Prestressed beam simulation method, system and equipment based on vector finite element

Abstract

The invention discloses a prestressed beam simulation method, system and device based on a vector type finite element, and relates to the technical field of finite element simulation, and the method comprises the steps: building a prestressed beam numerical model through a vector type finite element method; dispersing the prestressed beam numerical model into a plurality of nodes and a plurality of prestressed beam units; calculating a stiffness matrix of each prestressed beam unit; determining a control equation of each node based on the stiffness matrix; and solving the control equation of each node by adopting a central difference method, and updating the state of each node according to the solving result to finish the simulation of the prestressed beam. The method can achieve the simulation of the complex behaviors of the prestressed beam, and is high in calculation efficiency and good in calculation stability.

Description

Technical Field

[0001] This application relates to the field of finite element simulation technology, and particularly to a method, system and device for simulating prestressed beams based on vectorized finite elements. Background Art

[0002] At present, some progress has been made in the simulation research of prestressed beams by applying traditional methods in the industry. However, it is difficult for the existing calculation framework to simulate the complex behaviors of prestressed beams, and the calculation results of traditional methods do not converge and the calculation accuracy is not good. Summary of the Invention

[0003] The purpose of this application is to provide a method, system and device for simulating prestressed beams based on vectorized finite elements, which can accurately simulate prestressed beams containing prestressing tendons of any shape, improve the calculation efficiency and ensure the calculation stability.

[0004] To achieve the above purpose, this application provides the following solutions:

[0005] In the first aspect, this application provides a method for simulating prestressed beams based on vectorized finite elements, including:

[0006] Establishing a numerical model of a prestressed beam by using the vectorized finite element method;

[0007] Discretizing the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements;

[0008] Calculating the stiffness matrix of each prestressed beam element;

[0009] Determining the control equation of each node based on the stiffness matrix;

[0010] Solving the control equation of each node by using the central difference method and updating the state of each node according to the solution result to complete the simulation of the prestressed beam.

[0011] In the second aspect, this application provides a system for simulating prestressed beams based on vectorized finite elements, including:

[0012] A model construction module for establishing a numerical model of a prestressed beam by using the vectorized finite element method;

[0013] A discretization module for discretizing the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements;

[0014] A stiffness matrix calculation unit for calculating the stiffness matrix of each prestressed beam element;

[0015] A control equation determination module for determining the control equation of each node based on the stiffness matrix;

[0016] A solution and update module, which is used to solve the control equations of each node by using the central difference method, and update the states of each node according to the solution results, so as to complete the simulation of the prestressed beam.

[0017] In a third aspect, the present application provides a computer device, including: a memory, a processor, and a computer program stored on the memory and executable on the processor, wherein the processor executes the computer program to implement the above-mentioned prestressed beam simulation method based on the vectorized finite element.

[0018] According to the specific embodiments provided by the present application, the present application has the following technical effects:

[0019] The present application provides a prestressed beam simulation method, system and device based on the vectorized finite element. By establishing a numerical model of the prestressed beam using the vectorized finite element method, the design parameters of the prestressed beam structure can be optimized, facilitating the simulation of prestressed beams containing prestressing tendons of arbitrary shapes, effectively analyzing the large deformation and strong nonlinear problems of prestressed beams, significantly improving the calculation speed and result stability, and being applicable to the analysis of bridge structures with prestressing tendons in the field of civil engineering. Description of the Drawings

[0020] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings required to be used in the embodiments. Obviously, the following drawings are only some embodiments of the present application. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0021] Figure 1 It is a schematic flowchart of the prestressed beam simulation method based on the vectorized finite element provided by an embodiment of the present application;

[0022] Figure 2 It is a schematic diagram of the discretization of the prestressed beam numerical model;

[0023] Figure 3 It is a schematic diagram of the nodal displacements of the prestressed beam element;

[0024] Figure 4 It is a comparison diagram of the calculation results of the vectorized calculation result and the calculation result of the simplified formula of material mechanics;

[0025] Figure 5 It is a schematic structural diagram of a computer device provided by an embodiment of the present application. Detailed Embodiments

[0026] The following will clearly and completely describe the technical solutions in the embodiments of the present application in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present application without creative efforts shall fall within the protection scope of the present application.

