A method for reconstructing damage electrical impedance imaging of a fiber composite component embedded with a carbon nanotube film

Abstract

The present application relates to a kind of embedded carbon nanotube film fiber composite component damage resistance impedance imaging online accurate reconstruction method, comprising preparation including carbon nanotube film core sensing element composite component;Build acquisition platform including power supply, matrix switch, voltage acquisition instrument;The forward mathematical model of carbon nanotube film-fiber composite component considering conductivity orthogonal anisotropy is constructed;Calibrate the conductivity tensor of healthy component and contact impedance;Using one-step Gauss-Newton imaging algorithm, establish damage area priori;B-spline curve control point matrix is constructed, and damage area is mapped;Solve the jacobian matrix of voltage response to the change of control point weight, iteratively calculate control point weight, using line search to update damage area, until residual convergence.The present application can accurately reconstruct the specific position, area size, geometric characteristics of composite material structure damage, significantly reduce imaging artifact, provide technical guidance for composite material structure health monitoring.

Description

Technical Field

[0001] This invention relates to the field of composite material damage monitoring technology, and in particular to an online precise reconstruction method for damage electrical impedance imaging of fiber composite components embedded with carbon nanotube films. Background Technology

[0002] Fiber-reinforced composite materials are widely used in the aerospace field due to their advantages such as lightweight, high strength, and good designability. However, when composite materials are subjected to fatigue loads, impact loads, and extreme temperatures, they are prone to damage such as matrix cracking, interfacial debonding, and delamination. These types of damage are initially invisible to the naked eye, and if not detected and repaired in time, they will continue to expand, eventually leading to complete structural failure and seriously jeopardizing safety. However, traditional detection technologies have many limitations and are difficult to implement in-situ online monitoring of composite components. For example, methods such as tapping and visual inspection rely on the experience and judgment of professionals, resulting in low detection accuracy, poor stability, and an inability to quantitatively analyze the size of the damaged area. While industrial CT has high detection accuracy, it is difficult to adapt to large components. Although fiber Bragg grating sensors can be used for in-situ online monitoring of aerospace composite components, this method can only monitor the local state of the component, and the poor compatibility between optical fibers and composite materials, coupled with the fact that embedding them between the composite layers weakens their original strength, further complicates the situation.

[0003] Carbon nanotube films possess excellent flexible electrical sensing properties and exhibit good compatibility with fiber composites. Existing research has explored electrical impedance damage identification in carbon nanotube film-fiber reinforced composites; however, most damage reconstruction methods still rely on traditional electrical impedance imaging algorithms, resulting in large artifacts and low damage localization accuracy. Therefore, there is an urgent need in this field for a method that can achieve accurate online reconstruction of in-situ damage in large composite components. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides an online precise reconstruction method for damage in fiber-reinforced composite components using electrical impedance imaging with embedded carbon nanotube films. This method solves the problems of low detection accuracy and difficulty in achieving accurate in-situ online monitoring of damage in fiber-reinforced composite materials in existing technologies. This invention utilizes carbon nanotube films with electrosensing properties, combined with an improved electrical impedance imaging algorithm, to visualize the damaged area of ​​the component, enabling in-situ online monitoring of damage in fiber-reinforced composite components and improving the safety of related products.

[0005] To achieve the above technical objectives, the present invention provides the following technical solution: a method for online accurate reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films, comprising the following steps:

[0006] Electrodes are uniformly pasted around the edges of the carbon nanotube film, and epoxy resin structural film is symmetrically laid on the top and bottom sides. The film is then embedded into the interlayer of the fiber composite component, cured to form a carbon nanotube film-fiber composite component, and the flash is removed.

[0007] A data acquisition platform is established; the data acquisition platform includes a power supply, a voltage acquisition instrument, and a matrix switch, used to acquire the boundary voltage of the electrodes of the carbon nanotube film-fiber composite material component;

[0008] We construct the conductivity tensor of the conductive region of carbon nanotube thin film under healthy and damaged states, and construct a forward model of the healthy state with respect to the conductivity tensor and contact impedance.

[0009] An excitation current is passed through the electrodes of a carbon nanotube film-fiber composite component, and the boundary voltage under healthy conditions is collected. Based on the forward model of healthy conditions, the conductivity tensor and contact impedance under healthy conditions are calibrated.

[0010] Through-through damage was prepared on carbon nanotube film-fiber composite components, experimental differential voltage was calculated, and candidate damage prior regions were constructed on the carbon nanotube film-fiber composite components using a one-step Gauss-Newton imaging algorithm, area filtering and connected component analysis.

[0011] Construct a B-spline level set control point matrix, map the candidate damage prior region to the B-spline level set control point matrix, construct an initial control point vector, solve for the damage boundary, calculate the simulated differential voltage, and then obtain the weighted residual between the experimental differential voltage and the simulated differential voltage.

[0012] By combining weighted residuals to construct a control point optimization objective function to solve for the control point vector, a damage update equation is constructed to solve for the control point state update quantity to realize the update of the control point vector, and the final control point vector is obtained. Then, the final damage boundary, the final damage ratio distribution, and the final damage conductivity tensor are determined to complete the damage reconstruction of the carbon nanotube film-fiber composite component.

[0013] Optionally, the step of uniformly attaching electrodes around the edges of the carbon nanotube film, symmetrically laying epoxy resin structural adhesive films on the upper and lower sides, embedding them into the interlayer of the fiber composite component, curing to form a carbon nanotube film-fiber composite component, and removing flash includes:

[0014] Along the pulling direction of the carbon nanotube film manufacturing process, a carbon nanotube film with a prepreg area slightly larger than that required for manufacturing fiber composite components is cut.

[0015] Copper foil electrodes were uniformly and equally bonded to the edges of the carbon nanotube film using conductive silver paste, and the conductive silver paste was then heated and cured.

[0016] An epoxy resin structural film is symmetrically laid on the upper and lower sides of the carbon nanotube film after the copper foil electrode is attached, forming the core sensing element of the carbon nanotube film. The epoxy resin structural film has the same shape and area as the prepreg required for manufacturing fiber composite material components. It is laid in the middle area of ​​the upper and lower sides of the carbon nanotube film and does not directly contact the copper foil electrode.

[0017] Two layers of prepreg are laid according to the fiber composite component mold, and a carbon nanotube thin film core sensing element is laid between the two layers of prepreg.

[0018] The prepreg with carbon nanotube film core sensing element is fed into a hydraulic press, and the carbon nanotube film-fiber composite material component is cured according to the molding parameters of the prepreg.

[0019] The burrs on the cured carbon nanotube film-fiber composite component are removed by grinding. During the grinding process, the copper foil electrode is retained to ensure that the copper foil electrode can form a stable conductive path with the carbon nanotube film.

[0020] Optionally, the data acquisition platform is powered by a DC constant current power supply.

[0021] The data acquisition platform has its power supply current output terminal and voltage acquisition instrument measurement terminal connected to the row lines of a matrix switch, and the matrix switch column lines directly connected to the electrodes of the carbon nanotube film-fiber composite material component.

[0022] Optionally, the construction of conductivity tensors for the conductive regions of the carbon nanotube film in both healthy and damaged states, and the construction of a forward model of the healthy state relating the conductivity tensor to contact impedance, includes:

[0023] The current continuity equation for the internal potential of carbon nanotube films is constructed, mathematically expressed as:

[0024] ;

[0025] in, Indicates the spatial location within the conductive region of the carbon nanotube film; Indicates the spatial position within the carbon nanotube film The gradient-divergence differential operator is obtained; Indicates the spatial position within the carbon nanotube film The conductivity tensor at that point; Indicates the spatial position within the carbon nanotube film The potential at the point;

[0026] The conductivity tensor of the conductive region of the carbon nanotube film under healthy conditions is constructed as an orthogonal anisotropic conductivity tensor, mathematically represented as:

[0027] ;

[0028] in, The conductivity tensor represents the conductivity region of a carbon nanotube film under healthy conditions. , These represent the x- and y-components of the conductivity tensor in the material's principal coordinate system under healthy conditions, respectively. , These represent the xy and yx tangential components of the conductivity tensor in the material's principal coordinate system under healthy conditions.

[0029] A damage ratio function is introduced to describe the decrease in local conductivity of carbon nanotube film-fiber composite components after damage. The conductivity tensor of the conductive region of the carbon nanotube film under damage is constructed and mathematically represented as follows:

[0030] ;

[0031] in, Indicates the spatial position within the carbon nanotube film under damaged conditions. The conductivity tensor at that point; Spatial position within the carbon nanotube film Damage multiplier function at the location, When in the healthy state area , When located in the damaged area ;

[0032] A complete electrode model is used at the electrode boundary of the carbon nanotube film-fiber composite component, defining the first... Each electrode is , No. The potential of each electrode is , No. The contact resistance of each electrode is , No. The injection current of each electrode is The boundary conditions for the complete electrode model are: And it satisfies the current conservation and potential reference constraints. ;in Indicates the first Integral of the normal current on each electrode; These are the local spatial coordinates of the electrode surface; Represents the internal electric potential field of carbon nanotube films Along the outer normal The derivative; The conductivity tensor represents the electrical conductivity in a healthy or damaged state.

