A cuboid mesh based hypersingular panel method

Abstract

The application discloses a kind of based on cuboid grid's super singular volume integral solving method, including steps: the volume-area integral equation (VSIE) of metal-nonmagnetic medium mixed structure is established;Deduce the super singular volume integral to be handled;Super singular integral calculation is solved.This application first gives a kind of based on cuboid grid's super singular volume integral solving method, this method greatly improves the calculation precision of VSIE using MoM-Nystrom hybrid method solving.In the field of electromagnetic field simulation and analysis, the application provides an efficient and accurate numerical method for solving the volume integral equation in VSIE.Compared with existing algorithms, using MoM-Nystrom hybrid method to solve VSIE based on the solving method proposed in the application has made significant improvement in calculation efficiency, accuracy and other aspects, and provides a reliable theoretical tool and practical solution for electromagnetic property analysis of complex three-dimensional chip structure.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of three-dimensional chip structure analysis, and particularly relates to a super-singular volume integral solving method based on a cuboid grid. BACKGROUND

[0002] Packaging and testing are important links in the design and manufacture of three-dimensional integrated circuit chips. With the increasing complexity of three-dimensional integrated circuit design, the metal-dielectric mixed structure therein often has multi-scale characteristics, which puts higher requirements on electromagnetic analysis algorithms. Accurate extraction of key data such as parasitic parameters in such structures requires more advanced electromagnetic algorithms for accurate analysis.

[0003] In the prior art, the method of moments (MoM) is a widely used electromagnetic analysis method. However, the traditional method of moments relies on conformal grids for the subdivision and fitting of the target surface. When dealing with models with multi-scale characteristics, this dependence on conformal grids can cause over-dense or deformed grids in the transition zone, increasing the computational time and storage overhead of grid generation.

[0004] In contrast, the Nystrom method has the advantage of lower requirements for grids. When modeling using volume-surface integral equations (VSIEs) and solving using the Nystrom method or MoM-Nystrom hybrid method, non-conformal grids can be used to subdivide the target region. Chinese patent CN119089858A discloses a volume-surface integral equation hybrid solving method for three-dimensional chip structure analysis, which applies the Nystrom method to the solving of volume-surface integral equations. Due to the characteristics of the Nystrom method, such as not relying on conformal grids and having low requirements for the intrinsic parameters of the material inside the grid, the number of unknowns in the traditional method is greatly reduced when solving volume-surface integral equations, greatly reducing the difficulty of solving. However, the super-singular volume integral in the calculation of this method leads to its limited accuracy. SUMMARY

[0005] To improve the calculation accuracy of the MoM-Nystrom hybrid method for solving VSIE, the application proposes a super-singular volume integral solving method based on a cuboid grid.

[0006] The technical solution of the application is as follows:

[0007] A super-singular volume integral solving method based on a cuboid grid, comprising the steps of:

[0008] Step 1: establishing volume-surface integral equations (VSIE) for a metal-non-magnetic dielectric mixed structure;

[0009] Step 2 deduce hypersingular volume integrals to be processed;

[0010] Step 3 solve hypersingular integrals.

[0011] Beneficial effects:

[0012] When solving VSIEs using MoM-Nystrom hybrid method, hypersingular volume integral terms are introduced by dyadic Green's function. The present application firstly proposes the accurate calculation formula of these hypersingular integrals in cuboid integral domain. Based on the formula proposed by the present application, when solving VSIEs using MoM-Nystrom hybrid method, good calculation accuracy is achieved with less computing resources. In the field of electromagnetic field simulation and analysis, the present application provides an efficient and accurate numerical method for solving volume integral equations in VSIEs. Compared with existing algorithms, the MoM-Nystrom hybrid method for solving VSIEs based on the solving method proposed by the present application has made significant improvements in calculation efficiency, accuracy and other aspects, and provides a reliable theoretical tool and practical solution for electromagnetic characteristic analysis of complex three-dimensional chip structures. BRIEF DESCRIPTION OF DRAWINGS

[0013] Figure 1 The present application is a method flowchart.

[0014] Figure 2 The present application is a schematic diagram of hypersingular volume integrals in cuboid unit.

[0015] Figure 3 The present application is a schematic diagram of electromagnetic interconnection structure and antenna structure to be analyzed.

[0016] Figure 4 The present application is a calculation result of input impedance of the rectangular patch antenna structure shown in (a) of Figure 2 Figure 5 The present application is a calculation result of input impedance of the chip packaging related structure shown in (d) of Figure 2 DETAILED DESCRIPTION

[0017] The technical solutions provided by the present application will be further described below with specific embodiments and their accompanying drawings. The advantages and features of the present application will be clearer in combination with the following description.

