Post-processing method to improve the accuracy of displacement solution of intrinsic hybrid finite element method

Abstract

The present application discloses a post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method, and relates to the technical field of structural mechanics. Among them, the method includes: determining the values ​​of the stress tensor and displacement vector of the current unit at the integration point respectively by solving the linear algebraic equations by the intrinsic hybrid finite element method, and combining the values ​​of the k-1-order discontinuous Lagrangian element basis function at the integration point, the values ​​of the k+1-order discontinuous Lagrangian element basis function at the integration point and their derivative values ​​to reconstruct the unit coefficient matrix and unit load vector of the current unit, and solve the unit linear equations composed of the unit coefficient matrix and the unit load vector to obtain the post-processing displacement solution of the current unit, and the above post-processing process is repeated for each unit to finally obtain the overall post-processing displacement solution of the linear algebraic equations. The present application solves the technical problem of low displacement accuracy in the relevant intrinsic hybrid finite element method.

Description

Technical Field

[0001] The present application relates to the technical field of structural mechanics, and in particular to a post-processing method for improving the displacement solution accuracy of an intrinsic hybrid finite element method. Background Art

[0002] Linear elasticity mainly studies the stress, strain and displacement generated by linear elastic bodies under the influence of specific external factors, so as to solve problems such as stiffness, strength and stability in the design process of products or engineering structures. It has always been a key research direction for technicians in the fields of machinery, vehicles, civil engineering, aerospace, etc.

[0003] At present, the classical solutions to linear elasticity problems have limitations. When solving complex geometric domains and complex boundaries, analytical solutions are often not obtained, so various numerical solutions have emerged. Among them, the intrinsic mixed finite element method is an effective solution to linear elasticity problems. When solving, the displacement space (i.e., discontinuous element space) uses a k-1-order polynomial, while the stress space (i.e., the intrinsic mixed finite element space) uses a k-order polynomial, which makes the final displacement calculation accuracy one order lower than the stress calculation accuracy, resulting in poor displacement calculation accuracy.

[0004] To address the above-mentioned problems, no effective solution has been proposed yet. Summary of the invention

[0005] The embodiment of the present application provides a post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method, so as to at least solve the technical problem of low displacement solution accuracy of the related intrinsic hybrid finite element method.

[0006] According to one aspect of an embodiment of the present application, a post-processing method for improving the displacement solution accuracy of an intrinsic hybrid finite element method is provided, comprising the following solution steps:

[0007] Step S1, obtaining unit data, material parameters, and boundary conditions, wherein the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, material coefficients include: Young's modulus, Poisson's ratio, density, and the boundary conditions include: body force, stress constraint, and displacement constraint;

[0008] Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using an intrinsic mixed finite element method;

[0009] Step S3, for each unit, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-1 order discontinuous Lagrangian basis function at the integration point, the value of the k+1 order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1 order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3;

[0010] Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point;

[0011] Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix ;

[0012] Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector ;

[0013] Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit for:

[0014]

[0015] And using the first load vector , the second load vector Get the element load vector of the current element for:

[0016]

[0017] Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , get the post-processing displacement solution of the current unit;

[0018] Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain the overall post-processing displacement solution of the linear algebraic equation system Ax=b.

[0019] Optionally, the above step S2 includes the following steps:

[0020] Step S21, determining the type of boundary conditions, and processing different types of boundary conditions according to a preset processing method to obtain an overall load vector b and a stress constraint equation group Gx=s;

[0021] Step S22, for each unit, obtain the unit area, multiple integration points and the integral weight of each integration point of the current unit, and calculate the value of the k-order Lagrangian basis function at the integration point and the value of the k-1-order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S21 to obtain the unit area, multiple integration points and the integral weight of each integration point of the current unit;

[0022] Step S23, obtaining the tensor basis of the two-dimensional symmetric matrix space of the current unit, and combining the value of the k-order Lagrangian basis function at the integration point obtained in step S22 to obtain the value of the intrinsic mixed finite element basis function at the integration point;

[0023] Step S24, determining the overall degrees of freedom of the intrinsic mixed finite element basis function of the current unit;

[0024] Step S25, calculating the unit energy matrix using the values ​​of the intrinsic mixed finite element basis function at each integration point on the current unit, and constructing the coefficient matrix A in combination with step S24;

[0025] Step S26, calculating the derivative value of the k-th order Lagrangian basis function at the integration point, and multiplying it with the tensor basis obtained in step S23 to obtain the div value of the intrinsic mixed finite element basis function at the integration point;

[0026] Step S27, using the value of the k-1 order discontinuous Lagrangian basis function at the integration point obtained in step S22 and the div value of the intrinsic mixed finite element basis function at the integration point obtained in step S26 to calculate the unit divergence operator matrix, and append it to the coefficient matrix A;

[0027] Step S28, using the stress constraint equation group Gx=s, processing the coefficient matrix A obtained by looping steps S22 to S27 and the load vector b obtained in step S21 to obtain a linear algebraic equation group Ax=b;

[0028] Step S29, solving the linear algebraic equation system Ax=b to obtain stress solutions and displacement solutions.

[0029] Optionally, the above step S21 includes the following steps:

[0030] Step S211, determining the type of boundary conditions;

[0031] Step S212, when the type of boundary condition is body force, calling the displacement test function, and loading the integral result of the displacement test function and the body force into the load vector b;

[0032] Step S213, when the type of boundary condition is stress, calling the least square method to process the stress constraint to obtain the stress constraint equation group Gx=s;

[0033] Step S214, when the type of boundary condition is displacement constraint, the stress test function is called, and the integral result of the stress test function and the displacement constraint is loaded into the load vector b.

