forming a non-conformal physical interface.

[0041] S2. The upper-level geometrically complex subdomain is discretized using FEM. Specifically, a 9-node second-order quadrilateral element is used, with nodes evenly distributed at the four corners, midpoints of the four sides, and the center of the element. The shape function is a biquadratic polynomial.

[0042] The lower-level large-size propagation subdomain is discretized using SEM. Specifically, a 16-node fourth-order quadrilateral element is used, with nodes non-uniformly distributed at the element boundary and inside according to the GLL rule. The shape function is a bi-fourth-order Lagrange polynomial.

[0043] At this point, the interface The upper FEM side and the SEM side do not match in terms of node density, location and interpolation order.

[0044] S3. Set the FEM side as a slave face. SEM side is set as the main face Define a piecewise linear Lagrange multiplier subspace on the surface. High-order Gauss-Legendre numerical integrations were performed on the shape functions of the FEM and SEM side node elements to calculate the matrix and assemble it to obtain the Mortar interface coupling matrix.

[0045] S4. Globally assemble the local stiffness / mass matrices of the upper FEM subdomain and the lower SEM subdomain with the Mortar interface coupling matrix to construct the saddle point system equations of the global discrete system.

[0046] A sparse direct solver is used to solve the saddle point system at each frequency point of the above TF-SAW model, extracting the voltage and injection current across the electrode, calculating the admittance Y, and finally obtaining the frequency domain admittance amplitude curve.

[0047] S5. Based on the frequency domain admittance, extract key performance indicators such as the resonant frequency, anti-resonant frequency, effective electromechanical coupling coefficient, and quality factor (Q value) of the device to complete the device simulation evaluation.

[0048] To verify the accuracy and efficiency of the FEM-SEM hybrid simulation method of this invention, such as... Figure 4 As shown: This embodiment also sets up two sets of numerical models (Model A and Model B) for comparison with the hybrid model C of the present invention, namely: Model A (Traditional Global FEM Model): The entire domain is conformally discretized using 9-node quadrilateral low-order finite element (FEM Q9 element).

[0049] Model B (Ideal Global SEM Model): The entire domain is conformally discretized using 16-node high-order spectral elements (SEM Q16 units, including GLL non-uniform integration points). This model, by employing a global high-order orthogonal polynomial, greatly suppresses numerical dispersion in thick substrates, and its calculation results serve as a high-precision reference benchmark in this embodiment.

[0050] Model C (hybrid model of this invention): The geometrically complex subdomain containing electrodes and piezoelectric films above is discretized using FEM Q9 elements; the large-size propagation subdomain containing a thick substrate below is discretized using SEM Q16 elements; the interface between the two subdomains is coupled using the Mortar cross-order projection constraint of this invention.

[0051] Figure 5 The graph shows a comparison of the admittance curves calculated by the three models in the same frequency band. The performance differences of each model can be clearly observed from the curve characteristics in the graph: The admittance curve of Model A (Global FEM Q9) shows significant frequency deviation and amplitude differences compared to the high-precision reference model B (Global SEM Q16) near the high-frequency resonant point. This is due to the unavoidable numerical dispersion error generated by the low-order FEM elements when simulating the long-distance propagation of high-frequency acoustic waves in deep, thick substrates.

[0052] The frequency domain admittance curve of the FEM-SEM hybrid model C of this invention is compared with the admittance curve of the global high-order model B. The two are highly consistent in three core dimensions: resonant frequency localization, amplitude response intensity, and phase evolution law. The above effect mainly stems from the Mortar interface coupling mechanism's accurate modeling capability of the physical field transmission between the upper and lower subdomains. It retains the FEM's ability to resolve the geometric details and strong gradient fields of the top electrode / piezoelectric layer, and leverages the exponential approximation accuracy of SEM for long-wavelength acoustic propagation behavior in thick substrate regions. This fundamentally suppresses the numerical dispersion caused by low-order discretization in deep media by traditional global FEM.

[0053] Ultimately, the FEM-SEM hybrid simulation method of this invention achieves the accuracy level of a global high-order model without increasing the complexity of top-level modeling or sacrificing geometric adaptability. This provides a reliable simulation tool with both high fidelity and high computational efficiency for the design optimization of high-frequency broadband SAW devices.

[0054] Embodiments of the present invention also provide a computing device, including: a processor and a memory storing a computer program, wherein the computer program, when executed by the processor, performs the method described above. All implementations in the above method embodiments are applicable to this embodiment and can achieve the same technical effects.

[0055] Embodiments of the present invention also provide a computer-readable storage medium storing instructions that, when executed on a computer, cause the computer to perform the method described above. All implementations in the above method embodiments are applicable to this embodiment and can achieve the same technical effects.

[0056] The above are preferred embodiments of the present invention. It should be noted that those skilled in the art can make various equivalent modifications or substitutions without departing from the principle of the present invention. These possible changes and substitutions also fall within the scope of protection covered by the present invention.