[0027] The Vector Form Intrinsic Finite Element (VFIFE) is a brand-new finite element method proposed by Professor Ding Chengxian. This method uses vector theory, which can make the finite element program more concise and systematic, and can accurately simulate various structures, thereby predicting various mechanical behaviors of real structures. The core feature of the vector form finite element theory is that different structural forms, including plane and three-dimensional trusses, rigid frames, solids, and plates, etc., as well as complex mechanical behaviors, including large deformations, spatial motions, material nonlinearities, fractures, and collapses, etc., can all be processed with the same concept and process, having advantages that cannot be compared with traditional finite elements.

[0028] In the research of prestressed beams, the vector form finite element can obtain the cross-sectional stress distribution, deformation conditions, and failure mechanisms of prestressed beams under different load conditions. By using vector form finite element simulation in the design stage, the design parameters of the prestressed beam structure can be optimized. At the same time, the vector form finite element can also simulate the behaviors of prestressed beams during construction and use, such as simulating the application of tensile forces, structural cambers, and obtaining the frequencies and vibration modes of buildings or bridges.

[0029] The purpose of the present application is to provide a prestressed beam simulation method, system, and device based on the vector form finite element, which can accurately simulate prestressed beams containing prestressing tendons of any shape, improve the calculation efficiency, and ensure the calculation stability.

[0030] To make the above objects, features, and advantages of the present application more obvious and understandable, the present application will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0031] In an exemplary embodiment, as Figure 1 shown, a prestressed beam simulation method based on the vector form finite element is provided. This method is executed by a computer device, specifically, it can be executed alone by a computer device such as a terminal or a server, or jointly executed by a terminal and a server. In the embodiments of the present application, taking the application of this method to a server as an example for illustration, it includes the following steps S1 to S5. Among them:

[0032] S1: Establish a numerical model of the prestressed beam using the vector form finite element method.

[0033] Specifically, calculate the structural dimensions according to the actual structure of the prestressed beam and input them into the program. Input the material parameters corresponding to each part of the prestressed beam into the program according to the material performance test data. Determine the total duration and single time step of the program calculation, and run the program to establish a numerical model of the prestressed beam.

[0034] S2: Discretize the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements.

[0035] In a specific example, as Figure 2 shown, evenly divide the numerical model of the prestressed beam with 5 mass nodes, with a spacing of 0.25 m between each node. The elements between nodes are massless prestressed beam elements, so the length of each prestressed beam element is 0.25 m. Distribute the total mass of the prestressed beam evenly to each node, endow the characteristics of the materials of each part of the prestressed beam to all prestressed beam elements, and F1(t)-F5(t) represent the external forces on nodes 1-5.

[0036] S3: Calculate the stiffness matrix of each prestressed beam element. Specifically, it includes:

[0037] S31: Determine the strain matrix and the elastic matrix; the strain matrix includes the strain matrix of the prestressed beam element and the strain matrix of the prestressing tendon, and the elastic matrix includes the elastic matrix of the prestressed beam and the elastic matrix of the prestressing tendon.

[0038] As Figure 3 shown, first define the local coordinate system Define the displacement vector of the prestressed beam element in the coordinate system as where Δ e is the axial deformation of the element after deducting the rigid body displacement, is the rotation angle of node 1 in the prestressed beam element after deducting the rigid body displacement, is the rotation angle of node 2 in the prestressed beam element after deducting the rigid body displacement. Figure 3 In a is the position of node 1 before deformation, 1 d is the position of node 1 after deformation, 2 a is the position of node 2 before deformation, 2 d is the position of node 2 after deformation, and φ represents the angle between the prestressed beam element and the prestressing tendon.