[0034] Optionally, the above step S23 includes the following steps:

[0035] Step S231, determining the type of each Lagrangian interpolation point on the current unit, wherein the types of Lagrangian interpolation points include: unit vertex, unit internal point, edge internal point;

[0036] Step S232, determining the tensor basis of the two-dimensional symmetric matrix space according to the type of the Lagrangian interpolation point, wherein when the type of the Lagrangian interpolation point is a unit vertex and a unit internal point, obtaining the standard basis of the second-order symmetric tensor space; when the type of the Lagrangian interpolation point is an edge internal point, obtaining a unit tangent vector and a unit normal vector for each of the three unit edges of the current unit, and using a unit tangent vector and a unit normal vector for each edge to calculate the edge basis of the unit edge;

[0037] Step S233, combining the tensor basis of the two-dimensional symmetric matrix space with the value of the k-order Lagrangian basis function at the integration point obtained in step S22, to obtain the value of the intrinsic mixed finite element basis function at the integration point.

[0038] Optionally, the above step S4 includes the following steps:

[0039] Step S41, using the stress solution to linearly combine the values ​​of the intrinsic mixed finite element basis functions at the integration points obtained in S23, to obtain the values ​​of the stress tensors at the integration points;

[0040] Step S42, using the displacement solution, linearly combine the values ​​of the k-1 order Lagrangian element basis functions at the integration points to obtain the values ​​of the displacement vectors at the integration points.

[0041] According to another aspect of an embodiment of the present application, a computer program product is also provided, which includes a stored computer program, wherein when the computer program is executed by a processor, the above-mentioned post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method is implemented.

[0042] According to another aspect of an embodiment of the present application, a non-volatile storage medium is also provided, in which a program is stored, wherein when the program is running, the device where the non-volatile storage medium is located is controlled to execute the above-mentioned post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method.

[0043] According to another aspect of an embodiment of the present application, an electronic device is also provided, which includes: a memory and a processor, wherein the processor is used to run a program stored in the memory, wherein when the program is running, the above-mentioned post-processing method for improving the displacement solution accuracy of the intrinsic mixed finite element method is executed.

[0044] In the embodiment of the present application, the linear algebraic equations are first solved by the intrinsic mixed finite element method. The stress solution and displacement solution are obtained, and then based on the stress solution and displacement solution, the displacement solution with high precision is recalculated in the high-order (k+1) discontinuous Lagrangian space through post-processing method, thereby solving the technical problem of low displacement precision in the related intrinsic mixed finite element method. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] The drawings described herein are used to provide a further understanding of the present application and constitute a part of the present application. The illustrative embodiments of the present application and their descriptions are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:

[0046] Figure 1 It is a hardware structure block diagram of a computer terminal for implementing a post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method according to an embodiment of the present application;

[0047] Figure 2 It is a flow chart of an optional post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method according to an embodiment of the present application;

[0048] Figure 3 is a flow chart of an optional step S2 according to an embodiment of the present application;

[0049] Figure 4 is a flow chart of an optional step S21 according to an embodiment of the present application;

[0050] Figure 5 is a flowchart of an optional step S232 according to an embodiment of the present application;

[0051] Figure 6 is a flow chart of an optional step S24 according to an embodiment of the present application;

[0052] Figure 7is a schematic diagram of an optional working condition of a rectangular column cross section according to an embodiment of the present application;

[0053] Figure 8a is a schematic diagram of an optional grid model 1 according to an embodiment of the present application;

[0054] Figure 8b is a schematic diagram of an optional grid model 2 according to an embodiment of the present application;

[0055] Fig. 9 It is an optional displacement result cloud diagram solved by using a second-order Lagrangian triangular plane strain unit on the grid model 1 according to an embodiment of the present application;

[0056] Fig.10 It is an optional displacement result cloud diagram solved by using a third-order intrinsic hybrid finite element triangular plane strain unit on a grid model 2 according to an embodiment of the present application;

[0057] Fig.11 It is an optional displacement result cloud diagram of post-processing after solving by using a third-order intrinsic hybrid finite element triangular plane strain unit on the grid model 2 according to an embodiment of the present application;

[0058] Fig.12 It is a schematic diagram of the structure of an optional electronic device according to an embodiment of the present application. DETAILED DESCRIPTION

[0059] In order to enable those skilled in the art to better understand the solution of the present application, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present application.

[0060] It should be noted that the terms "first", "second", etc. in the specification and claims of the present application and the above-mentioned drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the data used in this way can be interchangeable where appropriate, so that the embodiments of the present application described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any of their variations are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device comprising a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0061] In addition, the relevant information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for display, data for analysis, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties. For example, an interface is set up between this system and the relevant users or organizations. Before obtaining relevant information, it is necessary to send an acquisition request to the aforementioned user or organization through the interface, and obtain relevant information after receiving the consent information fed back by the aforementioned user or organization.

[0062] Example 1

[0063] The method embodiments provided in the embodiments of the present application can be executed in a mobile terminal, a computer terminal or a similar computing device. Figure 1 The hardware structure block diagram of a computer terminal for implementing a post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method is shown. Figure 1 As shown, the computer terminal 10 may include one or more (102a, 102b, ..., 102n are used to illustrate) processors 102 (the processor 102 may include but is not limited to a processing device such as a microprocessor MCU or a programmable logic device FPGA), a memory 104 for storing data, and a transmission device 106 for communication functions. In addition, it may also include: a display, an input / output interface (I / O interface), a universal serial bus (USB) port (which may be included as one of the ports of the BUS bus), a network interface, a power supply and / or a camera. It can be understood by those skilled in the art that Figure 1 The structure shown is only for illustration and does not limit the structure of the above electronic device. Figure 1 More or fewer components as shown, or with Figure 1 Different configurations shown.

[0064] It should be noted that the one or more processors 102 and / or other data processing circuits described above may generally be referred to herein as "data processing circuits". The data processing circuits may be embodied in whole or in part as software, hardware, firmware, or any other combination thereof. In addition, the data processing circuit may be a single independent processing module, or may be incorporated in whole or in part into any of the other components in the computer terminal 10. As involved in the embodiments of the present application, the data processing circuit acts as a processor control (e.g., selection of a variable resistor terminal path connected to an interface).

