A prestressed beam simulation method, system and device based on vector finite element
By simulating prestressed beams using the vector finite element method, the problem of inaccurate simulation in existing technologies is solved, and efficient and stable simulation of prestressed beams is achieved, especially in bridge structural analysis.
Patent Information
- Application Number
- CN202510232966.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-28
- Publication Date
- 2025-12-05
- Estimated Expiration
- 2045-02-28
AI Technical Summary
Existing technologies struggle to accurately simulate the complex behavior of prestressed beams, resulting in non-convergent calculations and poor accuracy.
A numerical model of the prestressed beam was established using the vector finite element method. By discretizing the beam into multiple nodes and elements, the stiffness matrix was calculated, the governing equations were determined, and the central difference method was used to solve for the node states, thus completing the simulation of the prestressed beam.
It enables accurate simulation of prestressed tendons of arbitrary shapes, improves computational efficiency and result stability, and is applicable to bridge structure analysis in the field of civil engineering.
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Figure CN120163009B_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of finite element simulation technology, and in particular to a method, system and equipment for simulating prestressed beams based on vector finite element method. Background Technology
[0002] Currently, the industry has made some progress in simulating prestressed beams using traditional methods. However, the existing computational framework is difficult to simulate the complex behavior of prestressed beams, and the calculation results of traditional methods do not converge and have poor accuracy. Summary of the Invention
[0003] The purpose of this application is to provide a method, system, and device for simulating prestressed beams based on vector finite element method, which can accurately simulate prestressed beams containing prestressed tendons of arbitrary shapes, improve computational efficiency, and ensure computational stability.
[0004] To achieve the above objectives, this application provides the following solution:
[0005] Firstly, this application provides a method for simulating prestressed beams based on vector finite element method, including:
[0006] A numerical model of the prestressed beam was established using the vector finite element method.
[0007] The numerical model of the prestressed beam is discretized into multiple nodes and multiple prestressed beam elements;
[0008] Calculate the stiffness matrix of each prestressed beam element;
[0009] The governing equations for each node are determined based on the stiffness matrix.
[0010] The central difference method is used to solve the governing equations of each node, and the state of each node is updated based on the solution results to complete the simulation of the prestressed beam.
[0011] Secondly, this application provides a prestressed beam simulation system based on vector finite element method, comprising:
[0012] The model building module is used to establish a numerical model of a prestressed beam using the vector finite element method.
[0013] The discretization module is used to discretize the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements.
[0014] Stiffness matrix calculation unit, used to calculate the stiffness matrix of each prestressed beam element;
[0015] The control equation determination module is used to determine the control equations for each node based on the stiffness matrix.
[0016] The solution and update module is used to solve the governing equations of each node using the central difference method, and to update the state of each node based on the solution results, thus completing the simulation of the prestressed beam.
[0017] Thirdly, this application provides a computer device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described vector finite element-based prestressed beam simulation method.
[0018] According to the specific embodiments provided in this application, this application has the following technical effects:
[0019] This application provides a method, system, and device for simulating prestressed beams based on vector finite element method. By establishing a numerical model of the prestressed beam using the vector finite element method, the design parameters of the prestressed beam structure can be optimized. It is convenient for simulating prestressed beams containing prestressing tendons of arbitrary shapes. It can effectively analyze large deformation and strong nonlinear problems of prestressed beams, significantly improve the calculation speed and result stability, and is applicable to the analysis of bridge structures with prestressing tendons in the field of civil engineering. Attached Figure Description
[0020] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0021] Figure 1 A flowchart illustrating a prestressed beam simulation method based on vector finite element method provided in an embodiment of this application;
[0022] Figure 2 This is a schematic diagram of the discretization of the numerical model of a prestressed beam;
[0023] Figure 3 This is a schematic diagram of the nodal displacements of a prestressed beam element;
[0024] Figure 4 This is a comparison chart of the vector calculation results and the calculation results of the simplified formula in mechanics of materials.
[0025] Figure 5 This is a schematic diagram of the structure of a computer device provided in an embodiment of this application. Detailed Implementation
[0026] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.
