Quantitative analysis method and system of electric field of rod-shaped multilayer structure based on layer potential technique

Abstract

The application provides a rod-shaped multilayer structure electric field quantitative analysis method and system based on layer potential technology, and relates to the technical field of object detection. A multilayer structure geometric model is established, including an internal rod-shaped content, a middle layer region and an outer layer region; an electric potential control equation and a boundary transmission condition of the multilayer structure under a background electric field are constructed, an electric field boundary value problem is converted into an integral equation system on the multilayer boundary based on layer potential theory; layer potential operators in the integral equation system are subjected to asymptotic analysis, and a quantitative expression of a perturbation electric field is derived; according to the quantitative expression, geometric parameters and position information of the internal rod-shaped content are uniquely determined through one-time boundary electric field measurement data. Therefore, the perturbation electric field of the rod-shaped three-layer structure is accurately described, and a uniqueness theory that uniquely reverses and restores the rod-shaped content only by one-time boundary measurement is established.

Description

Technical Field

[0001] This invention relates to the field of object detection technology, and in particular to a method and system for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology. Background Technology

[0002] Multilayered composite structures with rod-like inclusions are core foundational structures in fields such as microelectromechanical systems (MEMS), semiconductor materials, and biochemical sensing. These structures typically consist of an outer layer as a substrate, a middle functional layer, and an inner layer of nanoscale rod-like inclusions. The geometric parameters and positions of the internal rod-like structures directly determine the overall device performance. However, the internal rod-like inclusions of such structures are typical "black boxes." They cannot be disassembled for direct observation, as disassembly would damage the original device performance and render it meaningless for practical applications. Furthermore, no equipment can penetrate the outer and middle layers to accurately detect their aspect ratio, dimensions, and positions. The internal structural state can only be reconstructed by measuring the external boundary electric field and combining it with theoretical models. This process, known as structural reconstruction or restoration, is a key research direction in this field.

[0003] Currently, research on the reconstruction of multilayer structures of rod-shaped inclusions based on electrostatic field boundary measurements has become a key approach to improve the design and detection accuracy of related devices. However, its technological development is still difficult to match the actual application requirements for precise and efficient structural detection.

[0004] Existing quantitative studies on perturbations of rod-shaped inclusions within electrostatic field systems are mostly limited to simple single- or double-layer structures, and have not yet extended to three-layer composite structures that are more suitable for practical applications. Furthermore, the few studies on multi-layer structures lack precise quantitative characterization methods for perturbation electric fields, making it impossible to clearly establish the correspondence between perturbation electric fields and parameters of the internal rod-shaped structure. This makes it difficult to effectively reconstruct the internal rod-shaped inclusions using external electric field measurement data, becoming a core bottleneck restricting the practical application of this technology. Summary of the Invention

[0005] To address the aforementioned issues, this invention proposes a quantitative analysis method and system for the electric field of rod-shaped multilayer structures based on layer potential technology. This method enables accurate characterization of the perturbation electric field of rod-shaped three-layer structures and establishes a unique theory that allows for the unique inverse deduction and reconstruction of rod-shaped inclusions using only a single boundary measurement.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] In a first aspect, the present invention provides a method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology, comprising:

[0008] Establish a multi-layered geometric model, including internal rod-shaped inclusions, intermediate layer regions, and outer layer regions;

[0009] We construct the potential control equations and boundary transport conditions of a multilayer structure under a background electric field, and transform the electric field boundary value problem into an integral equation system on the multilayer boundary based on the layer potential theory.

[0010] Asymptotic analysis of the layer potential operator in the integral equation system is performed to derive a quantitative expression for the perturbation electric field.

[0011] Based on the quantitative expression, the geometric parameters and location information of the internal rod-shaped inclusions are uniquely determined using boundary electric field measurement data.

[0012] As an optional implementation method, the establishment of the multi-layer structure geometric model specifically includes:

[0013] Define the central axis, length, and cross-sectional dimensions of the internal rod-shaped inclusion to form the rod-shaped geometric body and end structure;

[0014] The middle layer region completely surrounds the contents, and the outer layer region completely surrounds the middle layer region. Each layer boundary is a smooth closed curve, forming a three-layer nested structure from the inside to the outside.

