Node discrete type strain field analysis method

Abstract

The invention discloses a node discrete type strain field analysis method. The method comprises the following steps: (1) carrying out configuration decomposition on a structure solution domain to obtain an extreme value and a residue parameter which take a space coordinate as a variable; (2) constructing a virtual domain of definition, and realizing smooth analytic continuation based on the extreme value and the residue parameter to obtain an overall smooth imaginary form of the analytic continuation; (3) constructing a weight function and a shape function based on the overall smooth imaginary form of analysis continuation and the virtual domain of definition, and obtaining a moving least square approximation of a displacement field; (4) solving an original solution domain node discrete type strain field based on the smooth imaginary type of analysis continuation and the shape function; the method has the beneficial effects that the conditional number of the shape function and the derivative thereof is remarkably reduced, and the numerical solution stability is effectively improved; and meanwhile, an original problem domain strain solution is extracted in combination with a definition domain truncation strategy, so that the strain estimation precision of the node discrete type method at the boundary is effectively improved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of strain field analysis, and particularly relates to a node discrete type strain field analysis method. BACKGROUND

[0002] Strain estimation based on full-field displacement is a core technology in solid mechanics, material science and experimental mechanics, which is used to inversely deduce the strain distribution of an object through displacement field data. Accurate analysis of strain field is crucial for structural health monitoring, material performance evaluation, engineering failure prediction, etc., and how to accurately reconstruct the continuous strain field from discrete displacement measurement data (such as digital image correlation technology, optical fiber sensing or numerical simulation results) is very important. The methods of converting displacement field to strain field mainly include the following:

[0003] (1) Directly differentiating the displacement field or solving the strain field through difference;

[0004] (2) Solving the strain field by fitting the displacement field based on finite element smoothing denoising;

[0005] (3) Solving the strain by using least square fitting method on displacement field.

[0006] Among them, the direct difference method for solving the strain field is unstable, poor in reliability, and easy to expand the influence of the external environment; the strain field solving method based on finite element smoothing denoising involves mathematical operations of multiple processes, and the calculation process is complex and not easy to program; the least square fitting method for solving the strain field by fitting the displacement field through a quadratic surface and then differentiating the strain field, but due to the inability to grasp the deformation of the full field, it is impossible to select the fitting coefficients, and there is a certain limitation.

[0007] In view of the existing problems, a node discrete type strain field analysis method is proposed.

[0008] The above information disclosed in the background section of this specification is only used to understand the background of the present inventive concept, and therefore, it can include information which does not constitute prior art. SUMMARY

[0009] The present application aims to provide a node discrete type strain field analysis method to solve the problems raised in the background technology.

[0010] The steps of a node discrete type strain field analysis method include:

[0011] (1) Decompose the structure solution domain to obtain extreme values and residual parameters with spatial coordinates as variables;

[0012] (2) Construct a virtual domain, and realize smooth analytic continuation based on extreme value and residue parameter to obtain an overall smooth virtual configuration of analytic continuation;

[0013] (3) Based on the overall smooth virtual configuration of analytic continuation and the virtual domain, construct a weight function and a shape function to obtain a moving least square approximation of displacement field;

[0014] (4) Based on the smooth virtual configuration of analytic continuation and the shape function, solve the node discrete strain field of the original solving domain.

[0015] Preferably, the method for obtaining the extreme value and residue parameter comprises:

[0016] 1) For , the number of discrete points is , and the discrete interval is . The Hankel matrix is constructed by formula (1-3), and formula (1-3) is , =0,1,…, , and are the selected rows and columns of the Hankel matrix, respectively.

[0017] 2) The eigenvalues of the state matrix are extracted from the Hankel matrix in step 1) by singular value decomposition , there are n of them.

[0018] 3) The extreme value parameter with spatial coordinates as variables is obtained by formula (1-5) , and the residue parameter with spatial coordinates as variables is obtained by solving the linear equation system of formula (1-6) by least square method , and formula (1-6) is .

[0019] Preferably, the method for obtaining the state matrix is to construct the Hankel matrix in step 1), let , and singular value decomposition is performed on in the Hankel matrix to obtain the transformation matrix S1, the left singular vector matrix U and the right singular vector matrix V1, and the state matrix is represented by formula (1-4) .

