Supersingular volume solution method based on cuboid grid

Abstract

The invention discloses a supersingular volume integral solving method based on a cuboid grid. The method comprises the following steps: establishing a volume-area integral equation (VSIE) of a metal-non-magnetic medium mixed structure; deducing a to-be-processed supersingular volume fraction; and solving supersingular integral calculation. The invention provides a supersingular volume solution method based on a cuboid grid for the first time, and the method greatly improves the calculation precision of solving the VSIE by using a MoM-Nystrom hybrid method. In the field of electromagnetic field simulation and analysis, the invention provides an efficient and accurate numerical method for solving the volume integral equation in the VSIEs. Compared with an existing algorithm, on the basis of the solving method provided by the invention, the MoM-Nystrom hybrid method is used for solving the VSIE, so that the calculation efficiency, the precision and the like are remarkably improved, and a reliable theoretical tool and a practical solution are provided for electromagnetic property analysis of a complex three-dimensional chip structure.

Description

Technical Field

[0001] The invention belongs to the field of three-dimensional chip structure analysis, and in particular relates to a hypersingular volume integral solution method based on a cuboid grid. Background Art

[0002] Packaging and testing are important links in the design and manufacturing of 3D integrated circuit chips. As 3D integrated circuit design becomes increasingly complex, the metal-dielectric hybrid structure often has multi-scale characteristics, which puts higher demands on electromagnetic analysis algorithms. Accurately extracting key data such as parasitic parameters in such structures requires the use of more advanced electromagnetic algorithms for precise analysis.

[0003] In the prior art, the Method of Moments (MoM) is a widely used electromagnetic analysis method. However, the traditional MoM relies on conformal meshes to segment and fit the target surface. When dealing with models with multi-scale features, this reliance on conformal meshes can lead to over-dense or deformed meshes in the transition zone, increasing the computational time and storage overhead of mesh generation.

[0004] In contrast, the Nystrom method has the advantage of lower mesh requirements. When using volume-surface integral equations (VSIEs) for modeling and using the Nystrom method or the MoM-Nystrom hybrid method to solve, a non-conformal mesh can be used to segment the target area. Chinese patent CN119089858A discloses a hybrid solution method for volume-surface integral equations for three-dimensional chip structure analysis, which applies the Nystrom method to the solution of volume-surface integral equations. Since the Nystrom method has the characteristics of being independent of conformal meshes and having low requirements on the intrinsic parameters of the materials inside the mesh, the number of unknowns in the traditional method is greatly reduced when solving the volume-surface integral equations, which greatly reduces the difficulty of solving the problem. However, the hypersingular volume integrals that appear in the calculation of this method lead to its limited accuracy. Summary of the invention

[0005] In order to improve the calculation accuracy of MoM-Nystrom hybrid method for solving VSIE, the present invention proposes a super singular volume integral solution method based on cuboid grid.

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

[0007] A method for solving super singular volume integral based on cuboid grids, comprising the steps of:

[0008] Step 1: Establishing the volume-surface integral equation (VSIE) of the metal-nonmagnetic medium hybrid structure;

[0009] Step 2 derives the supersingular volume integral to be processed;

[0010] Step 3: Solve the hypersingular integral.

[0011] Beneficial effects:

[0012] When using the MoM-Nystrom hybrid method to solve VSIE, hypersingular volume integrals are introduced by the dyadic Green's function. The present invention proposes for the first time an accurate calculation formula for these hypersingular integrals in the rectangular volume integral domain. Based on the formula proposed in the present invention, when using the MoM-Nystrom hybrid method to solve VSIE, good calculation accuracy is achieved while occupying fewer computing resources. In the field of electromagnetic field simulation and analysis, the present invention provides an efficient and accurate numerical method for solving volume integral equations in VSIEs. Compared with existing algorithms, the use of the MoM-Nystrom hybrid method to solve VSIE on the basis of the solution method proposed in the present invention has achieved significant improvements in computational efficiency, accuracy, etc., and provides a reliable theoretical tool and practical solution for the electromagnetic characteristics analysis of complex three-dimensional chip structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 It is a schematic diagram of the process of the present invention.

[0014] Figure 2 It is a schematic diagram of the super singular volume targeted by the present invention being integrated on a cuboid unit.

[0015] Figure 3 Schematic diagram of the electromagnetic interconnection structure and antenna structure to be analyzed applicable to the present invention.

[0016] Figure 4 For example, based on the formula proposed by the present invention Figure 2 Calculation results of the input impedance of the rectangular patch antenna structure shown in (a). Figure 5 For example, based on the formula proposed by the present invention Figure 2 (d) shows the input impedance calculation results of the chip package related structure. DETAILED DESCRIPTION

[0017] The technical solution provided by the present application will be further described below in conjunction with specific embodiments and accompanying drawings. The advantages and features of the present application will become more apparent with the following description.