[0039] The strain matrix B of the prestressed beam element b and the strain matrix B of the prestressing tendon t , specifically see formula (1). The elastic matrix D of the prestressed beam element b and the elastic matrix D of the prestressing tendon t , specifically see formula (2).

[0040]

[0041] wherein, l is the length of the prestressed beam element, and l t is the length of the prestressed tendon within the prestressed beam element, are the coordinates in the local coordinate system, c is the cosine value of the angle between the prestressed tendon and the prestressed beam element, e1 and e2 are the distances between the left and right ends of the prestressed tendon and the prestressed beam element respectively, and E b is the stiffness of the prestressed beam element, and E t is the stiffness of the prestressed tendon.

[0042] S32: Determine the control equations of each prestressed beam element based on the strain matrix, the elastic matrix, and the displacement vectors of each prestressed beam element.

[0043] The strain energy expression of the prestressed beam element can be obtained by integrating formulas (1) and (2), as shown in formula (3) specifically. The variational principle is a generally applicable mathematical law in static (relatively stable state) things in nature, also known as the principle of least action. It is also a basic principle in physics, expressed by the variational method. According to the variational principle, the actual deformation of the structure makes the strain energy of the prestressed beam element take the minimum value, and based on this, the control equation can be obtained, as shown in formula (4) specifically. The stiffness matrix K of the prestressed beam element can be obtained from formula (4), e as shown in formula (5) specifically.

[0044]

[0045]

[0046] wherein, V b is the volume of the prestressed beam element, V t is the volume of the prestressed tendon, and dV is the volume element.

[0047] S33: Solve the control equations of each prestressed beam element to obtain the stiffness matrix of each prestressed beam element.

[0048] In this embodiment, E b = 4.16×10 10 N / m 2 and E t = 1.95×10 11 N / m 2 The cross-sectional width of the prestressed beam element is 0.12 m, and the cross-sectional height of the prestressed beam element is 0.24 m. Substituting the above values into formula (5), the element stiffness matrix K e can be obtained.

[0049] S4: Determine the control equations of each node based on the stiffness matrix. Specifically, it includes: calculating the internal force vector of each node based on the stiffness matrix; determining the control equations of each node based on the internal force vector and the external force vector.

[0050] Calculation formula for the internal force vector of a node:

[0051]

[0052] The motion of each node on the structure is described by a separate control equation, as shown in formula (7) specifically:

[0053]

[0054] In the formula, m i is the mass matrix of the i-th node, is the acceleration vector of the i-th node, is the external force vector of the i-th node, is the internal force vector of the i-th node.

[0055] S5: Solve the control equations of each node using the central difference method, and update the state of each node according to the solution results to complete the simulation of the prestressed beam.

[0056] The central difference method is based on replacing the derivative of displacement with respect to time by finite differences. The first derivative of displacement gives the velocity, and the second derivative of displacement gives the acceleration. The control equations of each node can be solved using the central difference method, as shown in formula (8) specifically:

[0057]

[0058] In the formula, x i,j is the displacement vector of the i-th node at the j-th time step, F i,j-1 is the resultant force vector of the i-th node at the (j - 1)-th time step, x i,j-1 is the displacement vector of the i-th node at the (j - 1)-th time step, x i,j-2 is the displacement vector of the i-th node at the (j - 2)-th time step, x i,-1 is the virtual displacement vector before the initial position of the i-th node, x i,0 is the initial displacement vector of the i-th node, is the initial velocity vector of the i-th node, F i,0 is the initial resultant force vector of the i-th node, h is the time step, ξ is the structural damping coefficient, and C1, C2 are intermediate variables.