[0020] Preferably, the method for obtaining the overall smooth virtual configuration comprises:

[0021] 1) The virtual domain is obtained by extension by formula (1-9), and formula (1-9) is ,

[0022] , wherein ​Indicates the first at the starting end One virtual definition point; Indicates the first term at the termination end One virtual definition point;

[0023] 2) Reconstruct the configuration of the extended interval domain obtained in step 1) using formula (1-10) to obtain the extended configuration of the virtual solution domain. Formula (1-10) is: In the formula and To decompose the extreme values ​​and residue parameters obtained from the discrete spatial domain configuration;

[0024] 3) Based on the extended configuration of the virtual solution domain obtained in step 2), the global smooth virtual form of the analytical extension is obtained by assembling it using formula (1-11). Overall smooth fictional Use formula (1-11) It means that, among them Formula (1-2) can be used. It means that in the formula This is the domain corresponding to the discrete spatial domain configuration.

[0025] Preferred methods for obtaining the weight function, shape function, and displacement field using moving least squares approximation include:

[0026] 1) For weighted residuals The norm is obtained by taking the minimum value using formula (1-18). coefficients, weighted residuals The norm is given by formula (1-14). It means that in the formula It is the weighting function; formula (1-18) is ;

[0027] 2) Combining formula (1-18) and equation (1-14), we obtain (1-19);

[0028] In the formula This represents the vector of displacement values ​​of each node within the support domain; and All are weight functions, where (1-20) (1-21)

[0029] The coefficient is expressed by formula (1-22) as follows: (1-22);

[0030] 3) The moving least square approximation of the displacement field is obtained by using formula (1-22) and formula (1-12), formula (1-12) is , wherein is the approximate displacement of point ; represents the number of nodes in the support domain of point ; represents the displacement component of the ith node in the support domain; represents the shape function of the ith node, and the moving least square approximation of the displacement field is represented by formula (1-23) , wherein is the basis function, and the shape function matrix is represented by formula (1-24) ;

[0031] 4) The moving least square approximation of the displacement field is obtained by formula (1-23) and formula (1-24), and is represented by formula (1-25) .

[0032] Preferably, the original method for obtaining the nodal discrete strain field of the solution domain comprises:

[0033] 1) The first-order derivative of the shape function obtained in step (3) is obtained by formula (1-26), wherein formula (1-26) is , and the second-order derivative of the shape function obtained in step (3) is obtained by formula (1-27), wherein formula (1-27) is ;

[0034] 2) The strain value of the global virtual domain is obtained by using formula (1-31) on the first-order derivative and the second-order derivative obtained in step 1) of the last step and the global smooth fictitious strain obtained in step (2) , wherein formula (1-31) is ; wherein is the axial strain, is the bending strain;

[0035] 3) The strain value of the global virtual domain obtained in step 2) of the last step is truncated according to the definition domain corresponding to the discrete spatial domain configuration , to obtain the nodal discrete strain field of the solution domain, and the nodal discrete strain field of the solution domain can be represented by formula (1-32), formula (1-32) is .

[0036] Preferably, the weight function is a cubic spline function, and the formula of the cubic spline function is ​(1-15), wherein , is the support size of the weight function, which is 4 times or more of the support size of the weight function.

[0037] Preferably, the weight function is a Gaussian function, and the Gaussian function formula is (1-16), wherein, is a shape parameter of the Gaussian function.

[0038] Preferably, the weight function is an exponential function, and the exponential function formula is (1-17), wherein, is a shape parameter of the exponential function.

[0039] Compared with the prior art, the present application has the beneficial effects that the condition number of the shape function and its derivative is significantly reduced, and the numerical solution stability is effectively improved; and the strain solution of the original problem domain is extracted by combining the domain truncation strategy, and the strain estimation accuracy of the node discrete method at the boundary is effectively improved. BRIEF DESCRIPTION OF DRAWINGS

[0040] Figure 1 is a cantilever beam model;

[0041] Figure 2 is the first three order mode shape functions;

[0042] Figure 3 is a complex exponential reconstruction displacement field comparison diagram based on spatial configuration;

[0043] Figure 4 is a two-way extended node discrete and domain diagram;

[0044] Figure 5 is a reconstructed displacement and analytical value relative error diagram;

[0045] Figure 6 is a node discrete and compact function construction diagram based on a virtual domain;

[0046] Figure 7 is a shape function and its derivative matrix condition number comparison diagram constructed by different methods;

[0047] Figure 8 Comparison diagram of estimated displacement and strain field based on direct MLS meshless method and the present application. DETAILED DESCRIPTION

[0048] With reference to the drawings and the embodiments of the present application, the technical solutions in the embodiments of the present application will be described clearly and completely. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments of the present application, all the other embodiments obtained by those skilled in the art without creative efforts are within the scope of the present application.