[0027] VectorForm Intrinsic Finite Element (VFIFE) is a novel finite element method proposed by Professor Ding Chengxian. This method utilizes vector theory, making finite element programs more concise and systematic, enabling accurate simulation of various structures and prediction of their mechanical behaviors. The core characteristic of vectorform finite element theory is that different structural forms, including planar and three-dimensional trusses, frames, solids, and plates / shells, as well as complex mechanical behaviors such as large deformations, spatial motion, material nonlinearity, fracture, and collapse, can all be handled using the same concepts and processes, offering advantages unmatched by traditional finite element methods.
[0028] In the study of prestressed beams, vector finite element method (DFEM) can obtain the stress distribution, deformation, and failure mechanism of prestressed beam sections under different load conditions. By using DFEM simulation in the design phase, the design parameters of prestressed beam structures can be optimized. Simultaneously, DFEM can also simulate the behavior of prestressed beams during construction and service, such as simulating the application of tension, structural camber, and obtaining the frequencies and mode shapes of buildings or bridges.
[0029] The purpose of this application is to provide a method, system, and device for simulating prestressed beams based on vector finite element method, which can accurately simulate prestressed beams containing prestressed tendons of arbitrary shapes, improve computational efficiency, and ensure computational stability.
[0030] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.
[0031] In one exemplary embodiment, such as Figure 1 As shown, a method for simulating prestressed beams based on vector finite element method is provided. This method is executed by a computer device, specifically by a terminal or server alone, or by both a terminal and a server. In this embodiment, the method is described using a server as an example, and includes the following steps S1 to S5. Wherein:
[0032] S1: A numerical model of the prestressed beam is established using the vector finite element method.
[0033] Specifically, the structural dimensions of the prestressed beam are calculated based on its actual structure and input into the program. The material parameters corresponding to each part of the prestressed beam are input into the program based on the material performance test data. The total calculation time and individual time step of the program are determined. The program is then run 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 instance, such as Figure 2 As shown, the numerical model of the prestressed beam is evenly divided into 5 nodes with mass, each node spaced 0.25m apart. The spaces between nodes are filled with massless prestressed beam elements, and each prestressed beam element has a length of 0.25m. The total mass of the prestressed beam is evenly distributed across the nodes, and the material properties of each part of the prestressed beam are assigned to all prestressed beam elements. F1(t)-F5(t) represent the external forces at nodes 1-5.
[0036] S3: Calculate the stiffness matrix of each prestressed beam element. Specifically, this includes:
[0037] S31: Determine the strain matrix and the elasticity matrix; the strain matrix includes the strain matrix of the prestressed beam element and the strain matrix of the prestressed tendon, and the elasticity matrix includes the elasticity matrix of the prestressed beam and the elasticity matrix of the prestressed tendon.
[0038] like Figure 3 As shown, a local coordinate system is first defined. Define prestressed beam elements in The displacement vector in the coordinate system is Where Δ e This represents the axial deformation of the element after deducting rigid body displacement. The rotation angle of node 1 in the prestressed beam element after deducting rigid body displacement. The rotation angle of node 2 in the prestressed beam element after deducting rigid body displacement. Figure 3 In the middle, 1 a The position of node 1 before deformation, 1 d The position of node 1 after deformation, 2 a The position of node 2 before deformation, 2 d φ represents the position of node 2 after deformation, and φ represents the angle between the prestressed beam element and the prestressed tendon.
[0039] Strain matrix B of prestressed beam element b And the strain matrix B of the prestressed tendons t For details, see formula (1), the elastic matrix D of the prestressed beam element. b And the elastic matrix D of the prestressed tendons t For details, see formula (2).
[0040]
[0041] In the formula, l is the length of the prestressed beam element, l t The length of the prestressing tendons within the prestressed beam element. Here, E represents the coordinates in the local coordinate system, c is the cosine 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. b E represents the stiffness of a prestressed beam element. t This refers to the stiffness of the prestressing tendons.
[0042] S32: Determine the governing equations for each prestressed beam element based on the strain matrix, the elasticity matrix, and the displacement vector 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). The variational principle is a universally applicable mathematical law in nature's static (relatively stable) phenomena, also known as the least action theorem. It is also a fundamental principle of physics, expressed using the variational method. According to the variational principle, the actual deformation of the structure causes the strain energy of the prestressed beam element to reach a minimum value, from which the governing equation can be obtained, as shown in formula (4). From formula (4), the stiffness matrix K of the prestressed beam element can be obtained. e For details, see formula (5).