[0015] As an alternative implementation, the construction of the potential control equations and boundary transport conditions for the multilayer structure under the background electric field, based on layer potential theory, transforms the electric field boundary value problem into a system of integral equations on the multilayer boundary, specifically including:

[0016] The electric field boundary value problem of multilayer structures is described by establishing the potential control equations and the interlayer current continuity condition within each layer region.

[0017] By using a single-layer potential to represent the potential distribution on each layer boundary, the boundary value problem is transformed into an integral expression of the boundary density function.

[0018] By applying the interlayer current continuity condition to the integral expression and combining it with the transmission conditions at each layer boundary, a closed integral equation system with respect to the multilayer boundary density function is constructed.

[0019] As an alternative implementation, the asymptotic analysis of the layer potential operator in the integral equation system to derive a quantitative expression for the perturbation electric field specifically includes:

[0020] By asymptotically expanding the single-layer potential operator and related integral operators, an approximate expression for the boundary density function is derived.

[0021] Substituting the approximate expression of the boundary density function in asymptotic form into the potential expression, we obtain the explicit asymptotic formula for the perturbation potential.

[0022] Based on the relationship between electric field and electric potential, a quantitative expression for the perturbation electric field is derived from the explicit asymptotic formula of the perturbation potential.

[0023] As an alternative implementation, the step of uniquely determining the geometric parameters and location information of the internal rod-shaped inclusion based on the quantitative expression and using first-order boundary electric field measurement data specifically includes:

[0024] Substitute the actual measured boundary disturbance electric field data into the quantitative expression;

[0025] By using the correspondence between the geometric parameters and the electric field distribution in the quantitative expression, the preliminary geometric parameters of the inclusions can be deduced, including the center position, axial direction, length, and cross-sectional dimensions.

[0026] Based on the preliminary geometric parameters, extreme point matching and direction analysis are used to uniquely determine all geometric and positional parameters of the inclusions.

[0027] Secondly, the present invention provides a quantitative analysis system for the electric field of a rod-shaped multilayer structure based on layer potential technology, comprising:

[0028] The modeling module is configured to create a multi-layered structural geometric model, including internal rod-shaped inclusions, intermediate layer regions, and outer layer regions.

[0029] The equation construction module is configured to construct the potential control equations and boundary transport conditions of the multilayer structure under the background electric field, and transform the electric field boundary value problem into an integral equation system on the multilayer boundary based on the layer potential theory.

[0030] The analysis module is configured to perform asymptotic analysis on the layer potential operator in the integral equation system and derive a quantitative expression for the perturbation electric field.

[0031] The solution module is configured to uniquely determine the geometric parameters and location information of the internal rod-shaped inclusions based on the quantitative expression and the boundary electric field measurement data.

[0032] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in the method for quantitative analysis of electric field of a rod-shaped multilayer structure based on layer potential technology as described in the first aspect.

[0033] Fourthly, the present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps in the method for quantitative analysis of electric field of a rod-shaped multilayer structure based on layer potential technology described in the first aspect.

[0034] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0035] This invention addresses the need for electric field analysis and internal structure detection of rod-shaped multilayer structures. By constructing a precise electric field analysis model using layer potential theory, it achieves a quantitative characterization of the perturbation electric field, overcoming the technical limitations of traditional methods in accurately quantifying the electric field of multilayer structures. This invention can uniquely determine the geometric parameters and location information of the internal rod-shaped inclusions using only a single boundary electric field measurement, significantly improving the efficiency and convenience of structural parameter detection without requiring multiple measurements or complex experimental operations. Furthermore, the potential control equations and integral equations established in this invention have strong adaptability, accurately adapting to the three-layer structural characteristics of the inner rod-shaped inclusions, intermediate layers, and outer layers. The analysis results are highly accurate, providing reliable theoretical support and technical methods for the design, detection, and performance optimization of rod-shaped multilayer structures in fields such as microelectromechanical systems (MEMS) and semiconductor materials.

[0036] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0037] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute a limitation thereof.

[0038] Figure 1 A main flowchart of a method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology provided in an embodiment of the present invention;

[0039] Figure 2 A thermogram showing the distribution of perturbation electric field intensity in a three-layer structure with a circular middle and outer layer under a horizontal background electric field, provided for an embodiment of the present invention.