[0049] An embodiment of the present application provides:

[0050] A node discrete strain field analysis method comprises the following steps:

[0051] 1. The structure solution domain is decomposed to obtain extreme value and residual parameters with spatial coordinates as variables, wherein the extreme value and residual parameters are obtained by the following method:

[0052] 1) when , the number of discrete points is , and the discrete interval is . The Hankel matrix is constructed by formula (1-3), and formula (1-3) is , wherein =0, 1, …, , and are selected rows and columns of the Hankel matrix, respectively.

[0053] 2) The eigenvalues of the state matrix are extracted from the Hankel matrix in step 1) by singular value decomposition. , There are n.

[0054] Specifically, the state matrix is obtained by the Hankel matrix constructed in the previous step, and , and singular value decomposition is performed on in the Hankel matrix to obtain a transformation matrix S1, a left singular vector matrix U and a right singular vector matrix V1, and the state matrix is represented by formula (1-4). .

[0055] 3) The extreme value parameters with spatial coordinates as variables are obtained by formula (1-5) , and the residual parameters with spatial coordinates as variables are obtained by solving the linear equations of formula (1-6) by the least square method. , and the formula (1-6) is . .

[0056] 2. constructing a virtual definition domain, realizing smooth analytic continuation based on the extreme value and the residual parameter, and obtaining an analytically continued overall smooth virtual configuration, and the method for obtaining the overall smooth virtual configuration is:

[0057] 1) using formula (1-9) to obtain the virtual definition domain, and the formula (1-9) is ,

[0058] wherein represents the first virtual definition point at the starting end; represents the first virtual definition point at the ending end;

[0059] 2) using formula (1-10) to reconstruct the configuration of the definition domain obtained in the above step 1), and obtaining the continuation configuration of the virtual solution domain, and the formula (1-10) is , wherein and are the extreme value and the residual parameter obtained by decomposing the discrete spatial domain configuration;

[0060] 3) based on the continuation configuration of the virtual solution domain obtained in the above step 2), assembling the analytically continued overall smooth virtual configuration by formula (1-11) , wherein the overall smooth virtual configuration is represented by formula (1-11) , wherein can be represented by formula (1-2) , wherein is the definition domain corresponding to the discrete spatial domain configuration.

[0061] 3. based on the analytically continued overall smooth virtual configuration and the virtual definition domain, constructing a weight function and a shape function, and obtaining a moving least squares approximation of a displacement field, and the method for obtaining the weight function, the shape function and the moving least squares approximation of the displacement field is:

[0062] 1) taking the minimum value of the weighted residual norm by formula (1-18) to obtain the coefficient, and the weighted residual norm is represented by formula (1-14) , wherein is a weight function; and the formula (1-18) is

[0063] Specifically, the weight function can be a cubic spline function, and the formula of the cubic spline function is (1-15), wherein ​​​​​, The size of the support domain is the weight function. Four times or more.

[0064] Specifically, the weighting function can also be a Gaussian function, and the formula for the Gaussian function is: (1-16), where, The shape parameters of the Gaussian function

[0065] Specifically, the weighting function is an exponential function, and the formula for the exponential function is: (1-17), where, is the shape parameter of the exponential function.

[0066] 2) Combining formula (1-18) and equation (1-14), we obtain (1-19);

[0067] In the formula This represents the vector of displacement values ​​of each node within the support domain; and Both are weight functions, where (1-20) (1-21)

[0068] The coefficient is expressed by formula (1-22) as follows: (1-22);

[0069] 3) Using formulas (1-22) and (1-12), the least squares approximation of the displacement field is obtained. Formula (1-12) is: In the formula For point Approximate displacement at; Point Supports the number of nodes in the domain; This represents the displacement component of the i-th node in the support domain; Let represent the shape function of the i-th node. The displacement field translation least squares approximation is obtained from formula (1-23). It means that in the formula These are basis functions, and the shape function matrix is ​​given by formula (1-24). express;

[0070] 4) The least squares approximation of the displacement field is derived from formulas (1-23) and (1-24) and formula (1-25). express.