[0033] The finite element method is used to discretize the current continuity equation and the boundary conditions of the complete electrode model, resulting in an algebraic forward model containing the finite element nodal potentials and electrode potentials. Then, from the electrode potential vector Extract the potential difference between adjacent electrodes to obtain the boundary measurement voltage. and forward mapping relationship ;in The expression is composed of the conductivity tensor φ and the contact resistance. The finite element system matrix obtained from assembly, This represents the vector of unknowns consisting of nodal potentials and electrode potentials in the finite element method. This indicates the injected current at the electrode. This represents the measurement channel selection matrix, used to extract the boundary voltage of the corresponding measurement channel from the electrode potential obtained from the finite element solution; This represents the forward mapping matrix, serving as the forward model for the health state.

[0034] Optionally, the calibration of the conductivity tensor and contact impedance under healthy conditions based on the forward model of healthy conditions includes:

[0035] Constructing the variables to be determined in the healthy state for:

[0036] ;

[0037] in, , , , These represent the components of the conductivity tensor to be calibrated under healthy conditions in the x-direction, y-direction, xy-tangential, and yx-tangential directions in the principal coordinate system of the material; Contact resistance; Indicates the transpose operation;

[0038] A bounded nonlinear least squares joint calibration method is used to construct the objective function and solve for the unknown variables. The objective function is defined as follows:

[0039] ;

[0040] in, Represents the variable to be determined The solution results; The measurement channel number is assigned to the upper boundary voltage of the electrode. This represents the total number of measurement channels for the upper boundary voltage of the electrode. Indicates the first electrode The voltage scaling factor for each measurement channel is defined as follows: , Indicates the calculation of the modulus. This indicates the operation of retrieving the maximum value. This indicates the median operation. Balance the weights for the median term; This indicates the number of electrodes collected under healthy conditions. The boundary voltage of each measurement channel is used as the experimental health voltage; Indicates the variable to be calculated The first health state forward model mapping obtained The theoretical boundary voltage of each measurement channel is used as the simulated health voltage. , These are the preset calibration variables and variables to be determined. The lower and upper limits, Indicates in Find the range of The expression with the smallest value ; The regularization term is defined as:

[0041] ;

[0042] in, To control and The penalty intensity for deviations from prior conductivity. , , respectively, are the prior conductivity in the x and y directions; To control and The penalty intensity for deviations from off-diagonal prior conductivity. It serves as a reference scale for off-diagonal prior conductivity; To control and The severity of punishment differs between them; This indicates the calculation of the logarithm square.

[0043] Optionally, the calculation of the experimental differential voltage uses a one-step Gauss-Newton imaging algorithm, area filtering, and connected component analysis to construct candidate damage prior regions on the carbon nanotube film-fiber composite component, including:

[0044] The boundary voltages measured before and after the initiation of penetrating damage are defined as the experimental healthy voltage. Experimental damage voltage The experimental differential voltage is calculated and mathematically expressed as follows:

[0045] ;

[0046] in, Indicates the experimental differential voltage;

[0047] The initial conductivity change estimate is obtained using a one-step Gauss-Newton imaging algorithm, mathematically expressed as:

[0048] ;

[0049] in, This represents an estimate of the initial conductivity change; This represents the partial derivative of conductivity change measured by applying a small conductivity perturbation to each element in the algebraic forward model under the background of conductivity tensor and contact impedance calibrated in the healthy state, and is used as the sensitivity matrix under the healthy state. For one step of Gauss-Newton regularization parameters; Indicates the transpose operation; The regularization matrix used in the initial exploration of Gauss-Newton; express The inverse matrix;

[0050] Estimating discrete changes in initial conductivity Interpolating to a regular grid yields the conductivity variation field, which can be mathematically represented as follows:

[0051] ;

[0052] in, This represents the conductivity variation field obtained through interpolation. , The x and y coordinates represent the elements within the field of conductivity change, respectively. For interpolation mapping operators;

[0053] The conductivity threshold is adaptively determined based on the elements within the field that change in conductivity. This process can be mathematically represented as follows:

[0054] ;

[0055] in, Indicates the conductivity threshold; This represents the threshold scaling factor, and its value range is... ; This indicates the operation of retrieving the maximum value; This indicates the operation of finding the minimum value; Indicates the calculation of the modulus;

[0056] A binary mask is constructed based on the conductivity threshold, mathematically represented as follows:

[0057] ;

[0058] in, Represents a binary mask. , Let x and y represent the x and y coordinates of the elements within the binary mask, respectively, and in the above formula... , ;

[0059] Perform area filtering on the binary mask, selecting pixels with areas smaller than a preset area filtering threshold. The region value is set to 0, and then connected component analysis is performed to obtain several candidate prior damage regions. This process is mathematically represented as follows:

[0060] ;

[0061] in, Indicates the area filtering operator; Represents the connected component analysis operator; Indicates the first One candidate damage prior region, This represents the total number of candidate damage prior regions.

[0062] Optionally, the step of constructing the B-spline level set control point matrix, mapping the candidate damage prior region to the B-spline level set control point matrix, constructing the initial control point vector, solving the damage boundary, and calculating the simulated differential voltage to obtain the weighted residual between the experimental differential voltage and the simulated differential voltage includes:

[0063] At the finite element center of the algebraic forward model, the level set function is represented by tensor product B-splines, mathematically as follows:

[0064] ;

[0065] in, Represents the control point matrix; Represents the control point matrix The first in Line number The column elements are the control point coefficients; This indicates the control point matrix. The control point vector obtained after vectorization; , These are the B-spline basis functions in the horizontal and vertical directions, respectively. , These represent the degrees of the B-spline in the horizontal and vertical directions, respectively. The first algebraic forward model The centers of each finite element are used to evaluate the level set function and the damage ratio of the current finite element. , The first The horizontal and vertical coordinates of the center of each finite element element; Represents the vector of the current control point The defined level set function is at the center of the finite element. The value at; , This represents decreasing the number of control points in the horizontal and vertical directions by 1, respectively. , These represent the control point indices in the horizontal and vertical directions, respectively. , Representing the centers of the finite element elements respectively Normalized coordinates in the horizontal and vertical directions;

[0066] The control point vectors are mapped to the level set function values ​​at the center of each finite element element, mathematically represented as:

[0067] ;

[0068] in, This represents the vector of level set function values ​​at the center of each finite element element; Represents the B-spline basis matrix;

[0069] Based on the B-spline level set control point matrix, the candidate damage prior region is mapped to the control point space to construct the initial control point vector. ;

[0070] Damage boundaries are determined based on level set functions, mathematically represented as follows:

[0071] ;

[0072] in, Indicates the damage boundary; Indicates the conductive region of the carbon nanotube thin film Internal spatial location;

[0073] Introducing the smoothing Heaviside function, the damage ratio of the finite element is calculated, mathematically expressed as:

[0074] ;

[0075] in, Indicates the first Damage ratio of each finite element; Indicates the first The level set function value at the center of the finite element element; This represents the scaling factor for the global conductivity of the damaged area; The smoothing heaviside function is defined as follows: ,in For boundary smoothing parameters, Represents the sine function; Indicates the calculation of the modulus; Pi;

[0076] The conductivity tensor of the finite element is calculated based on the damage ratio and mathematically expressed as:

[0077] ;

[0078] in, Indicates the first The conductivity tensor of a finite element;

[0079] Based on the simulated healthy voltage and simulated damaged voltage, the simulated differential voltage is calculated. The mathematical representation of the calculation process is as follows:

[0080] ;

[0081] in, Represents the vector of control points Global conductivity scaling factor of the damaged area Simulated differential voltage, The simulated health voltage is obtained by solving the forward model of the health state. For the forward model of health status based on control point vectors Global conductivity scaling factor of the damaged area The simulated damage voltage is obtained by solving the conductivity tensor of the mapping.

[0082] Define the weighted residual between the experimental differential voltage and the simulated differential voltage. ,in The residual weight matrix is ​​defined as follows: , Represents a diagonal matrix and a residual weight matrix. Inner element Defined as , This indicates the total number of effective boundary voltage measurement channels. Indicates the effective boundary voltage measurement channel number; , They represent the first Simulated health voltage and experimental health voltage for each channel.

[0083] Optionally, the step of mapping the candidate damage prior region to the control point space and constructing the initial control point vector includes:

[0084] The candidate damage prior region is initialized using a smoothed signed distance function, mathematically represented as:

[0085] ;

[0086] in, Indicates the first Initial control point vectors for candidate damage prior regions; Represents the hyperbolic tangent function; Indicates the first An equivalent envelope radius is the scale parameter on the control point grid after the area of ​​the candidate damage prior region is equivalent to a circle or envelope region. Indicates the first Distance matrix of each candidate damage prior region on the control point grid; This is a smoothing scale parameter used to avoid abrupt changes in the initial control points at the candidate region boundaries;

[0087] All candidate damage prior regions The initial control point vector is obtained by combining the elements using a union method, which can be mathematically represented as:

[0088] ;

[0089] in, Represents the initial control point vector; This means that the control point matrix of multiple candidate damage prior regions is taken as the maximum value point by point according to the control point position, which is equivalent to taking the union of multiple candidate damage prior regions.