Claims

1. A hybrid FEM-SEM simulation method for surface acoustic wave devices based on Mortar interface coupling, characterized in that, The steps include the following: S1. Construct the overall geometric model of the SAW device. Based on the geometric features and physical properties of the SAW device, in space, the SAW device is divided into two non-overlapping subdomains along the depth direction: the upper geometrically complex subdomain containing complex electrode structures and piezoelectric films, and the lower large-size propagation subdomain containing thick dielectric layers and silicon substrates. The two subdomains are separated by a non-conformal interface. S2, Physical Domain Decomposition: The upper-level geometrically complex subdomains are discretized using a highly geometrically adaptable FEM, and low-order shape functions and unstructured meshes are used to extract the complex physical boundaries of the upper-level structure. The lower large-size propagation subdomain is discretized using SEM, which has high precision and low degrees of freedom. Large-size regular spectral units are used, and high-order Lagrange interpolation polynomials and non-uniformly distributed GLL nodes are used inside the units to achieve a high-precision approximation of the wave field in the lower base region with very few degrees of freedom. S3. Constructing the Mortar cross-order projection interface: On the non-conformal interface between the upper FEM subdomain and the lower SEM subdomain, Mortar finite element constraints are introduced. By defining the Lagrange multiplier subspace, the cross-space projection integral between the low-order uniform node shape function and the high-order non-uniform node spectral shape function is calculated to construct the Mortar interface coupling matrix that connects the two different discretization schemes. S4. Global System Solution and Key Parameter Extraction: The local stiffness / mass matrices of the upper FEM subdomain and the lower SEM subdomain are globally assembled with the Mortar interface coupling matrix to construct a saddle point system equation set for the global discrete system. Solving this equation set yields the global displacement field, potential field, and frequency domain admittance characteristics of the SAW device.

2. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 1, characterized in that, In step S2, the unstructured mesh used for the upper geometrically complex subdomain is a triangular or tetrahedral mesh, and the low-order shape function used is the shape function corresponding to the second-order element, specifically the shape function of the second-order nine-node quadrilateral element, which is based on uniformly distributed nodes.

3. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 2, characterized in that, The lower-level large-size propagation subdomain is discretized using the spectral element method (SEM), specifically including: The thick substrate is divided into a small number of large-sized regular units, specifically high-order quadrilateral or hexahedral units. High-order Lagrange orthogonal polynomials are used as basis functions inside the large units, and the order of the high-order Lagrange interpolation polynomials is N≥2. Non-uniformly distributed GLL nodes are used, with nodes dense at the edge of the unit and sparse at the center, so that deep acoustic waves can be accurately simulated with a very small number of nodes.

4. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 1, characterized in that, In step S3, the specific process of constructing the Mortar cross-level projection interface is as follows: S31. Define the FEM side of the interface as the slave surface and the SEM side as the master surface; S32. Define a Lagrange multiplier field on the interface to represent the unknown force or charge flux on the interface. S33. Establish a weakly constrained integral equation. This integral equation requires calculating the inner product between the FEM shape function obtained by low-order polynomial interpolation and the SEM shape function obtained by high-order Lagrange polynomial interpolation. The interface integral calculation adopts the high-order Gauss-Legendre integration rule to ensure the accuracy of the interface coupling matrix calculation. S34. Discretize and numerically integrate the weakly constrained integral equations to transform the continuous constraints into algebraic equations containing cross-order coupling matrices of the FEM and SEM sides.

5. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 1, characterized in that, In step S4, the local stiffness / mass matrices of the upper FEM subdomain and the lower SEM subdomain are globally assembled with the Mortar interface coupling matrix to construct the saddle point system equations of the global discrete system, specifically including: During global assembly, the frequency domain dynamic control equations of the FEM and SEM subdomains are combined with the Mortar interface constraint equations from step S3 to form a global discrete system matrix. The global discrete system matrix includes the frequency domain dynamic stiffness matrix of the FEM subdomain, the frequency domain dynamic stiffness matrix of the SEM subdomain, the Mortar interface coupling matrix, and the Lagrange multiplier correlation matrix. The frequency domain dynamic stiffness matrix includes the stiffness matrix, damping matrix, mass matrix, angular frequency, and imaginary unit.

6. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 5, characterized in that, The local stiffness / mass matrices of the upper geometrically complex subdomain and the lower large-size propagation subdomain are globally assembled with the Mortar interface coupling matrix, and an electric excitation load vector is applied to the global discrete system. The electric excitation load vector is applied only to the electrode region in the upper geometrically complex subdomain, and the excitation source of the lower large-size propagation subdomain is set to 0. The global discrete system is solved using a sparse saddle point system linear algebra solver.

7. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 5, characterized in that, The sparse direct solver can be used to perform LU decomposition on the saddle point system and solve it back-substituted, or the preconditioned GMRES iterative method can be used with Schur supplemented preconditioners to accelerate convergence, or a block Gaussian elimination strategy can be used to first eliminate the λ variable and then solve the reduced FEM-SEM coupled system to obtain the numerical solutions of the global displacement field and electric potential field, and the port admittance can be calculated accordingly.

8. The FEM-SEM hybrid simulation method for surface acoustic wave devices based on Mortar interface coupling according to claim 1, characterized in that, It also includes S5 and performance parameter extraction: Based on the obtained potential and displacement of the electrode region, the frequency domain admittance of the device port is calculated, and then the performance indicators of the device, such as resonant frequency, anti-resonant frequency, effective electromechanical coupling coefficient and quality factor, are extracted.

9. A computing device, characterized in that, include: One or more processors; A storage device for storing one or more programs, which, when executed by one or more processors, cause the one or more processors to implement the FEM-SEM hybrid simulation method as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a program that, when executed by a processor, implements the FEM-SEM hybrid simulation method as described in any one of claims 1-8.

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