[0071] 4. Based on the smooth fictitious form of the analytical continuation and the shape function, solve for the discrete strain field at the nodes of the original solution domain. The method for obtaining the discrete strain field at the nodes of the original solution domain is as follows:

[0072] 1) Calculate the first derivative of the shape function obtained in step (3) using formula (1-26), where formula (1-26) is: The second derivative of the shape function obtained in step (3) is obtained using formula (1-27), where formula (1-27) is... ;

[0073] 2) Using formula (1-31), the strain values ​​of the global virtual domain are obtained by applying the first derivative, second derivative, and global smooth virtual domain obtained in step (2) to the global virtual domain. Formula (1-31) is In the formula For axial strain, For bending strain;

[0074] 3) Apply formula (1-32) to the strain value of the overall virtual domain obtained in step 2) above. According to the domain corresponding to the discrete spatial domain configuration After truncation, the discrete strain field at the solution domain nodes is obtained. The discrete strain field at the solution domain nodes can be expressed by formula (1-32). Formula (1-32) is... .

[0075] This invention provides an embodiment:

[0076] A schematic diagram of the cantilever beam model is attached. Figure 1 As shown, the length of the cantilever beam For 1 m, the elastic modulus 2.1×10 11 Pa, material density is 7850 kg / m³ 3 The cross-sectional dimensions are 0.025 m × 0.02 m, and the cross-sectional area is 5 × 10⁻⁶ m. -4 m 2 Moment of inertia of cross section It is 1.67 × 10 -8 m 4 The mode shape function of a cantilever beam can describe the most representative configuration of the entire structure in space, as shown in the attached figure. Figure 2 As shown. Circular frequency in structural dynamics. and the corresponding mode shape function The equations for the undamped free vibration of a cantilever beam (where p represents the mode order) can be derived from formulas (a)-(b), and are expressed as follows: (a) (b) (c)

[0077] In the formula, Indicates the bending displacement of the structure; Represents the roots of the frequency equation; Represents unit mass; defines amplitude parameters. .

[0078] Correspondingly, the second derivative of the mode shape function The bending strain corresponding to the spatial configuration of the structure is expressed as: (d)

[0079] (1) Perform configuration decomposition on the structural solution domain

[0080] In interval Under the given discreteness, the domain is first discretized along the length of the cantilever beam. Then, a displacement segment at the left and right boundaries of the cantilever beam described by formula (c) is selected to perform configurational decomposition on the structural solution domain, obtaining the extrema and residue parameters with spatial coordinates as variables. The decomposition and reconstruction results are attached. Figure 3 As shown, this application can accurately decompose the extreme parameter values ​​corresponding to the mode shape function, which also highly match the amplitude of the displacement configuration. It can also be seen that the extreme imaginary part and the analytical frequency value obtained by decomposition are both 7.855, indicating that this application can accurately decompose the extreme parameters.

[0081] (2) Constructing a global smooth virtual type of virtual domain and analytic extension

[0082] Similarly in the interval Given a certain degree of discreteness, a virtual domain extension is performed along the length direction of the cantilever beam, as shown in the attached figure. Figure 4 As shown, the extension length is determined based on the condition number of the truncated shape function and its matrix after extension, with the condition number of the second derivative of the shape function not exceeding 10. 3 For optimal performance, this embodiment selects 1 / 10 of the beam length for boundary extension, using formula (1-9). ( Indicates the first at the starting end One virtual definition point; Indicates the first term at the termination end The virtual domain [-0.2, 1.2] is obtained by extending the domain from the virtual definition points, and then further extended using formula (1-10). ( and To decompose the extreme values ​​and residue parameters of the discrete spatial domain, the configuration is reconstructed based on the extreme values, residue parameters, and virtual domain obtained from the decomposition, resulting in the extended configuration of the virtual solution domain. Then, the configuration is obtained through formula (1-11). ( Formula (1-2) can be used. express, Assemble the obtained extended configurations (where the domain corresponds to the discrete spatial domain configuration) to obtain the globally smooth fictitious form of the analytic extension, as shown in the appendix. Figure 5 As shown, the global smooth fictitious form of the analytic continuation and the analytic value described by formula (c) also maintain a high degree of agreement within the continuation domain.

[0083] The above results show that this application can accurately realize the global smooth virtual reconstruction based on analytical continuation and ensure the high-order continuity characteristics at the boundary, which can be further used to solve the nodal discrete strain field.