[0059] The entire calculation process is described as follows: First, set the total calculation time t n, solve the displacements, velocities, and accelerations of the virtual time nodes at t=-1 based on the displacements, velocities, and accelerations of the nodes at t=0, and calculate all nodes in a loop; then solve the displacements, velocities, and accelerations of the nodes at time t based on the displacements, velocities, and accelerations of the nodes at t-2 and t-1, and calculate all nodes in a loop; increment the time step by 1; calculate in a loop until the set calculation time t is reached n , and output the operation results.

[0060] Figure 4 is a comparison diagram of the calculation results of the vector calculation formula and the simplified formula of material mechanics. The relative error between the calculation results of the vector calculation formula and the simplified formula of material mechanics is 5.01%. Since the vector finite element can consider the prestress loss, its result is slightly smaller than the calculation result of the simplified formula of material mechanics.

[0061] Based on the same inventive concept, the embodiment of the present application also provides a prestressed beam simulation system based on vector finite element. The implementation solutions provided by this system to solve problems are similar to the implementation solutions described in the above method. Therefore, the specific limitations in one or more embodiments of the prestressed beam simulation system based on vector finite element provided below can refer to the limitations on the prestressed beam simulation method based on vector finite element in the above text, and will not be repeated here.

[0062] In an exemplary embodiment, a prestressed beam simulation system based on vector finite element is provided, including:

[0063] A model construction module for establishing a numerical model of a prestressed beam using the vector finite element method.

[0064] A discretization module for discretizing the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements.

[0065] A stiffness matrix calculation unit for calculating the stiffness matrices of the prestressed beam elements.

[0066] A control equation determination module for determining the control equations of each node based on the stiffness matrix.

[0067] A solution and update module for solving the control equations of each node using the central difference method and updating the states of each node according to the solution results to complete the simulation of the prestressed beam.

[0068] In an exemplary embodiment, a computer device is provided, including a memory and a processor. A computer program is stored in the memory, and when the processor executes the computer program, the steps in the above method embodiments are implemented. This computer device can be a server or a terminal, and its internal structure diagram can be as Figure 5As shown in the figure. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O), and a communication interface. Among them, the processor, the memory, and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Among them, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program, and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store data to be processed. The input / output interface of the computer device is used to exchange information between the processor and external devices. The communication interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, it implements a prestressed beam simulation method based on vector finite element.

[0069] Those skilled in the art can understand that Figure 5 the structure shown in the figure is only a block diagram of some structures related to the solution of this application, and does not constitute a limitation on the computer device to which the solution of this application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine some components, or have different component arrangements. In an exemplary embodiment, a computer device is provided, including a memory and a processor. A computer program is stored in the memory, and when the processor executes the computer program, the steps in the above method embodiments are implemented.

[0070] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use, and processing of relevant data need to comply with relevant regulations.

[0071] Those of ordinary skill in the art can understand that all or part of the processes in the methods of the above embodiments can be completed by instructing relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above methods. Among them, any reference to a memory, database, or other medium used in the various embodiments provided in the present application can include at least one of non-volatile and volatile memories. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc.

[0072] The databases involved in the various embodiments provided in the present application can include at least one of relational databases and non-relational databases. Non-relational databases can include distributed databases based on blockchain, etc., without limitation. The processors involved in the various embodiments provided in the present application can be general-purpose processors, central processors, graphics processors, digital signal processors, programmable logic devices, data processing logics based on quantum computing, etc., without limitation.

[0073] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the various technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered to be within the scope described in this specification.

[0074] Specific examples are used in this article to elaborate on the principles and implementation manners of the present application. The description of the above embodiments is only used to help understand the method and its core idea of the present application; at the same time, for those of ordinary skill in the art, according to the idea of the present application, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present application.

Claims

1. A prestressed beam simulation method based on vector finite element, characterized in that: include: The numerical model of prestressed beam is established by using vector finite element method; Discretizing the prestressed beam numerical model into a plurality of nodes and a plurality of prestressed beam units; Calculate the stiffness matrix of each prestressed beam element; Determining the control equations of each node based on the stiffness matrix; The central difference method is used to solve the control equations of each node, and the state of each node is updated according to the solution results to complete the simulation of the prestressed beam.