[0065] The memory 104 can be used to store software programs and modules of application software, such as the program instructions / data storage device corresponding to the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method in the embodiment of the present application. The processor 102 executes various functional applications and data processing by running the software programs and modules stored in the memory 104, that is, the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method of the above-mentioned application is realized. The memory 104 may include a high-speed random access memory, and may also include a non-volatile memory, such as one or more magnetic storage devices, flash memory, or other non-volatile solid-state memory. In some instances, the memory 104 may further include a memory remotely arranged relative to the processor 102, and these remote memories may be connected to the computer terminal 10 via a network. Examples of the above-mentioned network include, but are not limited to, the Internet, an intranet, a local area network, a mobile communication network, and a combination thereof.

[0066] The transmission device 106 is used to receive or send data via a network. The specific example of the above network may include a wireless network provided by a communication provider of the computer terminal 10. In one example, the transmission device 106 includes a network adapter (Network Interface Controller, NIC), which can be connected to other network devices through a base station so as to communicate with the Internet. In one example, the transmission device 106 can be a radio frequency (RF) module, which is used to communicate with the Internet wirelessly.

[0067] The display may be, for example, a touch screen liquid crystal display (LCD) that enables a user to interact with a user interface of the computer terminal 10 .

[0068] The embodiment of the present application provides a post-processing method for improving the accuracy of displacement solution of the intrinsic hybrid finite element method, aiming to obtain high-precision displacement calculation results in a high-order discontinuous element space. The above method provided by the present application can be widely used in the fields of machinery, vehicles, civil engineering, aerospace, ships, etc., and its specific application scenarios include but are not limited to: mechanical arm strength analysis, chassis frame stiffness analysis, tunnel deformation and damage analysis, rocket bracket lightweight analysis, helicopter hub deformation analysis, hull bending and torsional stiffness analysis, etc. It should be noted that the steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described can be executed in an order different from that here.

[0069] Figure 2 is a flow chart of an optional post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method according to an embodiment of the present application, such as Figure 2 As shown, the method comprises at least steps S1-S9, wherein:

[0070] Step S1, obtaining unit data, material parameters, and boundary conditions.

[0071] Among them, the above-mentioned unit data include: unit shape, node coordinates, unit quantity, unit dimension; the above-mentioned material coefficients include: Young's modulus, Poisson's ratio, density; the above-mentioned boundary conditions include: body force, stress constraint, displacement constraint.

[0072] Specifically, when simulating and analyzing two-dimensional structural mechanics, the mesh type, mesh density and mesh shape play a vital role in the accuracy of the simulation results. In order to simplify the structure to meet the accuracy requirements and quickly solve the simulation results, the mesh of the appropriate shape can be selected according to the actual situation to divide the stress area of ​​the elastic body to be analyzed. Among them, the mesh shape selected in the embodiment of the present application can be preferably a triangular mesh, and the number of units and the unit dimension obtained by the specific division can be determined according to the size and dimension of the divided mesh shape and the actual stress area. In addition, the material coefficient and boundary conditions can be determined according to the material coefficient and constraint conditions of the stress area of ​​the actual elastic body to be analyzed.

[0073] Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using the intrinsic mixed finite element method.

[0074] As an optional implementation, Figure 3 is a flow chart of an optional step S2 according to an embodiment of the present application, such as Figure 3 As shown, step S2 specifically includes the following steps S21-S29:

[0075] Step S21, determine the type of boundary conditions, and process different types of boundary conditions according to a preset processing method to obtain the overall load vector b and the stress constraint equation group Gx=s.

[0076] Loop through each unit and execute the following steps S22-S27:

[0077] Step S22, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-order Lagrangian basis function at the integration point and the value of the k-1-order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S21.

[0078] Specifically, the integration point (i.e., Lagrange interpolation point) of the current triangle unit K is recorded as , where for a given triangle unit K edge Points on (k is an integer greater than or equal to 3). According to the definition of Lagrange Basis Function, the k-order Lagrange Basis Function at the integral point can be calculated. The value at the point and the value of the k-1 order discontinuous Lagrangian basis function at the integration point, where the k-order Lagrangian basis function can be The value at , and the k-th Lagrangian basis function is at the integration point The value at satisfies the following conditions: At the point Take 1, and in the k-order Lagrange interpolation point set The other integral points in are set to 0. In addition, the integral weight and unit area obtained in step S22 can be used for the subsequent calculation of the unit energy matrix.

[0079] Step S23, obtaining the tensor basis of the two-dimensional symmetric matrix space of the current unit, and combining it with step S22 to obtain the value of the intrinsic mixed finite element basis function at the integration point.

[0080] Among them, the above two-dimensional symmetric matrix space is a vector space composed of all two-dimensional symmetric matrices, so the tensor basis of the two-dimensional symmetric matrix space can be understood as a set composed of several linearly independent two-dimensional symmetric matrices, and these two-dimensional symmetric matrices can represent all two-dimensional symmetric matrices in the space. Any second-order symmetric matrix can be represented by three specific symmetric matrices: , and Therefore, these three symmetric matrices are a set of bases in this space.

[0081] Step S24, determining the overall degrees of freedom of the intrinsic mixed finite element basis function of the current unit, that is, the degrees of freedom of the intrinsic mixed finite element on the current unit.

[0082] Step S25, calculating the unit energy matrix using the values ​​of the intrinsic mixed finite element basis functions at each integration point on the current unit, and constructing the coefficient matrix A in combination with step S24.

[0083] Specifically, Recorded as the integration point of the current triangle unit K, Denote as the integral weight of the corresponding integration point, S as the unit area of ​​the triangular unit K, and use the value of the intrinsic mixed finite element basis function at each integration point on the current unit, the corresponding integral weight and unit area to calculate the unit energy matrix through the following formula:

[0084]

[0085] Among them, the above denote the stress variable basis functions, and the above The expression can be written as:

[0086]

[0087] Where E represents Young's modulus, represents Poisson's ratio, represents the identity matrix and tr represents the trace.

[0088] Furthermore, a global coefficient matrix (generally a zero matrix or a unit matrix) is first initialized according to the overall degrees of freedom determined in step S24, where the order of the coefficient matrix is ​​the number of degrees of freedom; then, according to the mapping vector from the local degrees of freedom of each unit to the overall degrees of freedom (which specifies which row and column in the global coefficient matrix the local degrees of freedom of the unit corresponds to), the unit energy matrices of each unit are assembled into the global coefficient matrix, and finally the overall coefficient matrix is ​​obtained.