[0084] (3) Constructing the least squares approximation of the shape function and displacement field

[0085] From formula (1-24) The shape functions were calculated, as shown in the appendix. Figure 6 As shown, within the virtual domain [-0.2, 1.2], the shape function and its second derivative exhibit asymmetry and divergence at the boundary. Thanks to the extended virtual domain, the asymmetric shape function and its derivative are confined to the virtual domains [-0.2, 0] and [1, 1.2], while the shape function at each node within the domain [0, 1] displays good symmetry. Comparisons of the condition numbers of the shape function and its derivative matrix constructed using different methods are attached. Figure 7 As shown, the condition number of the shape function matrix constructed by the direct moving least squares (MLS) meshless method is less than 10, while the first and second derivative matrices of the shape functions increase sharply to 10. 11 The order of magnitude is significantly larger, exhibiting severe pathological behavior. In contrast, this application extends the virtual domain and truncates it within the [0,1] domain; the condition numbers of its shape functions and their derivative matrices are all below 10. 2 In numerical calculations, this can be defined as a well-state matrix. Furthermore, using formula (1-23)... ( (These are basis functions) and formula (1-24) The least squares approximation of the analytical displacement field is obtained, as shown in the appendix. Figure 8 As shown.

[0086] The above results show that this application can construct shape functions with lower condition numbers, which can accurately approximate existing displacement fields, thus ensuring the stability of strain solutions at the boundary from the perspective of numerical calculation principles.

[0087] (4) Solve the discrete strain field of the original solution domain nodes.

[0088] Using formula (1-26) Find the first derivative of the shape function obtained in step (3) using formula (1-27). Find the second derivative of the shape function obtained in the previous step (3) and apply formula (1-31). ( For axial strain, (For bending strain) to obtain the strain values ​​of the global smooth virtual domain for solving the global virtual domain. And through formula (1-32) The strain values ​​of the obtained global virtual domain According to the domain corresponding to the discrete space domain configuration The strain estimation results are truncated, retaining the strain estimation results for the corresponding interval of the original domain. Based on the direct MLS meshless method and the displacement and strain fields estimated in this application, a comparison with the analytical values ​​described by formula (d) is shown in the appendix. Figure 8 As shown.

[0089] From the appendix Figure 8 It is evident that this application can significantly improve the numerical oscillation problem at the boundary of the direct MLS meshless method, especially for the estimation of the strain field, controlling the estimation error at the strain field boundary to within 0.5%.

[0090] Working principle and usage process of this invention:

[0091] First, an analytical continuation is performed by introducing a complex exponential function decomposition and reconstruction method based on spatial configuration, constructing a function with... The proposed method establishes a smooth transition boundary between the continuous virtual domain and the structural solution domain. Furthermore, based on this method, the spatial distribution of shape functions within the structural solution domain in the virtual domain exhibits symmetry, significantly reducing the condition number of the shape functions and their derivatives, thus effectively improving the stability of the numerical solution. Simultaneously, by combining the domain truncation strategy to extract the strain solution of the structural solution domain, the strain estimation accuracy of the nodal discretization method at the boundary is effectively improved.

[0092] This invention addresses the numerical oscillations caused by singularity of shape functions and weight functions at the boundary of the structural solution domain in the traditional moving least squares meshless method. It constructs a virtual domain to improve the ill-conditioned nature of shape functions and their derivative matrices within the structural solution domain. Based on the discrete spatial domain configuration, a smooth analytical extension is performed to achieve a high-order continuous transition between the virtual domain and the structural solution domain, thus resolving the boundary singularity problem in nodal discrete strain estimation methods.

[0093] Although embodiments of the invention have been shown and described (see the detailed description above), it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for analyzing discrete strain fields at nodes, characterized in that, The steps of the analytical method include: (1) Perform configurational decomposition on the structural solution domain to obtain the extrema and residue parameters with spatial coordinates as variables; (2) Construct a virtual domain, and realize a smooth analytical extension based on the extreme value and the residue parameter to obtain the overall smooth virtual type of the analytical extension; (3) Based on the global smooth virtual form of the analytical continuation and the virtual domain, construct the weight function and shape function to obtain the moving least squares approximation of the displacement field; (4) Based on the smooth fictitious type of the analytical extension and the shape function, solve the original solution domain node discrete strain field.