[0018] A hypersingular volume integral solving method based on cuboid grid, comprising the following steps:

[0019] Step 1: establish volume-surface integral equation (VSIE) of metal-non-magnetic medium mixed structure

[0020] ​​Step 1.1. Formulate the area integral equation characterizing the electric field intensity of the metallic part

[0021] For the metallic part in the structure to be analyzed, the following area integral equation (1) is used to describe:

[0022]

[0023] where S denotes the surface of the metallic region to be analyzed, r denotes the position vector of an observation point on S, denotes the unit normal vector of S at r, E ex (r) denotes the value of the excitation field at r, r' denotes the position vector of the equivalent surface current, J S (r') denotes the surface current density at r'. i is the imaginary unit, ω is the angular frequency of the time-harmonic field under study, μ0 is the vacuum permeability. dS' denotes the area element on S at r'. denotes the value of the excitation field at r, r' denotes the position vector of the equivalent surface current, J

[0024] Step 1.2. Derive the volume-area integral equations (VSIEs) that can be used to describe the metallic-dielectric structure to be analyzed.

[0025] For the dielectric part in the structure to be analyzed, the following volume integral equations (2) and (3) are used to describe:

[0026]

[0027]

[0028] where V denotes the dielectric region to be analyzed, r denotes the position vector of an observation point within V, r' denotes the position vector of the equivalent volume current and the equivalent volume magnetic current in equations (2) and (3), respectively, J(r') and M V (r') denote the equivalent volume current and the equivalent volume magnetic current at r', respectively, ε0 is the vacuum permittivity. E(r) and H(r) denote the net electric field and the net magnetic field within the dielectric, E ex (r) denotes the value of the excitation field at r, H ex (r) denotes the value of the excitation field at r.

[0029] Step 1.3. Obtain the VSIEs by coupling the fields

[0030] For the dielectric region, in practice, there are often E ex = H ex = 0. E(r) and H(r) denote the net electric field and the net magnetic field within the dielectric, which have the following relationship with the volume current and the volume magnetic current within the dielectric region:

[0031] JV (r') = iω∈0(1 -∈ r )E(r') (4)

[0032] M V (r') = iωμ0(1 -μ r )H(r') (5)

[0033] ω is the angular frequency of the time-harmonic field under study, ε0 / μ0is the vacuum permittivity / magnetic permeability, ε r / μ r is the relative permittivity / relative permeability of the medium under study. r' is the position vector of the volume current / volume magnetic current in eq. (4) / (5), respectively.

[0034] Considering the net electric field and the net magnetic field on the metal-dielectric interface, the corresponding boundary conditions can be used to obtain:

[0035]

[0036]

[0037]

[0038] Assuming that the medium material under study is a non-magnetic material, M V (r') = 0. The corresponding eq. (6) and eq. (7) can be rewritten as follows:

[0039]

[0040]

[0041] The eq. (9) and eq. (10) are the required VSIEs.

[0042] Step 2 Derivation of the hypersingular volume integral to be processed

[0043] In eq. (10), the first term is an area integral equation, which is usually directly calculated by the method of moments. The second term is a volume integral equation. When the Nystrom method or the MoM-Nystrom hybrid method is used to solve it, the dyadic Green's function,

[0044]

[0045] where, represents the dyadic Green's function in vacuum, represents the unit dyadic, k0represents the wave number of the incident field under study in vacuum, g represents the scalar Green's function, and r represents the position vector of the observation point. r' represents the position vector of the source point.

[0046] The following formula is established:Figure 2 The body grid subdivision and coordinate system shown: △V is a cuboid element, and the three adjacent side lengths are set as a, b and c. Taking the centroid O of △V as the origin, taking the plane △S passing through O and parallel to a pair of parallel faces of △V as the uOv coordinate plane, and taking the normal direction of △S as the w axis direction, a local coordinate system (u, v, w) is established. In the local coordinate system (u, v, w), the double gradient operator is expanded to obtain the following formula,

[0047]

[0048] wherein

[0049]

[0050] The volume integral can be decomposed into a line integral along w and an area integral along △S. Among them, the line integral along w has regularity and can be calculated by numerical method; while the area integral along △S may be regular, nearly singular or singular, depending on the spatial relationship between the observation point and the source rectangle △S moving along the w axis. After regularization processing of the curved surface integral on △S by point subtraction, its closed form expression can be derived as the integral term of the line integral along w.