[0044]
[0045]
[0046] Among them, V b V is the volume of a prestressed beam element. t Let dV be the volume of the prestressed tendon, and dV be a volume element.
[0047] S33: Solve the governing equations for 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 E t =1.95×10 11 N / m 2 The cross-sectional width of the prestressed beam element is 0.12m, and the cross-sectional height of the prestressed beam element is 0.24m. Substituting these values into formula (5), the element stiffness matrix K can be obtained. e .
[0049] S4: Determine the governing equations for each node based on the stiffness matrix. Specifically, this includes: calculating the internal force vector of each node based on the stiffness matrix; and determining the governing equations for each node based on the internal and external force vectors.
[0050] Formula for calculating the internal force vector of a node:
[0051]
[0052] The motion of each node in the structure is described by a separate governing equation, as shown in equation (7):
[0053]
[0054] In the formula, m i Let be the quality matrix of the i-th node. Let be the acceleration vector of the i-th node. Let i be the external force vector at the i-th node. Let be the internal force vector of the i-th node.
[0055] S5: The central difference method is used to solve the governing 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.
[0056] The central difference method is based on replacing the derivative of displacement with respect to time with finite differences. The velocity is obtained by taking the first derivative of the displacement, and the acceleration is obtained by taking the second derivative. The central difference method can be used to solve the governing equations for each node, as shown in formula (8):
[0057]
[0058] In the formula, x i,j Let F be the displacement vector of the i-th node at time step j. i,j-1 Let x be the resultant force vector at the (j-1)th time step of the i-th node. i,j-1 Let x be the displacement vector of the i-th node at time step j-1. i,j-2 Let x be the displacement vector of the i-th node at time step j-2. i,-1 Let x be the virtual displacement vector before the initial position of the i-th node. i,0 Let be the initial displacement vector of the i-th node. Let F be the initial velocity vector of the i-th node. i,0 Let h be the initial resultant force vector at node i, h be the time step, ξ be the structural damping coefficient, and C1 and C2 be intermediate variables.
[0059] The entire calculation process is described as follows: First, the total calculation time t is set. nThe algorithm calculates the displacement, velocity, and acceleration of the node at the virtual time t=-1 based on the displacement, velocity, and acceleration of the node at time t=0, and iterates through all nodes. Then, it calculates the displacement, velocity, and acceleration of the node at time t based on the displacement, velocity, and acceleration of the nodes at times t-2 and t-1, and iterates through all nodes. The time step is incremented by 1; this process is repeated until the set calculation time t is reached. n Output the calculation result.
[0060] Figure 4 The graph shows a comparison between the results of the vector method calculation and the results of the simplified formula in mechanics of materials. The relative error between the vector method calculation and the simplified formula is 5.01%. Since the vector method can take into account the prestress loss, its result is slightly smaller than that of the simplified formula.
[0061] Based on the same inventive concept, this application also provides a prestressed beam simulation system based on vector finite element method. The solution provided by this system is similar to the solution described in the above method. Therefore, the specific limitations of one or more embodiments of the prestressed beam simulation system based on vector finite element method provided below can be found in the limitations of the prestressed beam simulation method based on vector finite element method above, and will not be repeated here.
[0062] In one exemplary embodiment, a prestressed beam simulation system based on vector finite element method is provided, comprising:
[0063] The model building module is used to establish a numerical model of a prestressed beam using the vector finite element method.
[0064] The discretization module is used to discretize the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements.
[0065] The stiffness matrix calculation unit is used to calculate the stiffness matrix of each prestressed beam element.
[0066] The control equation determination module is used to determine the control equations for each node based on the stiffness matrix.
[0067] The solution and update module is used to solve the governing equations of each node using the central difference method, and to update the state of each node based on the solution results, thus completing the simulation of the prestressed beam.