2. The method for analyzing a nodal discrete strain field according to claim 1, characterized in that, The methods for obtaining the extreme values ​​and the residue parameters include: 1) To When the number of discrete points is The discrete interval is Discrete spatial domain configuration The Hankel matrix is ​​constructed using formula (1-3), which is: In the formula =0,1,…, , and The rows and columns selected for the Hankel matrix; 2) Extract the eigenvalues ​​of the state matrix from the Hankel matrix obtained in step 1) using singular value decomposition. , The number of elements is n; 3) Using formula (1-5) Obtain the extreme value parameter with spatial coordinates as the variable. The linear equation system of formula (1-6) is solved using the least squares method, and the residue parameter with spatial coordinates as variables is obtained. The formula (1-6) is .

3. The method for analyzing a nodal discrete strain field according to claim 2, characterized in that, The state matrix is ​​obtained by constructing the Hankel matrix in step 1), let... and for the Hankel matrix Perform singular value decomposition to obtain the transformation matrix S1, the left singular vector matrix U, and the right singular vector matrix V1. The state matrix is ​​expressed by formula (1-4). express.

4. The method for analyzing a nodal discrete strain field according to claim 2, characterized in that, The method for obtaining the overall smooth virtual type includes: 1) The virtual domain is obtained by extending formula (1-9), which is: ， In the formula Indicates the first at the starting end One virtual definition point; Indicates the first term at the termination end One virtual definition point; 2) Reconstruct the virtual domain obtained in step 1) using formula (1-10) to obtain the extended configuration of the virtual solution domain. Formula (1-10) is as follows: In the formula and To decompose the extreme values ​​and residue parameters obtained from the discrete spatial domain configuration; 3) Based on the extended configuration of the virtual solution domain obtained in step 2), the overall smooth virtual form of the analytical extension is obtained by assembling it using formula (1-11). The overall smooth virtual type Use formula (1-11) It means that, among them Formula (1-2) can be used. It means that in the formula This is the domain corresponding to the discrete spatial domain configuration.

5. The method for analyzing a nodal discrete strain field according to claim 1, characterized in that, The method for obtaining the weight function, the shape function, and the displacement field by moving least squares approximation includes: 1) For weighted residuals The norm is obtained by taking the minimum value using formula (1-18). coefficients, the weighted residuals The norm is given by formula (1-14). It means that in the formula It is a weighting function; the formula (1-18) is... ; 2) Combining formula (1-18) and equation (1-14), we obtain （1-19）； In the formula This represents the vector of displacement values ​​of each node within the support domain; and All are weight functions, where （1-20） （1-21） The coefficient is expressed by formula (1-22) as follows: （1-22）； 3) Using formulas (1-22) and (1-12), the least squares approximation of the displacement field is obtained. Formula (1-12) is: In the formula For point Approximate displacement at; Point Supports the number of nodes in the domain; This represents the displacement component of the i-th node in the support domain; Let represent the shape function of the i-th node. The displacement field translation least squares approximation is obtained from formula (1-23). It means that in the formula These are basis functions, and the shape function matrix is ​​given by formula (1-24). express; 4) The least squares approximation of the displacement field is derived from formulas (1-23) and (1-24) and formula (1-25). express.

6. The method for analyzing a nodal discrete strain field according to claim 1, characterized in that, The original method for obtaining the discrete strain field at the nodal points of the solution domain includes: 1) Calculate the first derivative of the shape function obtained in step (3) using formula (1-26), where formula (1-26) is: The second derivative of the shape function obtained in step (3) is obtained using formula (1-27), where formula (1-27) is... ; 2) Using formula (1-31), the strain values ​​of the global virtual domain are obtained by applying the first derivative, second derivative, and global smooth virtual domain obtained in step (2) to the global virtual domain. Formula (1-31) is In the formula For axial strain, For bending strain; 3) Apply formula (1-32) to the strain value of the overall virtual domain obtained in step 2) above. According to the domain corresponding to the discrete spatial domain configuration After truncation, the discrete strain field at the solution domain nodes is obtained. The discrete strain field at the solution domain nodes can be expressed by formula (1-32). Formula (1-32) is... .

7. The method for analyzing a nodal discrete strain field according to claim 5, characterized in that, The weighting function is a cubic spline function, and the formula for the cubic spline function is: (1-15), where , The size of the support domain is the weight function. Four times or more.

8. The method for analyzing a nodal discrete strain field according to claim 5, characterized in that, The weighting function is a Gaussian function, and the formula for the Gaussian function is: (1-16), where, is the shape parameter of the Gaussian function.

9. The method for analyzing a nodal discrete strain field according to claim 5, characterized in that, The weighting function is an exponential function, and the formula for the exponential function is: (1-17), where, The shape parameter of the exponential function.

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