[0018] A method for solving super singular volume integral based on cuboid grid includes the following steps:

[0019] Step 1 Establish the volume-surface integral equation (VSIE) of the metal-nonmagnetic medium hybrid structure

[0020] Step 1.1 Establish the surface integral equation that characterizes the electric field strength of the metal part

[0021] For the metal part of the structure to be analyzed, the following surface integral equation (1) is used to describe it:

[0022]

[0023] Where: S represents the surface of the metal area to be analyzed, r represents the position vector of the observation point on the S surface, represents the unit normal vector of S at r, E ex (r) represents the value of the excitation field loaded on the metal part at r, r' represents the position vector of the equivalent surface current, J S (r′) represents the surface current density at r'. i is an imaginary unit, ω is the angular frequency of the harmonic field under study, and μ0 is the vacuum permeability. dS′ represents the area element at r' on S. It represents the size of the radiation field of the ideal point source r′ in free space at the observation point r.

[0024] Step 1.2 derives volume-surface integral equations (VSIEs) that can be used to describe the metal-dielectric structure to be analyzed.

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

[0026]

[0027]

[0028] Where: V represents the medium area to be analyzed, r represents the position vector of the observation point located in V, r' in formula (2) and formula (3) respectively represents the position vector of the equivalent body current and the position vector of the equivalent body magnetic current, J(r') and M V (r′) represents the equivalent volume current and equivalent volume magnetic current at r′, ε0 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 value of the excitation field of the external electric field of the medium at r, H ex (r) represents the value of the excitation field of the external magnetic field of the medium at r.

[0029] Step 1.3 Obtaining VSIEs through field coupling

[0030] 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, and they have the following relationship with the body current and body magnetic current in the medium area:

[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 co-field under study, ε0 / μ0 is the vacuum dielectric constant / vacuum permeability, ε r / μ r is the relative permittivity / relative permeability of the medium under study. r′ in formula (4) and formula (5) are the position vector of the body current and the position vector of the body magnetic current, respectively.

[0034] Considering the net electric field and net magnetic field on the metal-dielectric interface, using the corresponding boundary conditions, we can get:

[0035]

[0036]

[0037]

[0038] Assuming that the medium material under study is non-magnetic material, then M V (r′)=0. Formula (6) and formula (7) can be rewritten into the following forms:

[0039]

[0040]

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

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

[0043] In formula (10), the first term is the surface integral equation, which is usually directly calculated using the moment method. When solving the second volume integral equation using the Nystrom method or the MoM-Nystrom hybrid method, it is necessary to process the dyadic Green's function.

[0044]

[0045] in, represents the dyadic Green's function in vacuum, represents the unit vector, k0 represents the wave number of the incident field under study in vacuum, g represents the scalar Green's function, r represents the position vector of the observation point, and r' represents the position vector of the source point.

[0046] Establish as Figure 2 The volume meshing and coordinate system shown in the figure: △V is a rectangular parallelepiped unit, and the lengths of the three adjacent sides are set to 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 a local coordinate system (u, v, w). In the local coordinate system (u, v, w), the double gradient operator Expanding to get the following formula,

[0047]

[0048] in

[0049]

[0050] The volume integral can be decomposed into a line integral along w and a surface integral along △S. The line integral along w is regular and can be calculated numerically, while the surface 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 regularizing the surface integral on △S by singular point subtraction, its closed form expression can be derived as the integrand of the line integral along w.

[0051] Through this method, the supersingular volume integral to be calculated has the following form:

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] 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 applies, and the integral with this type of function as the integrand is called a supersingular volume integral.

[0059] Step 3: Solve the hypersingular integral

[0060] First, the hypersingular volume integral to be processed is converted into the form of outer layer line integral and inner layer surface integral. By using Stokes' theorem, the inner layer surface integral is further converted into perimeter line integral. By further variable substitution, the integral result of the hypersingular integral on the cuboid unit is finally obtained.

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

[0062]

[0063]

[0064]

[0065] I4=0

[0066]

[0067] I6=0

[0068] in:

[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 is 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) is a rectangular patch antenna structure, (b) is a corner-cut rectangular patch antenna structure, (c) is a cavity back-fed cross slot antenna structure, and (d) is a chip packaging structure containing 2 metal wires and 2 layers of dielectric substrate.

[0084] like Figure 4 As shown, the calculation results of the real part (blue circle mark) and the imaginary part (red diamond mark) of the patch antenna input impedance obtained using the supersingular 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 supersingular integral calculation formula proposed in the present invention.