[0051] By this method, the super-singular volume integral to be calculated has the following form,

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] wherein R=(u 2 +v 2 +w 2 ) 1 / 2 , and R=0 when u=v=w=0. The integral function of the above integral has the same asymptotic behavior as 1 / R 3 near R=0. The integral with this type of function as the integral function is called super-singular volume integral.

[0059] Step 3: Solve the super-singular integral

[0060] Firstly, the hypersingular volume integral to be processed is converted into the form of outer line integral and inner area integral. The inner area integral is further transformed into the line integral around the boundary by using Stokes theorem. Through further variable substitution, the integral result of hypersingular integral over a cuboid element is finally obtained.

[0061] The solution of the above integral is expressed as follows:

[0062]

[0063]

[0064]

[0065] I4= 0

[0066]

[0067] I6= 0

[0068] Wherein:

[0069]

[0070]

[0071] a i = (u i -u)(v i+1 -v i )-(v i -v)(u i+1 -u i )

[0072] b i = (u i+1 -u i )(u i -u)+(v i+1 -v i )(v i -v)

[0073] c i = (u i+1 -u i )(u i+1 -u)+(v i+1 -v i )(v i+1 -v)

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081]

[0082] Among them, u i+1 ,v i+1 Represents the local coordinates of the i+1th vertex of the rectangular cross section of the cuboid mesh, u i ,v i Represents the local coordinates of the i-th vertex of the rectangular section of the cuboid mesh, and w represents the w coordinate of the section in the local coordinate system. All variables can be based on u i+1 ,v i+1 ,u i ,v i ,w calculated, some of the variables calculated first also participate in the calculation basis of subsequent variables (such as l i and r i ).

[0083] The proposed method for solving the hypersingular integral can be widely used in electromagnetic parameter analysis of various metal-dielectric hybrid structures, including but not limited to Figure 3 The analysis objects are: (a) a rectangular patch antenna structure, (b) a corner-cut rectangular patch antenna structure, (c) a cavity-backfed cross-slot antenna structure, and (d) a chip package structure containing two metal wires and two layers of dielectric substrate.

[0084] like Figure 4 As shown in the figure, the calculation results of the real part (blue circle mark) and imaginary part (red diamond mark) of the patch antenna input impedance obtained using the hypersingular integral calculation formula proposed in the present invention are consistent with the calculation results of the mature commercial software FEKO (the blue implementation represents the real part, and the red dotted line represents the imaginary part), indicating the accuracy and reliability of the hypersingular integral calculation formula proposed in the present invention.

[0085] like Figure 5As shown, the calculation result of the S parameter of the chip packaging structure with 2 metal wires and 2 layers of dielectric substrate obtained by using the super-singular integral calculation formula provided in the application (the red line type represents the simulation result obtained by using the method of the application, the red circle mark corresponds to the S11 parameter, and the red diamond mark represents the S14 parameter) is consistent with the calculation result of the mature commercial software FEKO (the blue implementation with a circle represents the S11 parameter simulation result, and the blue implementation with a diamond mark represents the S14 simulation result) and the result of the conventional moment method (the black circle represents the S11 result, and the black diamond represents the S14 result), indicating the accuracy and reliability of the super-singular integral calculation formula provided in the application.

[0086] The above description is only a description of the preferred embodiments of the application, and is not any limitation on the scope of the application. Any modification or modification made by any person skilled in the art according to the above disclosed technical content shall be regarded as an equivalent effective embodiment, and shall fall within the scope of protection of the technical scheme of the application.