[0040] Figure 3 Thermodynamic diagram of the perturbation electric field intensity of a three-layer structure with an elliptical middle and outer layer under a horizontal background electric field provided for embodiments of the present invention;

[0041] Figure 4 Thermodynamic diagram of the perturbation electric field intensity of a three-layer structure with a circular middle layer and outer layer under a vertical background electric field provided in an embodiment of the present invention;

[0042] Figure 5 The thermal diagram of the perturbation electric field intensity of a three-layer structure with elliptical middle and outer layers under a vertical background electric field provided in an embodiment of the present invention. Detailed Implementation

[0043] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0044] Explanation of technical terms:

[0045] 1. The basic solution of Laplacian:

[0046] set up for The basic solution of the Laplacian is defined as follows:

[0047] ;

[0048] Point x is the field point. The location of the single-layer potential can be calculated within, outside, or on the boundary of region D. The independent variable in the equation.

[0049] 2. Single-layer potential operator:

[0050] For any bounded region The potential of a single layer is defined as:

[0051] ;

[0052] in, For the region The boundary; To define at the boundary Functions on; Indicates boundary The source point on; express It is a boundary Square-integrable functions on; Indicates boundary The arc length infinitesimal element on the surface.

[0053] 3. Jump Formula:

[0054] Single-layer potential operator The following jump formula is satisfied:

[0055] ;

[0056] In the formula, Represents the normal derivative at the boundary, subscript They represent External and internal (to) The limit of ) It is an identity operator.

[0057] 4. NP operators:

[0058] Taking layer D as an example, the Neumann-Poincaré operator (NP operator) is defined as follows:

[0059] ;

[0060] here, yes The unit external normal, express scalar product in.

[0061] 5. Layered Potential Techniques: Layered potential techniques are a class of mathematical methods used to solve boundary value problems of elliptic partial differential equations. The core idea is to construct a "layered potential" (a potential energy function in the form of a boundary integral) to transform the partial differential equation problem within the region into an integral equation on the boundary, thereby reducing the problem's dimensionality and simplifying the solution. Simply put, it "reduces" the problem of solving the field within the entire region into solving a "density function" on the boundary. It can be imagined as a technique of reconstructing the entire "onion" (physical field) using "onion skins" (boundary layers).

[0062] Example 1

[0063] like Figure 1 As shown, this embodiment discloses a method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology, including the following steps:

[0064] S1: Establish a multi-layered geometric model, including internal rod-shaped inclusions, intermediate layer regions, and outer layer regions;

[0065] S2: Construct the potential control equations and boundary transport conditions of the multilayer structure under the background electric field, and transform the electric field boundary value problem into an integral equation system on the multilayer boundary based on the layer potential theory.

[0066] S3: Perform asymptotic analysis on the layer potential operator in the integral equation system to derive a quantitative expression for the perturbation electric field;

[0067] S4: Based on the quantitative expression, the geometric parameters and location information of the internal rod-shaped inclusions are uniquely determined using the boundary electric field measurement data.

[0068] Next, combined Figure 1 This embodiment provides a detailed description of a quantitative analysis method for the electric field of a rod-shaped multilayer structure based on layer potential technology.

[0069] (I) Partial Differential Equation System

[0070] This embodiment considers a three-layer structure, where the inner layer F is a rod-shaped inclusion, and the middle layer E and the outer layer D are general regions.

[0071] First, a system of partial differential equations (PDEs) is constructed to describe the electrostatic potential distribution and electric field perturbation of the three-layer structure (FED) under a uniform background electric field. The conductivity equation is expressed as:

[0072] (1)

[0073] (2)

[0074] in, The potential governing equation for an electrostatic field is expressed in two-dimensional space. In this case, the divergence of the current density is 0, meaning there is no charge source, and the electric potential is... The conductivity is satisfied Electrostatic equilibrium conditions at that time; It is a far-field boundary condition, indicating that when moving away from the three-layer structure ( When ), the disturbance potential will with The rate of decay to 0 means that the potential at infinity approaches the background potential. . , , Let F represent the conductivity of the inner rod-shaped region F, the middle layer E (the region after removing the closure of F), and the outer layer D (the region after removing the closure of E), respectively. , ; yes A nontrivial harmonic function in represents the background potential; It is a solution to the partial differential equation (1), representing the total electric potential at any point x in space when the three-layer structure (FED) exists; the perturbation electric field is ; Let represent the conductivity of the space, where the conductivity of the uniform background space is 1. Equation (1) describes the electric field disturbance caused by the presence of the three-layer structure. This embodiment mainly considers the solution in (1). Quantitative properties and their relationship with , and The geometric relationship.