[0085] like Figure 5As shown, the calculation results of the S parameters of the chip packaging structure containing 2 metal wires and 2 dielectric substrates obtained by using the hypersingular integral calculation formula proposed in the present invention (the red line represents the simulation result obtained by the method of the present invention, the red circle mark corresponds to the S11 parameter, and the red diamond mark represents the S14 parameter) are consistent with the calculation results 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 results of the traditional moment method (the black circle represents the S11 result, and the black diamond represents the S14 result), indicating the accuracy and reliability of the hypersingular integral calculation formula proposed in the present invention.

[0086] The above description is only a description of the preferred embodiments of the present application, and is not intended to limit the scope of the present application. Any changes or modifications made by any person skilled in the art based on the above disclosed technical contents shall be deemed as equivalent effective embodiments and shall fall within the scope of protection of the technical solution of the present application.

Claims

1. A method for solving super singular volume integral based on cuboid grid, characterized in that: Includes steps: Step 1: Establishing volume-surface integral equations (VSIEs) of metal-nonmagnetic medium hybrid structure; Step 2 derives the supersingular volume integral to be processed; Step 3: Solve the hypersingular integral.

2. A method for solving super singular volume integral based on cuboid grid as claimed in claim 1, characterized in that: The step 1 comprises: Step 1.1 Establish the surface integral equation that characterizes 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, r represents the position vector of the observation point on the S surface, represents the unit normal vector of S at r, E ex (r) represents the value of the excitation field loaded on the metal part at r, r' represents the position vector of the equivalent surface current, J S (r′) represents the surface current density at r', i is an imaginary unit, ω is the angular frequency of the harmonic field under study, μ0 is the vacuum permeability, dS′ represents the area element at r' on S, It represents the size of the radiation field of the ideal point source r′ in free space at the observation point r; Step 1.2 Derive the volume-surface integral equations (VSIEs) describing 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, r represents the position vector of the observation point located in V, r' in formula (2) and formula (3) respectively represents the position vector of the equivalent body current and the position vector of the equivalent body magnetic current, J(r') and M V (r′) represents the equivalent volume current and equivalent volume magnetic current at r′, ε0 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 value of the excitation field of the external electric field of the medium at r, H ex (r) represents the value of the excitation field of the external magnetic field of the medium at r; 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 body current and body magnetic current in the medium area: ω is the angular frequency of the co-field under study, ε0 / μ0 is the vacuum dielectric constant / vacuum permeability, ε r / μ r is the relative permittivity / relative permeability of the medium under study, r′ in formula (4) and formula (5) are the position vector of the body current and the position vector of the body magnetic current respectively; Considering the net electric field and net magnetic field on the metal-dielectric interface, using the corresponding boundary conditions, we get: Assuming that the medium material under study is non-magnetic material, then M V (r′)=0, and rewrite formula (6) and formula (7) into the following form: Formulas (9) and (10) are the required VSIEs.

3. A method for solving super singular volume integral based on cuboid grid as claimed in claim 2, characterized in that: The step 2 comprises: In formula (10), the first term is the surface integral equation, which is directly calculated using the moment method. When solving the second volume integral equation using the Nystrom method or the MoM-Nystrom hybrid method, it is necessary to process the dyadic Green's function. in, represents the dyadic Green's function in vacuum, Represents the unit vector, k0 represents the wave number of the incident field under study in vacuum, g represents the scalar Green's function, r represents the position vector of the observation point, and r' 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 to 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, and establish the local coordinate system (u, v, w); in the local coordinate system (u, v, w), the double gradient operator Expanding to get the following formula, in The volume integral is decomposed into the line integral along w and the surface integral along △S. The line integral along w is regular and is calculated by numerical method. The surface integral along △S is 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 regularizing the surface integral on △S 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 applies, and the integral with this type of function as the integrand is called a supersingular volume integral.

4. A method for solving super singular volume integral based on cuboid grid as claimed in claim 3, characterized in that: The step 3 comprises: Firstly, the hypersingular volume integral to be processed is converted into the form of outer layer line integral and inner layer surface integral. Then, the inner layer surface integral is further converted into the perimeter line integral by using Stokes' theorem. Then, by further variable substitution, the integration result of the supersingular integral on the rectangular unit is finally obtained.

5. A method for solving super singular volume integral based on cuboid grid as claimed in claim 4, characterized in that: The step 3 comprises: The integral solution is expressed as follows: I4＝0 I6=0where: 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 cross section of the cuboid mesh, and w represents the w coordinate of the cross section in the local coordinate system.

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