Claims

1. A method for solving hypersingular volume integrals based on cuboid grids, characterized in that: Including steps: Step 1: Establish the volume-surface integral equations (VSIEs) of the metal-nonmagnetic medium hybrid structure. The VSIEs are as follows: In formula (9), S represents the surface of the metal area to be analyzed, represents the position vector of the observation point on the S surface, Indicates that S is The unit normal vector at E ex (r) represents the excitation field loaded on the metal part. The value of represents the position vector of the equivalent surface current, express The surface current density at Respectively The equivalent body current at , i is an imaginary unit, is the angular frequency of the time-harmonic field under investigation, is the vacuum permeability, Indicates S The area of ​​the infinitesimal element at represents an ideal point source The radiation field in free space at the observation point The size of the place; In formula (10), E(r) represents the net electric field in the medium, V represents the medium area to be analyzed, represents the position vector of the observation point located in V, represents the position vector of the equivalent body current; Step 2: Derivation of the hypersingular volume integral to be processed, including: In formula (10), the first term is the surface integral equation, which is directly calculated using the moment method; the second term is the volume integral equation, which needs to be solved by the Nystrom method or the MoM-Nystrom hybrid method, and the dyadic Green's function needs to be processed. in, represents the dyadic Green's function in vacuum, represents the unit dyadic, represents the wave number of the incident field under study in vacuum, g represents the scalar Green's function, represents the position vector of the observation point, Represents the position vector of the source point; establish the volume mesh subdivision and coordinate system: △V is a rectangular parallelepiped unit, and the lengths of the three adjacent sides are set as a, b and c; take the center of mass O of △V as the origin, take the plane △S passing through O and parallel to a pair of parallel faces of △V as the uOv coordinate plane, and then take the normal direction of △S as the w-axis direction to establish the local coordinate system (u, v, w); in the local coordinate system (u, v, w), the double gradient operator Expanding it gives the following formula: in The volume integral is decomposed into a line integral along w and a surface integral along ΔS. The line integral along w is regular and can be calculated numerically. The surface integral along ΔS is regular, nearly singular, or singular, depending on the spatial relationship between the observation point and the source rectangle ΔS moving along the w axis. After regularizing the surface integral on ΔS by singular point subtraction, its closed-form expression is derived as the integrand of the line integral along w. Through the above method, the supersingular volume integral to be calculated has the following form: Where R=(u 2 +v 2 +w 2 ) 1 / 2 , when u=v=w=0, R=0; the above integral integrand has the sum 1 / R near R=0 3 The same asymptotic behavior, the integral with this type of function as the integrand is called a supersingular volume integral; Step 3: Solve the hypersingular volume integral, including: First, the hypersingular volume integral to be processed is converted into the form of outer layer line integral and inner layer surface integral. Then, by using Stokes' theorem, the inner layer surface integral is further converted into perimeter line integral. Then, by further variable substitution, the integral result of the supersingular integral on the cuboid element is finally obtained; The integral solution is expressed as follows: in: in, represents the local coordinates of the i+1th vertex of the rectangular section of the cuboid mesh, represents the local coordinates of the i-th vertex of the rectangular section of the cuboid mesh, Represents the w coordinate of the section in the local coordinate system.

2. A method for solving hypersingular volume integrals based on a rectangular parallelepiped grid according to claim 1, characterized in that: The step 1 comprises: Step 1.1 Establish a surface integral equation to characterize the electric field strength of the metal part For the metal part of the structure to be analyzed, the following surface integral equation (1) is used to describe it: Where: S represents the surface of the metal area to be analyzed, represents the position vector of the observation point on the S surface, Indicates that S is The unit normal vector at E ex (r) represents the excitation field loaded on the metal part. The value of represents the position vector of the equivalent surface current, express The surface current density at , i is an imaginary unit, is the angular frequency of the time-harmonic field under investigation, is the vacuum permeability, Indicates S The area of ​​the infinitesimal element at represents an ideal point source The radiation field in free space at the observation point The size of the place; Step 1.2 Derive the volume-surface integral equations (VSIEs) used to describe the metal-dielectric structure to be analyzed. For the medium part of the structure to be analyzed, the following volume integral equations (2) and (3) are used to describe it: Where: V represents the medium area to be analyzed, represents the position vector of the observation point located in V, In formula (2) and formula (3), the position vector of the equivalent body current and the position vector of the equivalent body magnetic current are respectively expressed. and Respectively The equivalent volume current and equivalent volume magnetic current at is the vacuum dielectric constant; E(r) and H(r) represent the net electric field and net magnetic field in the medium, respectively. ex (r) represents the excitation field of the external electric field of the medium. The value of H ex (r) represents the excitation field of the external magnetic field of the medium The value of Step 1.3 Obtaining VSIEs through field coupling For the dielectric region, in practice, E ex = H ex = 0; E(r) and H(r) represent the net electric field and net magnetic field in the medium, which have the following relationship with the volume current and volume magnetic current in the medium area: is the angular frequency of the time-dependent field under study, / is the vacuum permittivity / vacuum permeability, / is the relative permittivity / relative permeability of the medium under study, In formula (4) and formula (5), are the position vector of body current and the position vector of body magnetic current respectively; Considering the net electric field and net magnetic field at the metal-dielectric interface and using the corresponding boundary conditions, we obtain: If the medium material under study is a non-magnetic material, then , rewrite formula (6) and formula (7) into the following forms: Formulas (9) and (10) are the required VSIEs.

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