[0075] (II) Mathematical Model

[0076] To describe the geometry of the nanorod (F), let E and D be... ( Bounded domain on ) This is a mathematical smoothness condition, meaning that the boundaries of E and D are continuous curves with a certain degree of smoothness; the connected complements are respectively , This indicates that E and D are simply connected regions without holes, and nanorods It is contained within E, and E is contained within D, forming a three-layer structure. Nanorods The detailed definition is as follows.

[0077] First, define nanorods. The central axis, set For a length of The coordinates of the straight line are defined as follows:

[0078] ;

[0079] Next, define the endpoints. , and normal vector .

[0080] Furthermore, nanorods Recorded as The decision, its main body Defined as:

[0081] ;

[0082] That is, in Based on this, extend a certain distance in both the upward and downward directions along the normal vector. Form a rectangular area. Used to describe cross-sectional dimensions; and Nanorods The two end caps are respectively... and Centered on, the radius is Two semicircles. More precisely:

[0083] ;

[0084] In this embodiment and in the following content, it is set and Let them be respectively , Caps for left and right ends , The boundary. Below, we define... and In particular, , It is the main body of nanorods The boundary, in which Defined as:

[0085]

[0086] In addition, using and Representing points respectively and exist The projection on the surface.

[0087] (III) Quantitative characterization of the perturbation electric field and reconstruction of inclusions in a three-layer structure

[0088] Based on the constructed three-layer structure, an integral equation system is obtained using layer potential techniques and boundary transport conditions and jump relations. Then, through asymptotic analysis, an accurate asymptotic formula is derived, enabling a quantitative characterization of the perturbation electric field. Next, using the obtained results, a unique theory for reconstructing nanorods using only a single measurement is established. Finally, the finite element method is used to generate a perturbation electric field image to verify the conclusions. Specifically:

[0089] First, the conductivity equation (1) is equivalent to the following transmission problem. The electric field of the entire space is decomposed into the electric fields of three layers F, E, and D, and the electric field rules of the layer boundaries are defined:

[0090] (3)

[0091] The first four formulas represent the electric field rules within each region. The first formula represents the Laplace operator; the last four formulas represent the continuity rules of current density at layer-to-layer boundaries.

[0092] Subsequently, based on the layer potential technique, the solution to the above equation (3) can be defined as:

[0093] (4)

[0094] in, As the background potential, , , These are the boundaries of layer F ( ), E-layer boundary ( ), D-layer boundary ( The single-layer potential of ) , , is the density function.

[0095] Obviously, the solution given in (4) satisfies the first three equations in (3).

[0096] Furthermore, the solution expression (4) lacks constraints on the layer-to-layer boundaries. Therefore, by combining the boundary transport conditions of the transport problem (3), formulas (5) and (6) are derived, ultimately yielding an integral equation system. Subsequently, the perturbation electric field and nanorod parameters are calculated based on this integral equation.

[0097] (5)

[0098] This includes FE boundary constraints, ED boundary constraints, and D-background space boundary constraints.

[0099] Using the jump formula and formula (5), we can further obtain the following integral:

[0100] (6)

[0101] Among them, single-layer potential operator The following jump formula is satisfied:

[0102]

[0103] In the formula, Represents the normal derivative at the boundary, subscript They represent External and internal (to) The limit of ) It is a D-level NP operator. Meanwhile,

[0104] (7)

[0105] make and define operators for:

[0106] (8)

[0107] , These are the NP operators for the E and F layers, respectively.

[0108] Then (6) can be written as

[0109] (9)

[0110] Then, based on the solvable system of integral equations (9), the solution is obtained. , , Substitute back into formula (4) to calculate the total potential. Then, the perturbation electric field is calculated, ultimately achieving quantitative characterization of the electric field and reconstruction of the nanorods.