[0090] Optionally, the step of constructing a control point optimization objective function using weighted residuals to solve for the control point vector, constructing a damage update equation to solve for the control point state update quantity to update the control point vector, obtaining the final control point vector, and then determining the final damage boundary, final damage ratio distribution, and final damage conductivity tensor to complete the damage reconstruction of the carbon nanotube film-fiber composite component includes:

[0091] Combined with weighted residuals Construct the control point optimization objective function :

[0092] ;

[0093] in, , This represents the current control point vector to be solved and the global conductivity scaling factor of the damage region; This represents the initial control point vector, which remains fixed as a regularization reference during the iterative optimization process; This represents the initial value for iterative optimization of the global conductivity scaling factor in the damaged region; , , These are the regularization parameters for amplitude constraint, smoothness constraint, and global scaling constraint, respectively. This represents the discrete difference Laplace smoothing operator on the control point grid; This represents the calculation of the square of the second norm of a vector;

[0094] Define the damage regularization matrix , ' represents the identity matrix, Indicates the transpose operation;

[0095] The Jacobian matrix of the simulated damage voltage with respect to the control point was calculated using the finite difference method.

[0096] Define the weighted control point Jacobian matrix , This represents the Jacobian matrix of the simulated damage voltage with respect to the control point. This represents the scaling factor of the simulated damage voltage with respect to the global conductivity of the damage region. The finite difference Jacobian matrix;

[0097] Define control point state vector The initial control point state vector is denoted as ;

[0098] Based on weighted residuals Damage regularization matrix Weighted control point Jacobian matrix Control point state vector and initial control point state vector The damage update equation is constructed and mathematically expressed as:

[0099] ;

[0100] in, Indicates the control point status update amount;

[0101] The control point state update amount is obtained based on the damage update equation. The control point update step size is determined by line search. Update the control point vectors; the control point update equation is:

[0102] ;

[0103] in, Indicates the updated control point status. , These represent the updated control point vector and the global conductivity scaling factor of the damaged region, respectively.

[0104] After each update, check if the objective function value of the control points has decreased. If it has not decreased, reduce the control point update step size λ and recalculate. , The iteration continues until the objective function value of the control points decreases or the preset maximum number of line searches is reached. After the iteration optimization is completed, the final control point vector is obtained. ;

[0105] The final damage boundary is determined based on the final control point vector. ;in Represents the final control point vector The defined level set function in the conductive region of the carbon nanotube thin film Internal spatial location The value at;

[0106] The final damage multiplier distribution is constructed based on the final control point vector as follows: ;in, Represents the control point state vector Next Center of each finite element element Final damage multiplier at the location; Represents a smoothing Heaviside function; Represents the final control point vector The determined level set function in the th Center of each finite element element The value at; This represents the scaling factor for the global conductivity of the final damaged area;

[0107] The final damage conductivity tensor is constructed as follows: ; Represents the control point state vector Next Center of each finite element element The final damage conductivity tensor at the point.

[0108] Optionally, the calculation of the Jacobian matrix of the simulated damage voltage with respect to the control point using the finite difference method includes:

[0109] For the There are 1 control point: If the global conductivity scaling factor of the damage region is updated simultaneously during damage reconstruction, then: ;

[0110] in, For the first Unit disturbance vector in the direction of each control point; The value is a small disturbance of the control point, and its range is [value range missing]. ; Indicates simulated damage voltage For the Control points Find the partial derivative;

[0111] Construct the Jacobian matrix from the partial derivatives of all control point directions. ;

[0112] in, This indicates the total number of control points.

[0113] By employing the above technical solution, the present invention provides a method for online accurate reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films, which has at least the following beneficial effects:

[0114] (1) The present invention uses a carbon nanotube thin film with good flexibility to construct the core electrical sensing element, which has good compatibility with fiber reinforced composite materials, has little impact on the mechanical properties of composite material structure, and is suitable for a variety of composite material components.

[0115] (2) The present invention uses control point matrix fitting of component differential voltage, which reduces the influence of initial manufacturing defects of component and underdeterminacy of solution, and improves damage reconstruction accuracy;

[0116] (3) The present invention adopts a one-step Gauss-Newton imaging algorithm, combined with conductivity threshold dynamic segmentation, to construct candidate damage prior regions, which significantly reduces the number of subsequent damage reconstruction iterations and improves the convergence speed, and can realize in-situ and online monitoring of damage to composite material components. Attached Figure Description

[0117] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings:

[0118] Figure 1 This is a flowchart of an online precise reconstruction method for damage electrical impedance imaging of fiber composite components embedded with carbon nanotube films according to the present invention;

[0119] Figure 2 This is a schematic diagram of the preparation process of carbon nanotube film-fiber composite material components in an embodiment of the present invention;

[0120] Figure 3 This is a schematic diagram of the data acquisition platform constructed in an embodiment of the present invention;

[0121] Figure 4 This is a schematic diagram of the boundary voltage measurement results collected in an embodiment of the present invention;

[0122] Figure 5 This is a schematic diagram illustrating the process of constructing candidate damage prior regions using the one-step Gauss-Newton imaging algorithm in an embodiment of the present invention.

[0123] Figure 6This is a flowchart of the damage reconstruction algorithm based on the proposed method in an embodiment of the present invention;

[0124] Figure 7 The weighted residual and loss function convergence curves and differential voltage correlation coefficient curves of the iterative reconstruction of the multi-damaged carbon nanotube film-fiber composite material component in this embodiment of the invention are shown.

[0125] Figure 8 This is a schematic diagram showing the reconstruction results of a multi-damaged carbon nanotube film-fiber composite material component in an embodiment of the present invention;

[0126] Figure 9 This is a comparison of the relative deviations in the damage reconstruction area of ​​the multi-damaged carbon nanotube film-fiber composite material components in the embodiments of the present invention;

[0127] Figure 10 This is a comparison of the center deviation distance of the damage reconstruction area of ​​the multi-damaged carbon nanotube film-fiber composite material component in the embodiments of the present invention.

[0128] In the figure, 1-fiber composite material component, 2-carbon nanotube film, 3-copper foil electrode, 4-epoxy resin structural film, 5-wind polishing pen, 6-host computer, 7-DC constant current power supply, 8-matrix switch, and 9-voltage acquisition instrument. Detailed Implementation

[0129] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. This will allow for a full understanding of how the present application uses technical means to solve technical problems and achieve technical effects, and to facilitate its implementation.

[0130] Those skilled in the art will understand that all or part of the steps in the implementation of the methods of the embodiments can be implemented by a program instructing related hardware. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0131] Please refer to Figures 1-10This illustration shows a specific implementation of the present embodiment. In this embodiment, a carbon nanotube thin film core sensing element is embedded within the interlayer of a fiber-reinforced composite component to construct a carbon nanotube thin film-fiber composite component. A data acquisition platform is built to collect boundary voltages. A forward model of the carbon nanotube thin film-fiber composite component considering orthogonal anisotropy of conductivity is constructed. The conductivity tensor and contact impedance of the electrodes under healthy conditions are calibrated. A one-step Gauss-Newton imaging algorithm is used to construct candidate damage prior regions on the carbon nanotube thin film-fiber composite component. The damage boundary is solved based on the B-spline curve control point matrix, and the weighted residual is calculated. A control point optimization objective function and a damage update equation are constructed and solved to obtain the final damage boundary, final damage magnification distribution, and final damage conductivity tensor. This enables in-situ online monitoring of damage to fiber-reinforced composite components, improving the safety of related products.

[0132] Please refer to Figure 1 This embodiment proposes an online accurate reconstruction method for damage electrical impedance to fiber composite components embedded with carbon nanotube films. The method includes the following steps:

[0133] S1. Co-curing carbon nanotube film-fiber composite material was prepared, and the flash was removed by manual sanding.

[0134] Electrodes are uniformly attached to the edges of the carbon nanotube film, and epoxy resin structural film is symmetrically laid on the top and bottom sides. The film is then embedded into the interlayer of the fiber composite component, cured to form a carbon nanotube film-fiber composite component, and the flash is removed.

[0135] As a preferred embodiment of step S1, it specifically includes:

[0136] S11. Along the pulling direction of the carbon nanotube film manufacturing process, cut a carbon nanotube film with a prepreg area slightly larger than that required for manufacturing fiber composite components to reduce the error caused by tangential conductivity.

[0137] S12. Prepare 16 conductive copper foils (10mm*40mm*0.2mm) as electrodes. The copper foil surface should be smooth and free of oil or other marks. Apply conductive silver paste with a volume resistivity of less than 0.015Ω·cm evenly to one end of the copper foil, with a coating length of approximately 10mm. Attach the 16 copper foil electrodes coated with conductive silver paste to the perimeter of the carbon nanotube film, 4 on each side, evenly distributed. Then place them in an oven at 60℃ for 120 minutes to fully cure the conductive silver paste and achieve reliable adhesion of the copper foil electrodes.

[0138] S13. Select an LJM 200 epoxy resin structural film with the same shape and area as the prepreg required for manufacturing the fiber composite component, and adhere it to the middle area of ​​the upper and lower surfaces of the carbon nanotube film, ensuring that the epoxy resin structural film does not directly contact the copper foil electrode. The LJM 200 epoxy resin structural film has good insulation properties, preventing potential conductivity of some fibers from damaging the original electrosensing network.