[0089] Step S26, calculate the derivative value of the k-th order Lagrangian basis function at the integration point, and multiply it with the tensor basis obtained in step S23 to obtain the div value of the intrinsic mixed finite element basis function at the integration point.

[0090] Specifically, the value of the k-order Lagrangian basis function at the integration point obtained in step S22 is differentiated to obtain the derivative value of the k-order Lagrangian basis function at the integration point. Then, according to the type of Lagrangian interpolation point (element vertex, element internal point, point on element edge), the derivative value of the k-order Lagrangian basis function at the integration point is multiplied by the tensor basis (standard basis, first-order symmetric matrix, second-order symmetric matrix) corresponding to the integration point type to obtain the div value (also known as the divergence value) of the intrinsic mixed finite element basis function at the integration point.

[0091] Step S27, using the value of the k-1 order discontinuous Lagrangian basis function at the integration point obtained in step S22 and the div value of the intrinsic mixed finite element basis function at the integration point obtained in step S26 to calculate the unit divergence operator matrix, and append it to the coefficient matrix A.

[0092] Step S28, using the stress constraint equation group Gx=s, the coefficient matrix A obtained by looping steps S22 to S27 and the load vector b obtained in step S21 are processed to obtain a linear algebraic equation group Ax=b.

[0093] Specifically, the linear algebraic equation Ax=b may be solved by a direct solution method or an iterative solution method, and no specific limitation is imposed on the solution method.

[0094] After the stress solution and displacement solution are obtained by solving the linear algebraic equation system Ax=b using the intrinsic mixed finite element method through the above steps S21-S28, since the displacement space (i.e., the discontinuous element space) uses a k-1 degree polynomial, and the stress space (i.e., the intrinsic mixed finite element space) uses a k degree polynomial, the displacement calculation accuracy of the final solution is one order lower than the stress calculation accuracy, resulting in poor displacement calculation accuracy. Therefore, the embodiment of the present application can further post-process the displacement solution of each unit, so each unit is looped and the following steps S3-S8 are executed:

[0095] Step S3, obtaining the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculating the value of the k-1th order discontinuous Lagrangian basis function at the integration point, the value of the k+1th order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1th order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3.

[0096] Specifically, the above-mentioned method of calculating the value of the k-1-order discontinuous Lagrangian basis function at the integration point and the value of the k+1-order discontinuous Lagrangian basis function at the integration point is the same as the method of calculating the value of the k-order Lagrangian basis function at the integration point in the above-mentioned step S22, so they will not be repeated here; the derivative value of the above-mentioned k+1-order discontinuous Lagrangian basis function at the integration point is the value of the result obtained after differentiating the k+1-order discontinuous Lagrangian basis function at the integration point.

[0097] Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point.

[0098] Specifically, assume that the stress solution is , the displacement solution is u, then the stress solution can be used to convert the intrinsic mixed finite element basis function into The values ​​at the integration point are linearly combined to obtain the stress tensor The value at can be written as:

[0099]

[0100] Among them, Stress solution In unit e, it corresponds to The components corresponding to the basis functions are represents the stress variable basis function.

[0101] Using the displacement solution, the k-1th order Lagrangian basis function is placed at the integration point The values ​​at the points are linearly combined to obtain the displacement vector at the integration point The value at can be written as:

[0102]

[0103] Among them, The displacement solution u corresponds to the first The components corresponding to the basis functions are represents the shift variable basis function.

[0104] Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix .

[0105] Specifically, the first load vector The expression can be written as:

[0106]

[0107] The coefficient matrix of the first unit above The expression can be written as:

[0108]

[0109] Among them, the above , They represent the symmetric derivatives of the k+1th order discontinuous Lagrangian basis functions respectively.

[0110] Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector .

[0111] Specifically, the second unit coefficient The expression can be written as:

[0112]

[0113] The second load vector The expression can be written as:

[0114]

[0115] Among them, represents the i-th integration point, Indicates the integral point The corresponding integral weight, S represents the unit area, represents the k+1th order discontinuous Lagrangian basis function, represents the displacement variable basis function, Denotes the displacement vector at the integration point The value at .

[0116] Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit, which can be written as:

[0117]

[0118] And using the first load vector , the second load vector Get the unit load vector of the current unit, and its expression can be written as:

[0119]

[0120] Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , and obtain the post-processing displacement solution of the current element.

[0121] Similarly, the above unit linear equations The solution method can also adopt a direct solution method or an iterative solution method. There is no specific restriction on the solution method here.

[0122] Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain a linear algebraic equation system The overall postprocessed displacement solution of .

[0123] Based on the scheme defined in the above steps S1 to S9, it can be known that, in the embodiment, the intrinsic hybrid finite element method is first used to solve the linear algebraic equation group Ax=b to obtain the stress solution and displacement solution, and then based on the stress solution and displacement solution, a post-processing method is used to recalculate the displacement solution with high precision in the high-order (k+1) discontinuous element space, thereby solving the technical problem of low displacement precision in the related intrinsic hybrid finite element method.

[0124] It should be noted that the solution provided in the embodiment of the present application only requires small-scale post-processing calculations on the divided grid units, so the post-processing calculations on each grid unit can be implemented in parallel or calculated sequentially, and the specific setting can be made according to the calculation complexity of the actual application scenario.

[0125] The technical solution provided in the above step S22 will be further described below.

[0126] Optionally, Figure 4 is a flow chart of an optional step S21 according to an embodiment of the present application, such as Figure 4 As shown, step S21 specifically includes the following steps S211-S214:

[0127] Step S211, determining the type of boundary condition.

[0128] The boundary conditions include at least one of the following: body force, stress constraint, and displacement constraint.

[0129] Step S212, when the type of boundary condition is body force, call the displacement test function, and load the integral result of the displacement test function and the body force into the load vector b.

[0130] Specifically, when the boundary condition type is body force, for the current triangular element K, assuming the body force is f, the integral result of the displacement test function v and the body force f can be calculated by the following formula, and the integral result is added to the load vector b:

[0131]

[0132] Step S213, when the type of boundary condition is stress, the least square method is called to process the stress constraint to obtain the stress constraint equation group Gx=s.