[0111] Specifically, density function , It has the following forms:

[0112] (10)

[0113] in:

[0114] ;

[0115] ;

[0116] .

[0117] Based on the operator in equation (8) , and It is reversible (wherein) , and From formula (7), combined with equation (6), we have:

[0118] ;

[0119] ;

[0120] Right now:

[0121]

[0122] remember Obviously Reversible.

[0123] The first equation of equation (6) Substituting into the equation above, we get:

[0124] (11)

[0125] in:

[0126] ;

[0127] To prove the operator It is reversible, still using Fredholm substitution, obviously the operator It is a compact operator; we only need to prove it below. It only needs to be a single shot. Let... ,Right now:

[0128]

[0129] Right now:

[0130]

[0131] in satisfy:

[0132]

[0133] Right now:

[0134]

[0135] Operators defined in equation (8) It is reversible, thus obtaining... .

[0136] F-layer NP operator The following gradual developments occur:

[0137] (12)

[0138] in, Indicates when hour, , That is, the size of this term is... Terms of the same or lower order are used to characterize the order of the remainder terms in an asymptotic expansion. They are defined as follows:

[0139] (13)

[0140] (14)

[0141] in, The characteristic function (also called the indicator function) is, that is, when hour, ;otherwise, . Indicates the side of the nanorod A small neighborhood near endpoint P, the distance between this region and endpoint P is on the order of magnitude. Similarly, It is a small neighborhood close to the other endpoint Q. Let x represent the unit outward normal vector at point x on the boundary. , , ; , , ; , This represents the scaled coordinates.

[0142] and:

[0143] ;

[0144] ;

[0145] Operator , Defined as:

[0146] (15)

[0147] Operator Satisfy the following formula:

[0148] (16)

[0149] in, The Peano remainder (a higher-order infinitesimal) is a term that is more than... A stronger depiction of infinitesimals. Let... and satisfy Then it is written as . , indicating when the parameter When the absolute value of the term approaches 0, it is an infinitesimal quantity, smaller than the constant term in the asymptotic expansion, and can be ignored. Indicates when At that time, this item and For infinitesimals of the same order, retain until First-order terms cannot be ignored.

[0150] Operator Defined as:

[0151] (17)

[0152] Indicates any one belonging to A function of space, that is, a function defined on an interval. A square-integrable function on the operator. It appears as an integrand (density function). In formula (17), there is a placeholder symbol indicating the object on which the operator acts, and its specific form (such as taking...). or The answer depends on the solution to the boundary value problem being considered.

[0153] Assuming density function ( ) is defined as:

[0154]

[0155] Then we have:

[0156] (18)

[0157] in Operator and Defined as:

[0158] (19)

[0159] The following equation holds true:

[0160] (20)

[0161] When function Defined as:

[0162] ;

[0163] In the above formula, a placeholder symbol represents the object on which the operator acts, its specific form (e.g., taking...). or The answer depends on the solution to the boundary value problem being considered.

[0164] Then we have:

[0165] (twenty one)

[0166] in .

[0167] When function Defined as:

[0168] ;

[0169] Then we have:

[0170] (twenty two)

[0171] in .

[0172] The following equation holds true:

[0173]

[0174] F-layer single-layer potential operator The following gradual developments occur:

[0175] (twenty three)

[0176] The density function defined in equation (10) It can be further simplified to:

[0177] (twenty four)

[0178] in:

[0179] (25)

[0180] (26)

[0181] (27)

[0182] because:

[0183] (28)

[0184] therefore:

[0185]

[0186] From equation (10), we can obtain ,Right now:

[0187] (29)

[0188] in satisfy Next, we will further analyze equation (29). Similar to the derivation above, according to equations (21)-(23), we have:

[0189]

[0190]

[0191] in:

[0192]

[0193]

[0194] therefore, Then equation (29) can be redefined as:

[0195] (30)

[0196] Finally, we get:

[0197]

[0198] The density function appears in formula (4), where Then we have:

[0199] (31)

[0200] in , and Defined by equations (25), (26), and (27), respectively.

[0201] From the first equation in equation (9), we get:

[0202]

[0203] Then, based on equations (18)-(19), (21)-(22), and (24), this result can be obtained.