[0139] S14. The fiber composite component is made of carbon fiber / epoxy T300 / 7901 prepreg with a single layer thickness of 0.2 mm and a layup design of [0 / 90 / 0 / 90]S. The upper and lower layers of prepreg are laid according to the fiber composite component mold, and a carbon nanotube thin film core sensing element is laid between the two layers of prepreg.

[0140] S15. The prepreg with carbon nanotube film core sensing element is fed into a hydraulic press. The temperature is raised from room temperature to 125°C at 2°C / min, held for 120 min, and then cooled to 65°C at 2°C / min. After that, it is allowed to cool naturally to complete the curing of the carbon nanotube film-fiber composite material component.

[0141] S16. Using a wind grinder with a diamond grinding head, manually grind away the burrs on the cured carbon nanotube film-fiber composite material component. During the grinding process, retain the copper foil electrode and the material in the area extending towards the component to ensure that the copper foil electrode can form a stable conductive path with the carbon nanotube film.

[0142] The above-mentioned preparation process for carbon nanotube film-fiber composite components can be referred to Figure 2 . Figure 2 The process includes three steps: prepreg laying, curing and molding, and grinding off the flash, ultimately forming a polished carbon nanotube film-fiber composite material component. Figure 2 The number 1 indicates a fiber composite component, the number 2 indicates a carbon nanotube film, the number 3 indicates a copper foil electrode, the number 4 indicates an epoxy resin structural film, and the number 5 indicates a wind-grinding pen.

[0143] This invention employs a core electrical sensing element constructed from a flexible carbon nanotube thin film, which exhibits good compatibility with fiber-reinforced composite materials, has minimal impact on the mechanical properties of the composite material structure, and is suitable for various types of composite material components.

[0144] S2. Build a data acquisition platform that includes power supply, matrix switch, and voltage acquisition instrument.

[0145] As a preferred embodiment of step S2, the data acquisition platform specifically includes: a power supply, a voltage acquisition instrument, and a matrix switch, used to acquire the boundary voltage of the electrodes of the carbon nanotube film-fiber composite material component. The power supply is a DC constant current power supply; the current output terminal of the power supply and the measurement terminal of the voltage acquisition instrument are connected to the row lines of the matrix switch, and the column lines of the matrix switch are directly connected to the electrodes of the carbon nanotube film-fiber composite material component.

[0146] In this embodiment, the current output terminal of the power supply and the measurement terminal of the voltage acquisition instrument are connected to the row lines of a 4*16 matrix switch, and the 16 electrodes of the carbon nanotube film-fiber composite material component are connected to the column lines of the matrix switch. Finally, the voltage acquisition instrument is connected to a host computer, and measurement commands are transmitted and data is read using optical fiber, such as... Figure 3 As shown, the data acquisition platform has been constructed. Figure 3 In the diagram, number 1 indicates the fiber composite component, number 3 indicates the copper foil electrode, number 6 indicates the host computer, number 7 indicates the DC constant current power supply, number 8 indicates the matrix switch, and number 9 indicates the voltage acquisition instrument.

[0147] S3. Construct a forward model of the health status of carbon nanotube thin film-composite components that considers orthogonal anisotropy of electrical conductivity.

[0148] We construct conductivity tensors for the conductive regions of carbon nanotube films under healthy and damaged states, and build a forward model of the healthy state relating the conductivity tensor to the contact impedance.

[0149] As a preferred embodiment of step S3, it specifically includes:

[0150] S31. The carbon nanotube film is equivalent to a two-dimensional conductive region. In this embodiment, the conductive region is a square region with a side length of 0.15m, and its vertex coordinates are (0.075, 0.075), (-0.075, 0.075), (-0.075, -0.075), (0.075, -0.075), and (0.075, -0.075). The maximum element size of the finite element is 0.002m, and 16 electrodes are arranged at the boundary. The width of each electrode is 0.01m, and the density coefficient around the electrodes is 2.

[0151] The internal electric potential of carbon nanotube films satisfies the following current continuity equation:

[0152] ;

[0153] in, Indicates the spatial location within the conductive region of the carbon nanotube film; Indicates the spatial position within the carbon nanotube film The gradient-divergence differential operator is obtained; Indicates the spatial position within the carbon nanotube film The conductivity tensor at that point; Indicates the spatial position within the carbon nanotube film The electric potential at that point.

[0154] S32. In the principal coordinate system of the material, the conductivity tensor of the conductive region of the carbon nanotube thin film under healthy conditions is constructed as an orthogonal anisotropic conductivity tensor, mathematically expressed as:

[0155] .

[0156] in, The conductivity tensor represents the conductivity region of a carbon nanotube film under healthy conditions. , These represent the x- and y-direction components of the conductivity tensor in the material's principal coordinate system under healthy conditions, respectively. , Let x and y represent the tangential components of the conductivity tensor in the material's principal coordinate system under healthy conditions, respectively. These components are typically taken as 0. .

[0157] S33. Introducing a damage ratio function to describe the decrease in local conductivity of carbon nanotube film-fiber composite components after damage, and constructing the conductivity tensor of the conductive region of the carbon nanotube film under damaged conditions, mathematically represented as:

[0158] ;

[0159] Among them, among them, Indicates the spatial position within the carbon nanotube film under damaged conditions. The conductivity tensor at that point; Spatial position within the carbon nanotube film Damage multiplier function at the location, When in the healthy state area , When located in the damaged area .

[0160] In this embodiment, based on the carbon nanotube thin film manufacturing process, the following is set up: , The prior value is 1.56S, and the contact resistance is... The prior value is set as Europe.

[0161] S34. Using a complete electrode model at the electrode boundary of the carbon nanotube film-fiber composite component, define the first... Each electrode is , No. The potential of each electrode is , No. The contact resistance of each electrode is , No. The injection current of each electrode is The boundary conditions for the complete electrode model are: And it satisfies the current conservation and potential reference constraints. ;in Indicates the first Integral of the normal current on each electrode; These are the local spatial coordinates of the electrode surface; Represents the internal electric potential field of carbon nanotube films Along the outer normal The derivative; The conductivity tensor represents the electrical conductivity in a healthy or damaged state.

[0162] S35. Using the finite element method, the current continuity equation and the boundary conditions of the complete electrode model are discretized to obtain an algebraic forward model containing the finite element nodal potentials and electrode potentials. Then, from the electrode potential vector Extract the potential difference between adjacent electrodes to obtain the boundary measurement voltage. and forward mapping relationship ;in The expression is composed of the conductivity tensor φ and the contact resistance. The finite element system matrix obtained from assembly, This represents the vector of unknowns consisting of nodal potentials and electrode potentials in the finite element method. This indicates the injected current at the electrode. This represents the measurement channel selection matrix, used to extract the boundary voltage of the corresponding measurement channel from the electrode potential obtained from the finite element solution; The forward mapping matrix is ​​used as the forward model for the healthy state, for calibration of the conductivity and contact impedance in the healthy state, and for calculation of the simulation voltage and residual in the damaged state in subsequent iterative reconstruction.

[0163] S4. Calibrate the conductivity tensor and contact impedance under healthy conditions.

[0164] An excitation current is passed through the electrodes of a carbon nanotube film-fiber composite component, and the boundary voltage under healthy conditions is collected. Based on the forward model of the healthy state, the conductivity tensor and contact impedance under healthy conditions are calibrated.

[0165] As a preferred embodiment of step S4, the calibration of the conductivity tensor and contact impedance under healthy conditions based on the forward model of healthy conditions specifically includes:

[0166] S41. Construct the variables to be determined under the healthy state. for:

[0167] ;

[0168] in, , , , These represent the components of the conductivity tensor to be calibrated under healthy conditions in the x-direction, y-direction, xy-tangential, and yx-tangential directions in the principal coordinate system of the material; Contact resistance; This indicates the transpose operation.

[0169] S42. Using a bounded nonlinear least squares joint calibration method fitted with a forward model of health status, construct the objective function and solve for the variables to be determined. For each set of candidate conductivity tensors and contact impedances, they are substituted into the healthy-state forward model. The simulated healthy voltage is calculated and its normalized residual is fitted to the experimental healthy voltage. The objective function is defined as:

[0170] ;

[0171] in, Represents the variable to be determined The solution results; The measurement channel number is assigned to the upper boundary voltage of the electrode. This represents the total number of measurement channels for the upper boundary voltage of the electrode. Indicates the first electrode The voltage scaling factor for each measurement channel is defined as follows: , Indicates the calculation of the modulus. This indicates the operation of retrieving the maximum value. This indicates the median operation. Balance the weights for the median term; This indicates the number of electrodes collected under healthy conditions. The boundary voltage of each measurement channel is used as the experimental health voltage; Indicates the variable to be calculated The first health state forward model mapping obtained The theoretical boundary voltage of each measurement channel is used as the simulated health voltage. , These are the preset calibration variables and variables to be determined. The lower and upper limits are used to keep the conductivity and contact resistance within physically reasonable and numerically stable ranges. Indicates in Find the range of The expression with the smallest value ; The regularization term is defined as:

[0172] ;

[0173] in, To control and The penalty intensity for deviations from prior conductivity. , These are the prior conductivity in the x and y directions, respectively. The prior conductivity in each direction needs to be obtained by measuring the conductivity of the initial carbon nanotube film or provided by the material manufacturer. To control and The penalty intensity for deviations from off-diagonal prior conductivity. It serves as a reference scale for off-diagonal prior conductivity; To control and The severity of punishment differs between them; This indicates the calculation of the logarithm square.