[0133] Specifically, when the boundary condition type is stress, firstly, the stress is given for each edge of the triangular unit, and the stress constraint equation group can be constructed according to the Lagrange interpolation points on the edge, and its expression is:

[0134]

[0135] Then, the obtained stress constraint equations are processed using the least squares method, and the corresponding expression is as follows:

[0136]

[0137] Finally, based on the processed stress constraint equations corresponding to all the above edges, the above-obtained processing results are added to the stress constraint equations to obtain the following equations:

[0138]

[0139] in, represents the unknown quantity that needs to be solved, g represents the number of unknown quantities, and m represents the number of linearly independent constraint equations.

[0140] Step S214, when the type of boundary condition is displacement constraint, the stress test function is called, and the integral result of the stress test function and the displacement constraint is loaded into the load vector b.

[0141] Specifically, when the boundary condition type is displacement constraint on all edges, for the edge E that meets the conditions in the current triangle unit K, let the displacement constraint be u, then the stress test function is calculated by the following formula: Calculate the integral of the displacement constraint and add it to the load vector b:

[0142]

[0143] Among them, the above n represents the unit normal vector on the edge E.

[0144] Therefore, by cyclically executing the above steps S211-S214, the load vector b and the stress constraint equation group Gx=s of the entire elastic body to be analyzed are obtained.

[0145] Optionally, the above step S23 may be further divided into the following steps S231-S233:

[0146] Step S231, determining the type of each Lagrangian interpolation point on the current unit, wherein the types of Lagrangian interpolation points include: unit vertex, unit internal point, edge internal point.

[0147] Step S232, determining the tensor basis of the two-dimensional symmetric matrix space according to the type of the Lagrangian interpolation point, wherein, when the type of the Lagrangian interpolation point is a unit vertex and a unit internal point, obtaining the standard basis of the second-order symmetric tensor space; when the type of the Lagrangian interpolation point is an edge internal point, obtaining a unit tangent vector and a unit normal vector for each of the three unit edges of the current unit, and using a unit tangent vector and a unit normal vector for each edge to calculate the edge basis of the unit edge.

[0148] Optionally, the above step S232 can also be performed as follows Figure 5 The specific steps are as follows:

[0149] Step 1: Get the current triangle unit;

[0150] Step 2: Cycle through the Lagrange interpolation points on the current triangle unit in turn;

[0151] Step 3: Determine the type of the Lagrangian interpolation point. When the type of the Lagrangian interpolation point is a unit vertex or a unit internal point, execute step 4; when the type of the Lagrangian interpolation point is an edge internal point, execute step 5.

[0152] Step 4: Obtain the standard basis of the two-dimensional symmetric tensor space.

[0153] Among them, the above standard basis can be , and A linear combination of .

[0154] Step 5: Get the unit tangent vector and unit normal vector of the three edges of the current triangle unit;

[0155] Step 6: Calculate the edge basis of each unit edge using the unit tangent vector and unit normal vector of each edge.

[0156] Among them, the edge basis of the above-mentioned unit edge includes: a first-type symmetric matrix spanned by the unit tangent vector of the unit edge, a type of symmetric matrix spanned by the unit normal vector of the unit edge and a type of symmetric matrix spanned together by the unit normal vector and unit tangent vector of the unit edge as a second-type symmetric matrix.

[0157] Specifically, the unit edge For example, for this unit edge The process of determining the edge basis is described as follows:

[0158] First, define the unit edge The unit normal vector on and the unit tangent vector .

[0159] Then, through the unit tangent vector We can define a symmetric matrix of the first kind with rank 1 and write it as , which can be expressed as: At the same time, the unit normal vector A class of symmetric matrices formed by Zhang and composed of unit normal vectors and the unit tangent vector The symmetric matrix of the second kind is the same as the symmetric matrix of the first kind. is an orthogonal complement relationship, and belongs to the second-order symmetric matrix space. The second-order symmetric matrix can be recorded as , which can be expressed as: , .

[0160] Therefore, we can use the first kind of symmetric matrix and the symmetric matrix of the second kind Together they form the edge basis of the two-dimensional symmetric matrix space. In the embodiment of the present application, in order to simplify the calculation, a set of the simplest trivial bases can be selected as the edge basis, and its expression is as follows: , and .

[0161] Step 7: Determine whether the loop reaches the last Lagrange interpolation point on the current triangle unit. If so, execute step 7, otherwise continue to execute step 3:

[0162] Step 8: Output the tensor basis of the two-dimensional symmetric matrix space corresponding to the type of each Lagrangian interpolation point on the current triangle unit.

[0163] Step S233, combining the tensor basis of the two-dimensional symmetric matrix space with the value of the k-order Lagrangian basis function at the integration point obtained in the above step S22, to obtain the value of the intrinsic mixed finite element basis function at the integration point.

[0164] Optionally, the above step S233 can also be divided into the following cases according to the type of Lagrange interpolation point:

[0165] When the type of Lagrange interpolation point is a unit vertex, the value of the intrinsic mixed finite element basis function at the integration point is obtained according to the product of the standard basis of the two-dimensional symmetric matrix space and the value of the k-order Lagrange element basis function corresponding to the current vertex at the integration point;

[0166] When the type of Lagrange interpolation point is an internal point of the unit, the value of the intrinsic mixed finite element basis function at the integration point is obtained according to the product of the standard basis of the two-dimensional symmetric matrix space and the value of the k-order Lagrange element basis function corresponding to the current internal point of the unit at the integration point;

[0167] When the type of Lagrange interpolation point is a point on the edge of the unit, the value of the intrinsic mixed finite element basis function at the integration point is obtained by multiplying the first-kind symmetric matrix of the two-dimensional symmetric matrix space with the value of the k-order Lagrange element basis function corresponding to the internal point of the current unit at the integration point;

[0168] When the type of Lagrangian interpolation point is a point on the edge of the unit, the value of the intrinsic mixed finite element basis function at the integration point is obtained based on the product of the second kind symmetric matrix of the two-dimensional symmetric matrix space and the value of the k-order Lagrangian basis function corresponding to the internal point of the current unit at the integration point.