[0204] Furthermore, If the density function appears in formula (4), then it satisfies the following equation:

[0205] (32)

[0206] According to equation (6), we have:

[0207]

[0208] For both sides of the above equation, simultaneously with respect to Integrating, we get:

[0209]

[0210] Limit points to Above, there is:

[0211]

[0212] Right now:

[0213] (33)

[0214] akin,

[0215] (34)

[0216] Finally, the result can be obtained according to equation (24).

[0217] Let be the solutions to partial differential equations (1) and (2). Then, ,have:

[0218] (35)

[0219] in , , , Given by (7), (25), (26) and (27) respectively.

[0220] The final solution reveals that, compared to studies on single-layer structures, the perturbation electric field is not only related to the inner nanorods but also influenced by the outer material. Figures 2 to 5 As shown, these are images of the perturbation electric field generated using the finite element method under a uniform background electric field. The color depth represents the intensity of the perturbation electric field. It can be observed that the perturbation electric field mainly appears near the high curvature part of the nanorod, and the intensity varies with the outer region, further verifying the quantitative analysis results obtained from the theoretical derivation. Figures 2 to 5 In the diagram, the color represents the "absolute value of the perturbation electric field", i.e., |uH|, with the unit being V.

[0221] By using (4) and along Taylor expansion yields:

[0222] (36)

[0223] From formula (31), we can obtain:

[0224] (37)

[0225] Similarly, there are:

[0226] (38)

[0227] From formula (32), we have:

[0228] (39)

[0229] because:

[0230]

[0231] Right now:

[0232]

[0233] From formulas (31) and (32), we have:

[0234] (40)

[0235] Finally, substitute equations (37) to (40) into equation (36) and obtain the result according to formula (24).

[0236] Next, consider applying the preceding quantitative results to establish a unique mechanism that allows for the unique recovery of rod-shaped inclusions using only a single boundary measurement. Based on rod-shaped inclusions... Define another bar Let the generating curve be... for middle Line segments on the axis. and It is a curve The two ends of the rod. Other definitions and Similarly, soon In the definition , , Replace with , , ,get The definition. Using... express The rod-shaped inclusion formed after rotation and translation, i.e. ,in For rotation matrix, For location.

[0237] NP operators satisfy ,in ,and .

[0238] set up and Consider two conductive rods. and The corresponding solutions to (1) are respectively compared with the given nontrivial background. Below and Related, where the parameters are defined in (2). Assumption:

[0239] (41)

[0240] in It includes and Given a bounded, simply connected Lipschitz field, the following equation holds:

[0241] .

[0242] From equation (41), we can see that exist Since it is established in the middle, it can also be obtained through unique extension. exist Established in China. Based on layer potential technology, and... The relevant solution (1) has the following form:

[0243] (42)

[0244] in , and .Depend on , and From the transmission conditions and hop relationships between them, we can obtain:

[0245] (43)

[0246] in Defined by equation (7). According to... and utilize , The first equation in equation (43) can be rewritten as:

[0247] (44)

[0248] Similar to the proof of formula (31), for ,in:

[0249] (45)

[0250] in The following equation holds true:

[0251] (46)

[0252] Similar to the proofs of formulas (32) and (35), we can obtain the following.

[0253] (47)

[0254] in:

[0255] (48)

[0256] Similar to the proofs of formulas (10) and (31), we have:

[0257] (49)

[0258] in:

[0259] (50)

[0260] (51)

[0261] and:

[0262] (52)

[0263] It is given in equation (48). Therefore, using equations (42), (48), and (50), we have:

[0264]

[0265] It is given in equation (48). It is known that for have:

[0266]

[0267] Then we can deduce:

[0268] (53)

[0269] in:

[0270]

[0271]

[0272] and:

[0273]

[0274] Next, we will analyze further (51). According to Taylor expansion, we have:

[0275] (54)

[0276] in By comparing the types of poles in equation (52), we can see that... If x takes two different directions, then Similarly, we can conclude that:

[0277]

[0278] therefore .