[0174] In this embodiment, Take 0.01, Take 1e-4, Take 1e-3; to make the conductivity tensor as close as possible to the symmetric tensor, set .

[0175] This step sets up the matrix switch path using the Sheffield protocol. First, a 1A DC current is applied to electrodes 1-2, and 13 sets of inter-electrode voltages are collected for 3-4, 4-5, ..., 15-16. Then, a 1A DC current is applied to electrodes 2-3, and 13 sets of electrode voltages are collected for 4-5, 5-6, ..., 16-1. This process is continued until all 16 sets of electrode excitation-measurement processes are completed, yielding a total of... The boundary voltage measurement results for each of the data points are as follows: Figure 4 As shown.

[0176] S5. Introduce damage initiation and use the one-step Gauss-Newton imaging algorithm to construct candidate damage prior regions.

[0177] Through-through damage was fabricated on carbon nanotube film-fiber composite components. Experimental differential voltages were calculated, and candidate damage prior regions were constructed on the carbon nanotube film-fiber composite components using a one-step Gauss-Newton imaging algorithm, area filtering, and connected component analysis.

[0178] As a preferred embodiment of step S5, it specifically includes:

[0179] S51. First, triangular, square, and circular penetration damages with well-defined geometric characteristics are prepared on carbon nanotube film-fiber composite components. The boundary voltages before and after penetration damage initiation are defined as the experimental healthy voltage. Experimental damage voltage The experimental differential voltage is calculated and mathematically expressed as follows:

[0180] ;

[0181] in, This represents the experimental differential voltage, which improves the convergence speed of subsequent damage reconstruction by constructing a damage prior.

[0182] S52. The initial conductivity change estimate is obtained using a one-step Gauss-Newton imaging algorithm, mathematically expressed as:

[0183] ;

[0184] in, This represents an estimate of the initial conductivity change; Given the conductivity tensor and contact impedance of the electrodes calibrated under healthy conditions, a small conductivity perturbation is applied to each element in the algebraic forward model by the healthy state forward model. The change in the measurement boundary voltage caused by the perturbation is calculated, thereby obtaining the partial derivative of the measurement boundary voltage with respect to the conductivity change of each element, which serves as the sensitivity matrix under healthy conditions. For one step of Gauss-Newton regularization parameters; Indicates the transpose operation; express The inverse matrix. The regularization matrix used in the initial Gauss-Newton study is typically constructed using a NOSER-type regularization prior, the basic idea of ​​which is based on the sensitivity matrix under healthy conditions. extract The diagonal information reflecting the sensitivity of each unit is used to form a diagonal regularization matrix. For regions with high sensitivity, this matrix imposes corresponding constraints to prevent a few high-sensitivity regions from over-responding in the initial inversion.

[0185] S53. Estimate the discrete initial conductivity changes Interpolation to A regular grid is used to obtain the conductivity variation field. This process can be mathematically represented as follows:

[0186] ;

[0187] in, This represents the conductivity variation field obtained through interpolation. , The x and y coordinates represent the elements within the field of conductivity change, respectively. It is an interpolation mapping operator used to transform discrete conductivity variations on finite element units into a regular mesh image field that facilitates thresholding and connected component analysis.

[0188] S54. Based on the positive and negative relationships of elements within the conductivity change field, the conductivity threshold is adaptively determined. This process is mathematically represented as follows:

[0189] ;

[0190] in, Indicates the conductivity threshold; This represents the threshold scaling factor, and its value range is... ; This indicates the operation of retrieving the maximum value; This indicates the operation of finding the minimum value; This indicates the calculation of the modulus.

[0191] S55. Construct a binary mask based on the conductivity threshold, mathematically represented as:

[0192] ;

[0193] in, Represents a binary mask. , Let x and y represent the x and y coordinates of the elements within the binary mask, respectively, and in the above formula... , .

[0194] S56. Perform area filtering on the binary mask, ensuring that the pixel area is less than a preset area filtering threshold. The region value is set to 0, and then connected component analysis is performed to obtain several candidate prior damage regions. This process is mathematically represented as follows:

[0195] ;

[0196] in, Indicates the area filtering operator; Represents the connected component analysis operator; Indicates the first One candidate damage prior region, This represents the total number of candidate damage prior regions.

[0197] In this embodiment, an area filtering threshold is set. After threshold segmentation, 1-3 candidate damage prior regions are obtained. The construction process can be found in [reference needed]. Figure 5Three types of damage were prepared on carbon nanotube film-fiber composite components: I. single triangular through-damage; II. triangular and square through-damage; III. triangular, square, and circular through-damage. Several candidate damage prior regions were obtained using a one-step Gauss-Newton imaging algorithm. The boundaries of the preliminary damage regions were obtained using the aforementioned area filtering operator and connected component analysis operator. A circular curve that can completely contain the boundary was used as the initial damage iteration boundary. Figure 5 The actual damage locations, one-step Gauss-Newton imaging algorithm results, and initial damage iteration boundaries for three damage types are presented respectively. By constructing a damage prior, the convergence speed of subsequent accurate reconstruction is reduced.

[0198] This invention employs a one-step Gauss-Newton imaging algorithm, combined with dynamic segmentation based on conductivity threshold, to construct candidate damage prior regions. This significantly reduces the number of subsequent damage reconstruction iterations and improves the convergence speed, enabling in-situ, online monitoring of damage in composite material components.

[0199] S6. Construct the B-spline level set control point matrix and map the candidate damage prior region.

[0200] Construct a B-spline level set control point matrix, map the candidate damage prior region to the B-spline level set control point matrix, construct an initial control point vector, solve for the damage boundary, calculate the simulated differential voltage, and then obtain the weighted residual between the experimental differential voltage and the simulated differential voltage.

[0201] As a preferred embodiment of step S6, it specifically includes:

[0202] S61. At the finite element center of the algebraic forward model, the level set function is represented by the tensor product B-spline, mathematically as follows:

[0203] ;

[0204] in, Represents the control point matrix; Represents the control point matrix The first in Line 1 The column elements are the control point coefficients; This indicates the control point matrix. The control point vector obtained after vectorization; , These are the B-spline basis functions in the horizontal and vertical directions, respectively. , These represent the degrees of the B-spline in the horizontal and vertical directions, respectively. The first algebraic forward model The centers of each finite element are used to evaluate the level set function and the damage ratio of the current finite element. , The first The horizontal and vertical coordinates of the center of each finite element element; Represents the vector of the current control point The defined level set function is at the center of the finite element. The value at; , This represents decreasing the number of control points in the horizontal and vertical directions by 1, respectively. , These represent the control point indices in the horizontal and vertical directions, respectively. , Representing the centers of the finite element elements respectively The coordinates after normalization in the horizontal and vertical directions.

[0205] The control point vectors are mapped to the level set function values ​​at the center of each finite element element, mathematically represented as:

[0206] ;

[0207] in, This represents the vector of level set function values ​​at the center of each finite element element; This represents the B-spline basis matrix, used to integrate control point vectors. The mapping is to the level set function value at the center of the finite element element.

[0208] In this embodiment, the settings are as follows: , Therefore, the size of the B-spline level set control point matrix is: .

[0209] S62. Based on the B-spline level set control point matrix, the candidate damage prior region is mapped to the control point space to construct the initial control point vector.

[0210] As a preferred embodiment of step S62, mapping the candidate damage prior region to the control point space and constructing the initial control point vector includes:

[0211] S621. The candidate damage prior region is initialized using a smoothed signed distance function, mathematically expressed as:

[0212] ;

[0213] in, Indicates the first Initial control point vectors for candidate damage prior regions; Represents the hyperbolic tangent function; Indicates the first An equivalent envelope radius is the scale parameter on the control point grid after the area of ​​the candidate damage prior region is equivalent to a circle or envelope region. Indicates the first Distance matrix of each candidate damage prior region on the control point grid; This is a smoothing scale parameter used to avoid abrupt changes in the initial control points at the candidate region boundaries. It is preset based on empirical values ​​and is usually set to a value of [value missing]. .

[0214] S622, All candidate damage prior regions The initial control point vector is obtained by combining the elements using a union method, which can be mathematically represented as:

[0215] ;

[0216] in, Represents the initial control point vector; This means that the control point matrix of multiple candidate damage prior regions is taken as the maximum value point by point according to the control point position, which is equivalent to taking the union of multiple candidate damage prior regions.

[0217] S63. Damage boundaries are determined based on level set functions, mathematically represented as follows:

[0218] ;

[0219] in, Indicates the damage boundary; Indicates the conductive region of the carbon nanotube thin film The spatial location within the damage boundary. The zero-level set of the damage boundary is used for subsequent calculations of the smoothing Heaviside function, element damage multiplier, damage conductivity tensor, simulated damage voltage, and final boundary output.