[0169] Specifically, , and are respectively recorded as the two-dimensional force area of ​​the thin plate The unit vertex set, the internal point set on the unit edge, and the internal point set of the unit are all the triangular units divided in the unit. is the value of the Lagrangian basis function at a certain integration point, where represents the k-order Lagrangian finite element, then the value of the intrinsic mixed finite element basis function at the integration point can be determined according to the following rules:

[0170] For a given , then the value of the intrinsic mixed finite element basis function at the integration point can be written as:

[0171]

[0172] For a given , then the value of the intrinsic mixed finite element basis function at the integration point can be written as:

[0173]

[0174] For a given is an internal point on edge E, and E is a unit and unit The common edge of , then the value of the intrinsic mixed finite element basis function at the integration point can be written as:

[0175]

[0176] For a given , then the value of the intrinsic mixed finite element basis function at the integration point can be written as:

[0177]

[0178] Among them, the value of the above intrinsic mixed finite element basis function at the integration point is , , are continuous; To be discontinuous.

[0179] Optionally, Figure 6 is a flow chart of an optional step S24 according to an embodiment of the present application, such as Figure 6 As shown, step S24 specifically includes the following multiple sub-steps:

[0180] Step 1: Input multiple intrinsic mixed finite element basis functions;

[0181] Step 2: loop through each intrinsic mixed finite element basis function;

[0182] Step 3: Determine the type of the Lagrangian interpolation point corresponding to the intrinsic mixed finite element basis function, wherein when the type of the Lagrangian interpolation point is a point on the edge of the unit, execute the fourth step; when the type of the Lagrangian interpolation point is a unit vertex and a point inside the unit, execute the sixth step;

[0183] Step 4: Determine whether the intrinsic mixed finite element basis function is composed of pure tangential vectors. If so, execute step 5, otherwise execute step 6;

[0184] Step 5: Determine the overall degrees of freedom of the intrinsic mixed finite element basis function of the current unit according to the order of discontinuous degrees of freedom;

[0185] Step 6: Determine the overall degrees of freedom of the intrinsic mixed finite element basis function of the current unit according to the order of continuous degrees of freedom;

[0186] Step 7: Determine whether the loop reaches the last intrinsic mixed finite element basis function. If so, execute step 8, otherwise execute step 3;

[0187] Step 8: Integrate the overall degrees of freedom of the intrinsic hybrid finite element basis functions obtained by continuous degree of freedom sequence and discontinuous degree of freedom sequence.

[0188] Optionally, the specific implementation of step S28 includes the following steps:

[0189] Step 1: Simplify the stress constraint equation group Gx=s and obtain the equivalent constraint ,in, represents the m-dimensional constraint vector, represents an n-dimensional free vector, is a constant vector representing m dimensions, m and n represent the number of constraint variables and free variables respectively, and m+n=g. It should be noted that in the above step S102, Gaussian elimination method can be used to simplify the stress constraint equation group Gx=s;

[0190] Step 2: Adjust the order of x, and the adjusted x is:

[0191]

[0192] Step 3: Block processing is performed on the coefficient matrix A obtained by looping through the above steps S22 to S27 and the load vector b obtained in step S21, and the coefficient matrix A after block processing is obtained as follows:

[0193]

[0194] And the load vector b after block processing is:

[0195]

[0196] Step 4: Rearrange the coefficient matrix A after block processing to obtain the adjusted coefficient matrix A:

[0197]

[0198] The load vector b after block processing is rearranged to obtain the adjusted load vector b:

[0199]

[0200] Step 5: Solve the linear algebraic equations , and obtain the stress solution and displacement solution.

[0201] For example, the following will be Figure 7 The rectangular cylinder shown is subjected to stiffness analysis and the results are compared using commercial software to prove that the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method provided in the embodiment of the present application can effectively improve the displacement solution accuracy.

[0202] Figure 7 is a schematic diagram of an optional rectangular column cross section according to an embodiment of the present application, such as Figure 7 As shown, the width of the rectangular cylinder is 10 mm, and the height of the cylinder is much greater than the width of its cross section. At the same time, the left side of the cylinder is fixed, and the right side is subjected to a uniform load of 100 MPa in the x direction. Among them, the material parameters of the cylinder are Young's modulus E=1000 MPa and Poisson's ratio v=0.3.

[0203] Considering that the longitudinal length of the column is much larger than its cross-sectional size, and the external load acts parallel to the cross-sectional area, and the external load and constraint do not change along the length direction, it can be regarded as a plane strain problem. At this time, a triangular mesh can be used to divide the two-dimensional force area of ​​the simplified rectangular column, thereby minimizing the number of meshes.

[0204] Specifically, the verification ideas are as follows:

[0205] 1. In Figure 8a The displacement results are solved by using the second-order Lagrangian triangular plane strain element (Abaqus-CPE6) on the mesh model 1 (the number of elements is 2178) shown, and the obtained displacement results are regarded as the benchmark values;

[0206] 2. Figure 8b The displacement results are solved by using the third-order intrinsic hybrid finite element triangular plane strain element on the mesh model 2 (the number of elements is 200) shown, and the displacement results are post-processed using the method provided in this application. The displacement results before and after post-processing are compared with the benchmark values, respectively, to prove that the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method can effectively improve the displacement solution accuracy.

[0207] in, Figure 9-11 The displacement result cloud diagrams using the second-order Lagrangian triangular plane strain unit on mesh model 1, the displacement result cloud diagram using the third-order intrinsic hybrid finite element triangular plane strain unit on mesh model 2, and the displacement result cloud diagram after post-processing using the third-order intrinsic hybrid finite element triangular plane strain unit on mesh model 2 are respectively shown. In addition, the above-mentioned reference values ​​and displacement calculation results before and after post-processing are summarized in a table, and the following Table 1 is obtained.

[0208] Table 1

[0209]

[0210] Therefore, it can be seen from the above Table 1 that the displacement calculation is also performed using the grid model 2. The displacement calculation accuracy obtained by the post-processing method provided in the embodiment of the present application is reduced from 0.0930% to 0.0907% compared with the benchmark value, thereby significantly improving the displacement solution accuracy.