[0279] This specific embodiment applies layer potential technology to the electric field analysis of a rod-shaped three-layer structure, constructing an electric potential control equation and boundary transport conditions adapted to multi-layer structures, filling the technical gap in quantitative analysis of the electric field of multi-layer structures. Secondly, through asymptotic analysis of the layer potential operator, a quantitative expression for the perturbation electric field is derived, establishing a clear correspondence between the perturbation electric field and the internal structural parameters, solving the problem of unclear correlation between the electric field and structural parameters of multi-layer structures. In addition, it achieves the unique determination of the parameters of the internal rod-shaped inclusions under a single boundary measurement, breaking through the technical bottleneck of traditional methods requiring multiple measurements and low reconstruction efficiency, and providing a new technical path for structural reconstruction research in this field.

[0280] Example 2

[0281] This embodiment provides a quantitative analysis system for the electric field of a rod-shaped multilayer structure based on layer potential technology, including:

[0282] The modeling module is configured to create a multi-layered structural geometric model, including internal rod-shaped inclusions, intermediate layer regions, and outer layer regions.

[0283] The equation construction module is configured to construct the potential control equations and boundary transport conditions of the multilayer structure under the background electric field, and transform the electric field boundary value problem into an integral equation system on the multilayer boundary based on the layer potential theory.

[0284] The analysis module is configured to perform asymptotic analysis on the layer potential operator in the integral equation system and derive a quantitative expression for the perturbation electric field.

[0285] The solution module is configured to uniquely determine the geometric parameters and location information of the internal rod-shaped inclusions based on the quantitative expression and the boundary electric field measurement data.

[0286] Example 3

[0287] This embodiment provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps in the method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology as described in Embodiment 1 above.

[0288] Example 4

[0289] This embodiment provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the steps in the method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology as described in Embodiment 1 above.

[0290] The steps or modules involved in Embodiments 2 to 4 above correspond to those in Embodiment 1. For specific implementation details, please refer to the relevant description section of Embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.

[0291] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology, characterized in that, include: Establish a multi-layered geometric model, including internal rod-shaped inclusions, intermediate layer regions, and outer layer regions; We construct the potential control equations and boundary transport conditions of a multilayer structure under a background electric field, and transform the electric field boundary value problem into an integral equation system on the multilayer boundary based on the layer potential theory. Asymptotic analysis of the layer potential operator in the integral equation system is performed to derive a quantitative expression for the perturbation electric field. Based on the quantitative expression, the geometric parameters and location information of the internal rod-shaped inclusions are uniquely determined using a single boundary electric field measurement. The establishment of the multi-layered geometric model specifically includes: defining the central axis, length, and cross-sectional dimensions of the internal rod-shaped inclusion to form the rod-shaped geometric body and end structure; setting the middle layer region to completely surround the inclusion, the outer layer region to completely surround the middle layer region, and the boundaries of each layer to be smooth closed curves, forming a three-layer nested structure from the inside to the outside. Define nanorods The central axis, set For a length of The coordinates of the straight line are defined as follows: Define endpoints , normal vector ; Furthermore, nanorods Recorded as Decision, main body Defined as: That is, in Based on this, extend a certain distance upwards and downwards along the normal vector. Form a rectangular area. Used to describe cross-sectional dimensions; and Nanorods The end caps, respectively with and Centered on, with radius as a semicircle; and as follows: ； in, , ;make , Caps for left and right ends , Boundary; definition and In particular, , Nanorods as the main body The boundary, in which for: ； use and Representing points respectively and exist Projection on; The step of uniquely determining the geometric parameters and positional information of the internal rod-shaped inclusion based on the quantitative expression and a single boundary electric field measurement specifically includes: substituting the actually measured boundary disturbance electric field data into the quantitative expression; deducing the preliminary geometric parameters of the inclusion, including the center position, axial direction, length, and cross-sectional dimensions, through the correspondence between the geometric parameters and the electric field distribution in the quantitative expression; and uniquely determining all geometric and positional parameters of the inclusion based on the preliminary geometric parameters using extreme point matching and direction analysis.

2. The method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology as described in claim 1, characterized in that, The construction of the potential control equations and boundary transport conditions for the multilayer structure under the background electric field, based on layer potential theory, transforms the electric field boundary value problem into a system of integral equations on the multilayer boundary, specifically including: The electric field boundary value problem of multilayer structures is described by establishing the potential control equations and the interlayer current continuity condition within each layer region. By using a single-layer potential to represent the potential distribution on each layer boundary, the boundary value problem is transformed into an integral expression of the boundary density function. By applying the interlayer current continuity condition to the integral expression and combining it with the transmission conditions at each layer boundary, a closed integral equation system with respect to the multilayer boundary density function is constructed.