[0220] S64. Introducing the smoothing Heaviside function to calculate the damage ratio of finite element elements, mathematically expressed as:

[0221] ;

[0222] in, Indicates the first Damage ratio of each finite element; Indicates the first The level set function value at the center of the finite element element; This represents the scaling factor for the global conductivity of the damaged area, i.e., the reduction in conductivity of the damaged area to its original value. times, and have , The closer the value is to 0, the closer the damaged area is to insulation. The smoothing heaviside function is defined as follows: ,in For boundary smoothing parameters, Represents the sine function; Indicates the calculation of the modulus; Pi is the mathematical constant of a circle.

[0223] In this embodiment, in this example, the boundary smoothing parameter The value gradually decays from the initial value to the lower limit, defined as... , To perform the maximum value operation, This represents the iteration number of the outer B-spline level set. The conductivity of the damaged region is expressed using a global scaling factor. In this embodiment, it is fixed at 0.002.

[0224] S65. The conductivity tensor of the finite element is calculated based on the damage ratio, and its mathematical expression is:

[0225] ;

[0226] in, Indicates the first The conductivity tensor of a finite element; , These represent the x- and y-direction components of the conductivity tensor in the material's principal coordinate system under healthy conditions.

[0227] S66. Based on the simulated healthy voltage and simulated damaged voltage, calculate the simulated differential voltage. The mathematical representation of the calculation process is as follows:

[0228] ;

[0229] in, Represents the vector of control points Global conductivity scaling factor of the damaged area Simulated differential voltage, The simulated health voltage is obtained by solving the forward model of the health state. For the forward model of health status based on control point vectors Global conductivity scaling factor of the damaged area The simulated damage voltage is obtained by solving the conductivity tensor of the mapping.

[0230] S67. Define the weighted residual between the experimental differential voltage and the simulated differential voltage. ,in The residual weight matrix is ​​defined as follows: , Represents a diagonal matrix and a residual weight matrix. Inner element Defined as , This indicates the total number of effective boundary voltage measurement channels. This indicates the effective boundary voltage measurement channel number, which is the number of boundary voltage data points after removing data from channels adjacent to the excitation electrode from the voltage data measured by the boundary electrode. , They represent the first Simulated health voltage and experimental health voltage for each channel.

[0231] This invention uses a control point matrix to fit the differential voltage of the component, which reduces the influence of initial manufacturing defects in the component and the underdeterminacy of the solution, thereby improving the accuracy of damage reconstruction.

[0232] S7. Calculate the Jacobian matrix and iterate the control point weights until the residuals converge.

[0233] By combining weighted residuals to construct a control point optimization objective function to solve for the control point vector, a damage update equation is constructed to solve for the control point state update quantity to realize the update of the control point vector, and the final control point vector is obtained. Then, the final damage boundary, the final damage ratio distribution, and the final damage conductivity tensor are determined to complete the damage reconstruction of the carbon nanotube film-fiber composite component.

[0234] As a preferred embodiment of step S7, it specifically includes:

[0235] S71, Combined with weighted residuals Construct the control point optimization objective function :

[0236] ;

[0237] in, , This represents the current control point vector to be solved and the global conductivity scaling factor of the damage region; This represents the initial control point vector, which remains fixed as a regularization reference during the iterative optimization process; This represents the initial value for iterative optimization of the global conductivity scaling factor in the damaged region; , , These are the regularization parameters for amplitude constraint, smoothness constraint, and global scaling constraint, respectively. This represents the discrete difference Laplace smoothing operator on the control point grid; This represents the calculation of the square of the second norm of a vector.

[0238] In this embodiment, take , , .

[0239] S72, Define the damage regularization matrix , ' represents the identity matrix, This indicates the transpose operation.

[0240] S73. To avoid the inapplicability of the analytical Jacobian matrix in anisotropic tensor form, the finite difference method is used to calculate the Jacobian matrix of the simulated damage voltage with respect to the control points, and then the weighted control point Jacobian matrix is ​​defined. , Represents the simulated damage voltage versus control point vector The finite difference Jacobian matrix, This represents the scaling factor of the simulated damage voltage with respect to the global conductivity of the damage region. Finite difference Jacobian matrix, residual weight matrix The channel mismatch between the simulated health voltage and the experimental health voltage is determined once and remains unchanged in the outer iteration; it is used for both residual calculation and weighted Jacobian matrix.

[0241] As a preferred embodiment of step S73, the calculation of the Jacobian matrix of the simulated damage voltage to the control point using the finite difference method specifically includes:

[0242] S731, regarding the first There are 1 control point: If the global conductivity scaling factor of the damage region is updated simultaneously during damage reconstruction, then: .

[0243] in, For the first Unit disturbance vector in the direction of each control point; The value is a small disturbance of the control point, and its range is [value range missing]. In this embodiment, we take ; Indicates simulated damage voltage For the Control points Find the partial derivative.

[0244] S732, construct the Jacobian matrix from the partial derivatives of all control point directions. ;

[0245] in, This indicates the total number of control points.

[0246] S74. Define the control point state vector The initial control point state vector is denoted as .

[0247] S75, Based on weighted residuals Damage regularization matrix Weighted control point Jacobian matrix Control point state vector and initial control point state vector The damage update equation is constructed and mathematically expressed as:

[0248] ;

[0249] in, This indicates the control point status update amount.

[0250] S76. Based on the damage update equation, obtain the control point state update amount. The control point update step size is determined by line search. Update the control point vectors; the control point update equation is:

[0251] ;

[0252] in, Indicates the updated control point status. , These represent the updated control point vector and the global conductivity scaling factor of the damaged region, respectively.

[0253] Based on the definition of the objective function for control point optimization, the search criteria are as follows: after each trial update, check if the objective function value decreases. If it does not decrease, reduce the control point update step size λ and recalculate. , The outer iterative optimization continues until the objective function value of the control points decreases or reaches the preset maximum number of line searches, at which point the final control point vector is obtained. .

[0254] In this embodiment, an initial setting is implemented. If the objective function value of the control points does not decrease, then let (symbol" "" indicates "update to"), and the line search will attempt a maximum of 8 times.

[0255] The convergence curves of the weighted residuals and loss function values ​​in this embodiment can be referred to... Figure 7 The left side, where the left vertical axis indicates the weighted residual and the right vertical axis indicates the loss function value; the differential voltage correlation coefficient curve in this embodiment can be referred to Figure 7 Right side. From Figure 7 It can be seen that the convergence condition is reached within 20 iterations in this embodiment, and the correlation coefficient between the simulated differential voltage and the experimental differential voltage approaches 1 as the number of iterations increases, proving that the damage reconstruction results are accurate and meet the requirements of online monitoring.

[0256] S77. Determine the final damage boundary based on the final control point vector: ;in Represents the final control point vector The defined level set function in the conductive region of the carbon nanotube thin film Internal spatial location The value at that location.

[0257] S78. The final damage multiplier distribution is constructed based on the final control point vector as follows: ;in, Represents the control point state vector Next Center of each finite element element Final damage multiplier at the location; Represents a smoothing Heaviside function; Represents the final control point vector The determined level set function in the th Center of each finite element element The value at; This represents the scaling factor for the global conductivity of the final damaged area.

[0258] S79. Construct the final damage conductivity tensor as follows: ; Represents the control point state vector Next Center of each finite element element The final damage conductivity tensor at the point.

[0259] This allows us to obtain the damage conductivity tensor and damage boundary image after damage reconstruction.

[0260] The damage distribution reconstruction results in this embodiment can be referred to... Figure 8 . Figure 8 The actual damage locations for three damage types (I, II, III) are compared with the reconstruction results of this method.

[0261] The algorithm flow for damage reconstruction based on the proposed method can be found in [reference needed]. Figure 6First, a finite element model of the electrosensing network of the composite material component is established. Bounded nonlinear least squares joint calibration of the conductivity and electrode contact impedance of the carbon nanotube film-fiber composite component in the X and Y directions is used to obtain a forward model of the healthy state consistent with the actual experimental specimen. Then, the centers of the finite element elements are extracted and a mapping matrix is ​​constructed to reduce the underdeterminacy of the solution. Based on this, the Gauss-Newton one-step difference method is used to perform a preliminary inversion of the experimental differential voltage to quickly locate the damage range. Then, effective damage regions are screened through threshold segmentation and connected component analysis, the initial damage range is marked, and it is mapped to the control point matrix as the initial value and calculation parameters for subsequent iterations. In the iterative reconstruction stage, a damage conductivity field is constructed based on the current damage region, and forward modeling is performed to obtain the simulated differential voltage. The difference between this and the experimental differential voltage is used to obtain the weighted residual. The influence of small changes in boundary parameters on the weighted residual is further evaluated, the boundary correction direction is calculated, and the damage region is updated through line search. The above process is repeated until the residual meets the convergence condition or the number of iterations is reached. Reaching the preset maximum value The final output is the damage reconstruction result.