[0211] Example 2

[0212] According to an embodiment of the present application, a computer program product is also provided, which includes a stored computer program, wherein when the computer program is executed by a processor, the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method in Example 1 is implemented.

[0213] Optionally, the computer program executes the following steps:

[0214] Step S1, obtaining unit data, material parameters, and boundary conditions, wherein the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, material coefficients include: Young's modulus, Poisson's ratio, density, and the boundary conditions include: body force, stress constraint, and displacement constraint;

[0215] Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using an intrinsic mixed finite element method;

[0216] Step S3, for each unit, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-1 order discontinuous Lagrangian basis function at the integration point, the value of the k+1 order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1 order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3;

[0217] Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point;

[0218] Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix ;

[0219] Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector ;

[0220] Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit for:

[0221]

[0222] And using the first load vector , the second load vector Get the element load vector of the current element :

[0223]

[0224] Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , get the post-processing displacement solution of the current unit;

[0225] Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain the overall post-processing displacement solution of the linear algebraic equation system Ax=b.

[0226] Example 3

[0227] According to an embodiment of the present application, a non-volatile storage medium is also provided, in which a program is stored, wherein when the program is running, the device where the non-volatile storage medium is located is controlled to execute the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method in Example 1.

[0228] Optionally, the device where the non-volatile storage medium is located implements the following steps by running the program:

[0229] Step S1, obtaining unit data, material parameters, and boundary conditions, wherein the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, material coefficients include: Young's modulus, Poisson's ratio, density, and the boundary conditions include: body force, stress constraint, and displacement constraint;

[0230] Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using an intrinsic mixed finite element method;

[0231] Step S3, for each unit, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-1 order discontinuous Lagrangian basis function at the integration point, the value of the k+1 order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1 order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3;

[0232] Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point;

[0233] Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix ;

[0234] Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector ;

[0235] Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit for:

[0236]

[0237] And using the first load vector , the second load vector Get the element load vector of the current element :

[0238]

[0239] Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , get the post-processing displacement solution of the current unit;

[0240] Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain the overall post-processing displacement solution of the linear algebraic equation system Ax=b.

[0241] Example 4

[0242] According to an embodiment of the present application, a processor is also provided, which is used to run a program, wherein when the program is running, the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method in Example 1 is executed.

[0243] Optionally, the following steps are performed when the program is running:

[0244] Step S1, obtaining unit data, material parameters, and boundary conditions, wherein the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, material coefficients include: Young's modulus, Poisson's ratio, density, and the boundary conditions include: body force, stress constraint, and displacement constraint;

[0245] Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using an intrinsic mixed finite element method;

[0246] Step S3, for each unit, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-1 order discontinuous Lagrangian basis function at the integration point, the value of the k+1 order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1 order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3;

[0247] Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point;

[0248] Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix ;

[0249] Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector ;

[0250] Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit for:

[0251]

[0252] And using the first load vector , the second load vector Get the element load vector of the current element :

[0253]

[0254] Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , get the post-processing displacement solution of the current unit;

[0255] Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain the overall post-processing displacement solution of the linear algebraic equation system Ax=b.

[0256] Example 5

[0257] According to an embodiment of the present application, an electronic device is also provided, wherein: Fig.12 is a schematic diagram of the structure of an optional electronic device according to an embodiment of the present application, such as Fig.12 As shown, the electronic device includes one or more processors; a memory for storing one or more programs, which, when the one or more programs are executed by the one or more processors, enables the one or more processors to run the programs, wherein the programs are configured to execute the post-processing method for improving the displacement solution accuracy of the intrinsic mixed finite element method in the above-mentioned embodiment 1 when running.

[0258] Optionally, the processor is configured to implement the following steps by executing a computer program:

[0259] Step S1, obtaining unit data, material parameters, and boundary conditions, wherein the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, material coefficients include: Young's modulus, Poisson's ratio, density, and the boundary conditions include: body force, stress constraint, and displacement constraint;

[0260] Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using an intrinsic mixed finite element method;

[0261] Step S3, for each unit, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-1 order discontinuous Lagrangian basis function at the integration point, the value of the k+1 order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1 order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3;

[0262] Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point;

[0263] Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix ;

[0264] Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector ;

[0265] Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit for:

[0266]

[0267] And using the first load vector , the second load vector Get the element load vector of the current element :

[0268]

[0269] Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , get the post-processing displacement solution of the current unit;

[0270] Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain the overall post-processing displacement solution of the linear algebraic equation system Ax=b.

[0271] The serial numbers of the above-mentioned embodiments of the present application are for description only and do not represent the advantages or disadvantages of the embodiments.

[0272] In the above embodiments of the present application, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, please refer to the relevant description of other embodiments.

[0273] In the several embodiments provided in this application, it should be understood that the disclosed technical content can be implemented in other ways. Among them, the device embodiments described above are only schematic. For example, the division of the units can be a logical function division. There may be other division methods in actual implementation. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, indirect coupling or communication connection of units or modules, which can be electrical or other forms.

[0274] The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed on multiple units. Some or all of the units may be selected according to actual needs to achieve the purpose of the present embodiment.

[0275] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0276] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present application, or the part that contributes to the relevant technology or all or part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium, including a number of instructions to enable a computer device (which can be a personal computer, server or network device, etc.) to execute all or part of the steps of the method described in each embodiment of the present application. The aforementioned storage medium includes: U disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), mobile hard disk, disk or optical disk, etc. Various media that can store program codes.

[0277] The above is only a preferred implementation of the present application. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present application. These improvements and modifications should also be regarded as the scope of protection of the present application.