2. The prestressed beam simulation method based on vector finite element according to claim 1 is characterized in that: After the prestressed beam numerical model is discretized into a plurality of nodes and a plurality of prestressed beam units, the method further includes: setting boundary conditions and load conditions.

3. The prestressed beam simulation method based on vector finite element according to claim 1 is characterized in that: Calculate the stiffness matrix of each prestressed beam unit, including: Determine a strain matrix and an elastic matrix; the strain matrix includes a strain matrix of a prestressed beam unit and a strain matrix of a prestressed tendon, and the elastic matrix includes an elastic matrix of a prestressed beam and an elastic matrix of a prestressed tendon; Determining the control equation of each prestressed beam unit based on the strain matrix, the elastic matrix and the displacement vector of each prestressed beam unit; The control equations of each prestressed beam unit are solved to obtain the stiffness matrix of each prestressed beam unit.

4. The prestressed beam simulation method based on vector finite element according to claim 3 is characterized in that: The strain matrix of the prestressed beam unit and the strain matrix of the prestressed tendon are expressed as follows: The expressions of the elastic matrix of the prestressed beam and the elastic matrix of the prestressed tendon are: Among them, B b is the strain matrix of the prestressed beam element, B t is the strain matrix of the prestressed tendons, l is the length of the prestressed beam unit, l t is the length of the prestressed tendon in the prestressed beam unit, s is The ratio of l, and For prestressed beam elements coordinates in the coordinate system, c is the cosine value of the angle between the prestressed tendon and the prestressed beam unit, e1 and e2 are the distances between the left and right ends of the prestressed tendon and the prestressed beam unit, D b is the elastic matrix of the prestressed beam, D t is the elastic matrix of the prestressed tendon, E b is the stiffness of the prestressed beam unit, E t is the stiffness of the prestressed tendons.

5. The prestressed beam simulation method based on vector finite element according to claim 4 is characterized in that: The calculation formula of the stiffness matrix of the prestressed beam element is: Among them, K e is the stiffness matrix of the prestressed beam element, V b is the volume of the prestressed beam unit, V t is the volume of the prestressed tendon, and dV is the volume element.

6. The prestressed beam simulation method based on vector finite element according to claim 1 is characterized in that: The control equations of each node are determined based on the stiffness matrix, specifically including: Calculating the internal force vector of each node based on the stiffness matrix; The control equation of each node is determined based on the internal force vector and the external force vector.

7. The prestressed beam simulation method based on vector finite element according to claim 6 is characterized in that: The calculation formula of the node's internal force vector is: Among them, F int is the internal force vector of the node, K e is the stiffness matrix of the prestressed beam element, For prestressed beam elements Displacement vector in the coordinate system.

8. The prestressed beam simulation method based on vector finite element according to claim 6 is characterized in that: The expression of the node's control equation is: Among them, m i is the mass matrix of the ith node, is the acceleration vector of the ith node, F i ext is the external force vector of the ith node, F i int is the internal force vector of the i-th node.

9. A prestressed beam simulation system based on vector finite element, characterized in that: include: Model building module, used to establish the numerical model of prestressed beams using vector finite element method; A discrete module, used for discretizing the prestressed beam numerical model into a plurality of nodes and a plurality of prestressed beam units; Stiffness matrix calculation unit, used to calculate the stiffness matrix of each prestressed beam unit; A control equation determination module, used for determining the control equation of each node based on the stiffness matrix; The solving and updating module is used to solve the control equations of each node using the central difference method, and update the status of each node according to the solution results to complete the simulation of the prestressed beam.

10. A computer device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the prestressed beam simulation method based on vector finite element according to any one of claims 1 to 8.

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