[0262] Therefore, this application achieves accurate reconstruction of damage characteristics of carbon nanotube film-fiber composite components by constructing a B-spline curve control point matrix and fitting a damage distribution image. Under three damage types, the relative deviation of the damage reconstruction area and the distance of the center deviation of the damage reconstruction region between the triangular, square, and circular damage features and the traditional Tikhonov reconstruction algorithm are compared. Figure 9 , Figure 10 As shown, it can be seen that compared with the traditional Tikhonov reconstruction method, the reconstruction accuracy of this method is significantly reduced in terms of both the reconstruction area accuracy and the position accuracy of the three types of damage features: I. single triangular penetrating damage; II. triangular and square penetrating damage; and III. triangular, square, and circular penetrating damage. The artifact elimination effect is significant.

[0263] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of those different embodiments or examples.

[0264] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus or device (such as a computer-based system, a processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0265] The above embodiments provide a detailed description of the present invention. Specific examples have been used to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of the present invention. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films, characterized in that, include: Electrodes are uniformly pasted around the edges of the carbon nanotube film, and epoxy resin structural film is symmetrically laid on the top and bottom sides. The film is then embedded into the interlayer of the fiber composite component, cured to form a carbon nanotube film-fiber composite component, and the flash is removed. A data acquisition platform is established; the data acquisition platform includes a power supply, a voltage acquisition instrument, and a matrix switch, used to acquire the boundary voltage of the electrodes of the carbon nanotube film-fiber composite material component; We construct the conductivity tensor of the conductive region of carbon nanotube thin film under healthy and damaged states, and construct a forward model of the healthy state with respect to the conductivity tensor and contact impedance. An excitation current is passed through the electrodes of a carbon nanotube film-fiber composite component, and the boundary voltage under healthy conditions is collected. Based on the forward model of healthy conditions, the conductivity tensor and contact impedance under healthy conditions are calibrated. Through-through damage was prepared on carbon nanotube film-fiber composite components, experimental differential voltage was calculated, and candidate damage prior regions were constructed on the carbon nanotube film-fiber composite components using a one-step Gauss-Newton imaging algorithm, area filtering and connected component analysis. Construct a B-spline level set control point matrix, map the candidate damage prior region to the B-spline level set control point matrix, construct an initial control point vector, solve for the damage boundary, calculate the simulated differential voltage, and then obtain the weighted residual between the experimental differential voltage and the simulated differential voltage. By combining weighted residuals to construct a control point optimization objective function to solve for the control point vector, a damage update equation is constructed to solve for the control point state update quantity to realize the update of the control point vector, and the final control point vector is obtained. Then, the final damage boundary, the final damage ratio distribution, and the final damage conductivity tensor are determined to complete the damage reconstruction of the carbon nanotube film-fiber composite component.

2. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 1, characterized in that: The process involves uniformly attaching electrodes around the edges of the carbon nanotube film, symmetrically laying epoxy resin structural films on both the top and bottom surfaces, embedding them into the interlayer of the fiber composite component, curing to form a carbon nanotube film-fiber composite component, and removing burrs, including: Along the pulling direction of the carbon nanotube film manufacturing process, a carbon nanotube film with a prepreg area slightly larger than that required for manufacturing fiber composite components is cut. Copper foil electrodes were uniformly and equally bonded to the edges of the carbon nanotube film using conductive silver paste, and the conductive silver paste was then heated and cured. An epoxy resin structural film is symmetrically laid on the upper and lower sides of the carbon nanotube film after the copper foil electrode is attached, forming the core sensing element of the carbon nanotube film. The epoxy resin structural film has the same shape and area as the prepreg required for manufacturing fiber composite material components. It is laid in the middle area of ​​the upper and lower sides of the carbon nanotube film and does not directly contact the copper foil electrode. Two layers of prepreg are laid according to the fiber composite component mold, and a carbon nanotube thin film core sensing element is laid between the two layers of prepreg. The prepreg with carbon nanotube film core sensing element is fed into a hydraulic press, and the carbon nanotube film-fiber composite material component is cured according to the molding parameters of the prepreg. The burrs on the cured carbon nanotube film-fiber composite component are removed by grinding. During the grinding process, the copper foil electrode is retained to ensure that the copper foil electrode can form a stable conductive path with the carbon nanotube film.

3. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 1, characterized in that: The data acquisition platform is powered by a DC constant current power supply. The data acquisition platform has its power supply current output terminal and voltage acquisition instrument measurement terminal connected to the row lines of a matrix switch, and the matrix switch column lines directly connected to the electrodes of the carbon nanotube film-fiber composite material component.

4. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 1, characterized in that: The construction of conductivity tensors for the conductive regions of carbon nanotube films in healthy and damaged states, and the construction of a forward model of the healthy state relating the conductivity tensor to contact impedance, include: The current continuity equation for the internal potential of carbon nanotube films is constructed, mathematically expressed as: ； in, Indicates the spatial location within the conductive region of the carbon nanotube film; Indicates the spatial position within the carbon nanotube film The gradient-divergence differential operator is obtained; Indicates the spatial position within the carbon nanotube film The conductivity tensor at that point; Indicates the spatial position within the carbon nanotube film The potential at the point; The conductivity tensor of the conductive region of the carbon nanotube film under healthy conditions is constructed as an orthogonal anisotropic conductivity tensor, mathematically represented as: ； in, The conductivity tensor represents the conductivity region of a carbon nanotube film under healthy conditions. , These represent the x- and y-components of the conductivity tensor in the material's principal coordinate system under healthy conditions, respectively. , These represent the xy and yx tangential components of the conductivity tensor in the material's principal coordinate system under healthy conditions. A damage ratio function is introduced to describe the decrease in local conductivity of carbon nanotube film-fiber composite components after damage. The conductivity tensor of the conductive region of the carbon nanotube film under damage is constructed and mathematically represented as follows: ； in, Indicates the spatial position within the carbon nanotube film under damaged conditions. The conductivity tensor at that point; Spatial position within the carbon nanotube film Damage multiplier function at the location, When in the healthy state area , When located in the damaged area ; A complete electrode model is used at the electrode boundary of the carbon nanotube film-fiber composite component, defining the first... Each electrode is , No. The potential of each electrode is , No. The contact resistance of each electrode is , No. The injection current of each electrode is The boundary conditions for the complete electrode model are: And it satisfies the current conservation and potential reference constraints. ;in Indicates the first Integral of the normal current on each electrode; These are the local spatial coordinates of the electrode surface; Represents the internal electric potential field of carbon nanotube films Along the outer normal The derivative; The conductivity tensor represents the electrical conductivity in a healthy or damaged state. The finite element method is used to discretize the current continuity equation and the boundary conditions of the complete electrode model, resulting in an algebraic forward model containing the finite element nodal potentials and electrode potentials. Then, from the electrode potential vector Extract the potential difference between adjacent electrodes to obtain the boundary measurement voltage. and forward mapping relationship ;in The expression is composed of the conductivity tensor φ and the contact resistance. The finite element system matrix obtained from assembly, This represents the vector of unknowns consisting of nodal potentials and electrode potentials in the finite element method. This indicates the injected current at the electrode. This represents the measurement channel selection matrix, used to extract the boundary voltage of the corresponding measurement channel from the electrode potential obtained from the finite element solution; This represents the forward mapping matrix, serving as the forward model for the health state.

5. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 1, characterized in that: The calibration of the conductivity tensor and contact impedance under healthy conditions based on the forward model of healthy conditions includes: Constructing the variables to be determined in the healthy state for: ； in, , , , These represent the components of the conductivity tensor to be calibrated under healthy conditions in the x-direction, y-direction, xy-tangential, and yx-tangential directions in the principal coordinate system of the material; Contact resistance; Indicates the transpose operation; A bounded nonlinear least squares joint calibration method is used to construct the objective function and solve for the unknown variables. The objective function is defined as follows: ； in, Represents the variable to be determined The solution results; The measurement channel number is assigned to the upper boundary voltage of the electrode. This represents the total number of measurement channels for the upper boundary voltage of the electrode. Indicates the first electrode The voltage scaling factor for each measurement channel is defined as follows: , Indicates the calculation of the modulus. This indicates the operation of retrieving the maximum value. This indicates the median operation. Balance the weights for the median term; This indicates the number of electrodes collected under healthy conditions. The boundary voltage of each measurement channel is used as the experimental health voltage; Indicates the variable to be calculated The first health state forward model mapping obtained The theoretical boundary voltage of each measurement channel is used as the simulated health voltage. , These are the preset calibration variables and variables to be determined. The lower and upper limits, Indicates in Find the range of The expression with the smallest value ; The regularization term is defined as: ； in, To control and The penalty intensity for deviations from prior conductivity. , , respectively, are the prior conductivity in the x and y directions; To control and The penalty intensity for deviations from off-diagonal prior conductivity. It serves as a reference scale for off-diagonal prior conductivity; To control and The severity of punishment differs between them; This indicates the calculation of the logarithm square.

6. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 4, characterized in that: The calculated experimental differential voltage was constructed using a one-step Gauss-Newton imaging algorithm, area filtering, and connected component analysis to create candidate damage prior regions on the carbon nanotube film-fiber composite component, including: The boundary voltages measured before and after the initiation of penetrating damage are defined as the experimental healthy voltage. Experimental damage voltage The experimental differential voltage is calculated and mathematically expressed as follows: ； in, Indicates the experimental differential voltage; The initial conductivity change estimate is obtained using a one-step Gauss-Newton imaging algorithm, mathematically expressed as: ； in, This represents an estimate of the initial conductivity change; This represents the partial derivative of conductivity change measured by applying a small conductivity perturbation to each element in the algebraic forward model under the background of conductivity tensor and contact impedance calibrated in the healthy state, and is used as the sensitivity matrix under the healthy state. For one step of Gauss-Newton regularization parameters; Indicates the transpose operation; The regularization matrix used in the initial exploration of Gauss-Newton; express The inverse matrix; Estimating discrete changes in initial conductivity Interpolating to a regular grid yields the conductivity variation field, which can be mathematically represented as follows: ； in, This represents the conductivity variation field obtained through interpolation. , The x and y coordinates represent the elements within the field of conductivity change, respectively. For interpolation mapping operators; The conductivity threshold is adaptively determined based on the elements within the field that change in conductivity. This process can be mathematically represented as follows: ； in, Indicates the conductivity threshold; This represents the threshold scaling factor, and its value range is... ; This indicates the operation of retrieving the maximum value; This indicates the operation of finding the minimum value; Indicates the calculation of the modulus; A binary mask is constructed based on the conductivity threshold, mathematically represented as follows: ； in, Represents a binary mask. , Let x and y represent the x and y coordinates of the elements within the binary mask, respectively, and in the above formula... , ; Perform area filtering on the binary mask, selecting pixels with areas smaller than a preset area filtering threshold. The region value is set to 0, and then connected component analysis is performed to obtain several candidate prior damage regions. This process is mathematically represented as follows: ； in, Indicates the area filtering operator; Represents the connected component analysis operator; Indicates the first One candidate damage prior region, This represents the total number of candidate damage prior regions.

7. The online precise reconstruction method for damage electrical impedance tomography of fiber composite components embedded with carbon nanotube films according to claim 6, characterized in that: The process involves constructing a B-spline level set control point matrix, mapping candidate damage prior regions onto this matrix, constructing initial control point vectors, solving for damage boundaries, calculating simulated differential voltages, and obtaining the weighted residual between experimental and simulated differential voltages. This includes: At the finite element center of the algebraic forward model, the level set function is represented by tensor product B-splines, mathematically as follows: ； in, Represents the control point matrix; Represents the control point matrix The first in Line number The column elements are the control point coefficients; This indicates the control point matrix. The control point vector obtained after vectorization; , These are the B-spline basis functions in the horizontal and vertical directions, respectively. , These represent the degrees of the B-spline in the horizontal and vertical directions, respectively. The first algebraic forward model The centers of each finite element are used to evaluate the level set function and the damage ratio of the current finite element. , The first The horizontal and vertical coordinates of the center of each finite element element; Represents the vector of the current control point The defined level set function is at the center of the finite element. The value at; , This represents decreasing the number of control points in the horizontal and vertical directions by 1, respectively. , These represent the control point indices in the horizontal and vertical directions, respectively. , Representing the centers of the finite element elements respectively Normalized coordinates in the horizontal and vertical directions; The control point vectors are mapped to the level set function values ​​at the center of each finite element element, mathematically represented as: ； in, This represents the vector of level set function values ​​at the center of each finite element element; Represents the B-spline basis matrix; Based on the B-spline level set control point matrix, the candidate damage prior region is mapped to the control point space to construct the initial control point vector. ; Damage boundaries are determined based on level set functions, mathematically represented as follows: ； in, Indicates the damage boundary; Indicates the conductive region of the carbon nanotube thin film Internal spatial location; Introducing the smoothing Heaviside function, the damage ratio of the finite element is calculated, mathematically expressed as: ； in, Indicates the first Damage ratio of each finite element; Indicates the first The level set function value at the center of the finite element element; This represents the scaling factor for the global conductivity of the damaged area; The smoothing heaviside function is defined as follows: ,in For boundary smoothing parameters, Represents the sine function; Indicates the calculation of the modulus; Pi; The conductivity tensor of the finite element is calculated based on the damage ratio and mathematically expressed as: ； in, Indicates the first The conductivity tensor of a finite element; Based on the simulated healthy voltage and simulated damaged voltage, the simulated differential voltage is calculated. The mathematical representation of the calculation process is as follows: ； in, Represents the vector of control points Global conductivity scaling factor of the damaged area Simulated differential voltage, The simulated health voltage is obtained by solving the forward model of the health state. For the forward model of health status based on control point vectors Global conductivity scaling factor of the damaged area The simulated damage voltage is obtained by solving the conductivity tensor of the mapping. Define the weighted residual between the experimental differential voltage and the simulated differential voltage. ,in The residual weight matrix is ​​defined as follows: , Represents a diagonal matrix and a residual weight matrix. Inner element Defined as , This indicates the total number of effective boundary voltage measurement channels. Indicates the effective boundary voltage measurement channel number; , They represent the first Simulated health voltage and experimental health voltage for each channel.

8. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 7, characterized in that: The step of mapping the candidate damage prior region to the control point space and constructing the initial control point vector includes: The candidate damage prior region is initialized using a smoothed signed distance function, mathematically represented as: ； in, Indicates the first Initial control point vectors for candidate damage prior regions; Represents the hyperbolic tangent function; Indicates the first An equivalent envelope radius is the scale parameter on the control point grid after the area of ​​the candidate damage prior region is equivalent to a circle or envelope region. Indicates the first Distance matrix of each candidate damage prior region on the control point grid; This is a smoothing scale parameter used to avoid abrupt changes in the initial control points at the candidate region boundaries; All candidate damage prior regions The initial control point vector is obtained by combining the elements using a union method, which can be mathematically represented as: ； in, Represents the initial control point vector; This means that the control point matrix of multiple candidate damage prior regions is taken as the maximum value point by point according to the control point position, which is equivalent to taking the union of multiple candidate damage prior regions.

9. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 7, characterized in that: The process involves constructing a control point optimization objective function using weighted residuals to solve for the control point vector, constructing a damage update equation to solve for the control point state update quantity, and obtaining the final control point vector. This process then determines the final damage boundary, the final damage ratio distribution, and the final damage conductivity tensor, thus completing the damage reconstruction of the carbon nanotube film-fiber composite component. This includes: Combined with weighted residuals Construct the control point optimization objective function : ； in, , This represents the current control point vector to be solved and the global conductivity scaling factor of the damage region; This represents the initial control point vector, which remains fixed as a regularization reference during the iterative optimization process; This represents the initial value for iterative optimization of the global conductivity scaling factor in the damaged region; , , These are the regularization parameters for amplitude constraint, smoothness constraint, and global scaling constraint, respectively. This represents the discrete difference Laplace smoothing operator on the control point grid; This represents the calculation of the square of the second norm of a vector; Define the damage regularization matrix , ' represents the identity matrix, Indicates the transpose operation; The Jacobian matrix of the simulated damage voltage with respect to the control point was calculated using the finite difference method. Define the weighted control point Jacobian matrix , This represents the Jacobian matrix of the simulated damage voltage with respect to the control point. This represents the scaling factor of the simulated damage voltage with respect to the global conductivity of the damage region. The finite difference Jacobian matrix; Define control point state vector The initial control point state vector is denoted as ; Based on weighted residuals Damage regularization matrix Weighted control point Jacobian matrix Control point state vector and initial control point state vector The damage update equation is constructed and mathematically expressed as: ； in, Indicates the control point status update amount; The control point state update amount is obtained based on the damage update equation. The control point update step size is determined by line search. Update the control point vectors; the control point update equation is: ； in, Indicates the updated control point status. , These represent the updated control point vector and the global conductivity scaling factor of the damaged region, respectively. After each update, check if the objective function value of the control points has decreased. If it has not decreased, reduce the control point update step size λ and recalculate. , The iteration continues until the objective function value of the control points decreases or the preset maximum number of line searches is reached. After the iteration optimization is completed, the final control point vector is obtained. ; The final damage boundary is determined based on the final control point vector. ;in Represents the final control point vector The defined level set function in the conductive region of the carbon nanotube thin film Internal spatial location The value at; The final damage multiplier distribution is constructed based on the final control point vector as follows: ;in, Represents the control point state vector Next Center of each finite element element Final damage multiplier at the location; Represents a smoothing Heaviside function; Represents the final control point vector The determined level set function in the th Center of each finite element element The value at; This represents the scaling factor for the global conductivity of the final damaged area; The final damage conductivity tensor is constructed as follows: ; Represents the control point state vector Next Center of each finite element element The final damage conductivity tensor at the point.

10. The method for online precise reconstruction of damage electrical impedance to fiber composite components embedded with carbon nanotube films according to claim 9, characterized in that: The calculation of the Jacobian matrix of the simulated damage voltage with respect to the control point using the finite difference method includes: For the There are 1 control point: If the global conductivity scaling factor of the damage region is updated simultaneously during damage reconstruction, then: ; in, For the first Unit disturbance vector in the direction of each control point; The value is a small disturbance of the control point, and its range is [value range missing]. ; Indicates simulated damage voltage For the Control points Find the partial derivative; Construct the Jacobian matrix from the partial derivatives of all control point directions. ; in, This indicates the total number of control points.

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