Claims

1. A post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method, characterized in that: The method comprises the following solving steps: Step S1, obtaining unit data, material parameters, and boundary conditions, wherein the unit data includes: unit shape, node coordinates, unit quantity, unit dimension, material coefficients include: Young's modulus, Poisson's ratio, density, and the boundary conditions include: body force, stress constraint, and displacement constraint; Step S2, obtaining stress solutions and displacement solutions obtained by solving the linear algebraic equation system Ax=b using an intrinsic mixed finite element method; Step S3, for each unit, obtain the unit area, multiple integration points and the integration weight of each integration point of the current unit, and calculate the value of the k-1 order discontinuous Lagrangian basis function at the integration point, the value of the k+1 order discontinuous Lagrangian basis function at the integration point and the derivative value of the k+1 order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S1, wherein k is an integer greater than or equal to 3; Step S4, using the stress solution and the displacement solution to obtain the values ​​of the stress tensor and displacement vector of the current unit at the integration point; Step S5, calculating the first load vector using material parameters, the value of the stress tensor of the current unit at the integration point, and the derivative value of the k+1 order discontinuous Lagrangian element basis function at the integration point , and use the derivative values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point to calculate the first unit coefficient matrix ; Step S6, using the values ​​of the k+1th order discontinuous Lagrangian basis function at the integration point and the values ​​of the k-1th order discontinuous Lagrangian basis function at the integration point to calculate the second unit coefficient matrix , and use the value of the displacement vector of the current unit at the integration point and the value of the k-1 order discontinuous Lagrangian element basis function at the integration point to calculate the second load vector ; Step S7, using the first unit coefficient matrix , the coefficient matrix of the second unit Get the unit coefficient matrix of the current unit for: ； And using the first load vector , the second load vector Get the element load vector of the current element for: ； Step S8, solve the unit coefficient matrix and the element load vector The unit linear equation system , get the post-processing displacement solution of the current unit; Step S9, based on the post-processing displacement solutions of each unit obtained by looping steps S3 to S8, obtain the overall post-processing displacement solution of the linear algebraic equation system Ax=b.

2. The method according to claim 1, characterized in that: The step S2 comprises the following steps: Step S21, determining the type of boundary conditions, and processing different types of boundary conditions according to a preset processing method to obtain an overall load vector b and a stress constraint equation group Gx=s; Step S22, for each unit, obtain the unit area, multiple integration points and the integral weight of each integration point of the current unit, and calculate the value of the k-order Lagrangian basis function at the integration point and the value of the k-1-order discontinuous Lagrangian basis function at the integration point in combination with the unit data and material parameters obtained in step S21 to obtain the unit area, multiple integration points and the integral weight of each integration point of the current unit; Step S23, obtaining the tensor basis of the two-dimensional symmetric matrix space of the current unit, and combining the value of the k-order Lagrangian basis function at the integration point obtained in step S22 to obtain the value of the intrinsic mixed finite element basis function at the integration point; Step S24, determining the overall degrees of freedom of the intrinsic mixed finite element basis function of the current unit; Step S25, calculating the unit energy matrix using the values ​​of the intrinsic mixed finite element basis function at each integration point on the current unit, and constructing the coefficient matrix A in combination with step S24; Step S26, calculating the derivative value of the k-th order Lagrangian basis function at the integration point, and multiplying it with the tensor basis obtained in step S23 to obtain the div value of the intrinsic mixed finite element basis function at the integration point; Step S27, using the value of the k-1 order discontinuous Lagrangian basis function at the integration point obtained in step S22 and the div value of the intrinsic mixed finite element basis function at the integration point obtained in step S26 to calculate the unit divergence operator matrix, and append it to the coefficient matrix A; Step S28, using the stress constraint equation group Gx=s, processing the coefficient matrix A obtained by looping steps S22 to S27 and the load vector b obtained in step S21 to obtain a linear algebraic equation group Ax=b; Step S29, solving the linear algebraic equation system Ax=b to obtain stress solutions and displacement solutions.

3. The method according to claim 2, characterized in that The step S21 comprises the following steps: Step S211, determining the type of boundary conditions; Step S212, when the type of boundary condition is body force, calling the displacement test function, and loading the integral result of the displacement test function and the body force into the load vector b; Step S213, when the type of boundary condition is stress, calling the least square method to process the stress constraint to obtain the stress constraint equation group Gx=s; Step S214, when the type of boundary condition is displacement constraint, the stress test function is called, and the integral result of the stress test function and the displacement constraint is loaded into the load vector b.

4. The method according to claim 2, characterized in that: The step S23 comprises the following steps: Step S231, determining the type of each Lagrangian interpolation point on the current unit, wherein the types of Lagrangian interpolation points include: unit vertex, unit internal point, edge internal point; Step S232, determining the tensor basis of the two-dimensional symmetric matrix space according to the type of the Lagrangian interpolation point, wherein when the type of the Lagrangian interpolation point is a unit vertex and a unit internal point, obtaining the standard basis of the second-order symmetric tensor space; when the type of the Lagrangian interpolation point is an edge internal point, obtaining a unit tangent vector and a unit normal vector for each of the three unit edges of the current unit, and using a unit tangent vector and a unit normal vector for each edge to calculate the edge basis of the unit edge; Step S233, combining the tensor basis of the two-dimensional symmetric matrix space with the value of the k-order Lagrangian basis function at the integration point obtained in step S22, to obtain the value of the intrinsic mixed finite element basis function at the integration point.

5. The method according to claim 1, characterized in that The step S4 comprises the following steps: Step S41, using the stress solution to linearly combine the values ​​of the intrinsic mixed finite element basis functions at the integration points obtained in S23, to obtain the values ​​of the stress tensors at the integration points; Step S42, using the displacement solution, linearly combine the values ​​of the k-1 order Lagrangian element basis functions at the integration points to obtain the values ​​of the displacement vectors at the integration points.

6. A computer program product, characterized in that include: A computer program, wherein when the computer program is executed by a processor, it implements the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method as described in any one of claims 1 to 5.

7. A non-volatile storage medium, characterized in that: The non-volatile storage medium stores a computer program, wherein the device where the non-volatile storage medium is located executes the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method as described in any one of claims 1 to 5 by running the computer program.

8. An electronic device, characterized in that: include: A memory and a processor, wherein the processor is used to run a program stored in the memory, wherein when the program is run, the post-processing method for improving the displacement solution accuracy of the intrinsic hybrid finite element method as described in any one of claims 1 to 5 is executed.

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