3. The method for quantitative analysis of the electric field of a rod-shaped multilayer structure based on layer potential technology as described in claim 1, characterized in that, The asymptotic analysis of the layer potential operator in the integral equation system to derive a quantitative expression for the perturbation electric field specifically includes: By asymptotically expanding the single-layer potential operator and related integral operators, an approximate expression for the boundary density function is derived. Substituting the approximate expression of the boundary density function in asymptotic form into the potential expression, we obtain the explicit asymptotic formula for the perturbation potential. Based on the relationship between electric field and electric potential, a quantitative expression for the perturbation electric field is derived from the explicit asymptotic formula of the perturbation potential.

4. A quantitative analysis system for the electric field of a rod-shaped multilayer structure based on layer potential technology, characterized in that, include: The modeling module is configured to create a multi-layered structural geometric model, including internal rod-shaped inclusions, intermediate layer regions, and outer layer regions. The equation construction module is configured to construct the potential control equations and boundary transport conditions of the multilayer structure under the background electric field, and transform the electric field boundary value problem into an integral equation system on the multilayer boundary based on the layer potential theory. The analysis module is configured to perform asymptotic analysis on the layer potential operator in the integral equation system and derive a quantitative expression for the perturbation electric field. The solution module is configured to uniquely determine the geometric parameters and position information of the internal rod-shaped inclusions based on the quantitative expression and the boundary electric field measurement data. The establishment of the multi-layered geometric model specifically includes: defining the central axis, length, and cross-sectional dimensions of the internal rod-shaped inclusion to form the rod-shaped geometric body and end structure; setting the middle layer region to completely surround the inclusion, the outer layer region to completely surround the middle layer region, and the boundaries of each layer to be smooth closed curves, forming a three-layer nested structure from the inside to the outside. Define nanorods The central axis, set For a length of The coordinates of the line are defined as: Define endpoints , normal vector ; Furthermore, nanorods Recorded as Decision, main body Defined as: That is, in Based on this, extend a certain distance upwards and downwards along the normal vector. Form a rectangular area. Used to describe cross-sectional dimensions; and Nanorods The end caps, respectively with and Centered on, with radius as a semicircle; and as follows: ； in, , ;make , Caps for left and right ends , Boundary; definition and In particular, , Nanorods as the main body The boundary, in which for: ； use and Representing points respectively and exist Projection on; The step of uniquely determining the geometric parameters and positional information of the internal rod-shaped inclusion based on the quantitative expression and a single boundary electric field measurement specifically includes: substituting the actually measured boundary disturbance electric field data into the quantitative expression; deducing the preliminary geometric parameters of the inclusion, including the center position, axial direction, length, and cross-sectional dimensions, through the correspondence between the geometric parameters and the electric field distribution in the quantitative expression; and uniquely determining all geometric and positional parameters of the inclusion based on the preliminary geometric parameters using extreme point matching and direction analysis.

5. The quantitative analysis system for electric field of a rod-shaped multilayer structure based on layer potential technology as described in claim 4, characterized in that, The construction of the potential control equations and boundary transport conditions for the multilayer structure under the background electric field, based on layer potential theory, transforms the electric field boundary value problem into a system of integral equations on the multilayer boundary, specifically including: The electric field boundary value problem of multilayer structures is described by establishing the potential control equations and the interlayer current continuity condition within each layer region. By using a single-layer potential to represent the potential distribution on each layer boundary, the boundary value problem is transformed into an integral expression of the boundary density function. By applying the interlayer current continuity condition to the integral expression and combining it with the transmission conditions at each layer boundary, a closed integral equation system with respect to the multilayer boundary density function is constructed.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps in the method for quantitative analysis of electric field of rod-shaped multilayer structures based on layer potential technology as described in any one of claims 1-3.

7. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps in the method for quantitative analysis of electric field of rod-shaped multilayer structures based on layer potential technology as described in any one of claims 1-3.

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