[0068] In one exemplary embodiment, a computer device is provided, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps in the above-described method embodiments. The computer device may be a server or a terminal, and its internal structure diagram may be as follows: Figure 5As shown, this computer device includes a processor, memory, input / output interfaces (I / O), and a communication interface. The processor, memory, and I / O interfaces are connected via a system bus, and the communication interface is also connected to the system bus via the I / O interfaces. The processor provides computational and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system, computer programs, and a database. The internal memory provides the environment for the operation of the operating system and computer programs stored in the non-volatile storage media. The database stores data to be processed. The I / O interfaces are used for exchanging information between the processor and external devices. The communication interface is used for communicating with external terminals via a network connection. When the computer program is executed by the processor, it implements a vector-based finite element method for simulating prestressed beams.
[0069] Those skilled in the art will understand that Figure 5 The structures shown are merely block diagrams of some structures related to the present application and do not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than shown in the figures, or combine certain components, or have different component arrangements. In an exemplary embodiment, a computer device is provided, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps in the above-described method embodiments.
[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 used for analysis, data stored, data displayed, 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 the relevant data must comply with relevant regulations.
[0071] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, databases, or other media used in the embodiments provided in this application can include at least one of non-volatile and volatile memory. 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), magnetic 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 take many forms, such as Static Random Access Memory (SRAM) or Dynamic Random Access Memory (DRAM).
[0072] The databases involved in the embodiments provided in this application may include at least one type of relational database and non-relational database. Non-relational databases may include, but are not limited to, blockchain-based distributed databases. The processors involved in the embodiments provided in this application may be general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic devices, quantum computing-based data processing logic devices, etc., and are not limited to these.
[0073] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.
[0074] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.
Claims
1. A method for simulating prestressed beams based on vector finite element method, characterized in that, include: A numerical model of the prestressed beam was established using the vector finite element method. The numerical model of the prestressed beam is discretized into multiple nodes and multiple prestressed beam elements; Calculate the stiffness matrix of each prestressed beam element; specifically, this includes: determining the strain matrix and elasticity matrix; the strain matrix includes the strain matrix of the prestressed beam element and the strain matrix of the prestressing tendon, and the elasticity matrix includes the elasticity matrix of the prestressed beam and the elasticity matrix of the prestressing tendon; determining the governing equations of each prestressed beam element based on the strain matrix, the elasticity matrix, and the displacement vector of each prestressed beam element; solving the governing equations of each prestressed beam element to obtain the stiffness matrix of each prestressed beam element; The governing equations for each node are determined based on the stiffness matrix. The central difference method is used to solve the governing 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. The expressions for the strain matrix of the prestressed beam element and the strain matrix of the prestressed tendon are as follows: The expressions for the elastic matrix of the prestressed beam and the elastic matrix of the prestressed tendon are as follows: Among them, B b Let B be the strain matrix of the prestressed beam element. t Let l be the strain matrix of the prestressed tendon, and l be the length of the prestressed beam element. t Let s be the length of the prestressed tendon within the prestressed beam element. The ratio of l to l and For prestressed beam elements in In the coordinate system, c is the cosine of the angle between the prestressing tendon and the prestressed beam element, e1 and e2 are the distances between the left and right ends of the prestressing tendon and the prestressed beam element, respectively, and D is the coordinate of the prestressing tendon. b Let D be the elastic matrix of the prestressed beam. t Let E be the elastic matrix of the prestressed tendon. b E represents the stiffness of a prestressed beam element. t The stiffness of the prestressing tendons; The formula for calculating the stiffness matrix of a prestressed beam element is: Among them, K e V is the stiffness matrix of a prestressed beam element. b V is the volume of a prestressed beam element. t Let dV be the volume of the prestressed tendon, and dV be a volume element.
2. The prestressed beam simulation method based on vector finite element method according to claim 1, characterized in that, After discretizing the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements, the method further includes setting boundary conditions and load conditions.
3. The prestressed beam simulation method based on vector finite element method according to claim 1, characterized in that, The governing equations for each node are determined based on the stiffness matrix, specifically including: The internal force vector of each node is calculated based on the stiffness matrix; The control equations for each node are determined based on the internal force vector and the external force vector.
4. The prestressed beam simulation method based on vector finite element method according to claim 3, characterized in that, The formula for calculating the internal force vector of a node is: Among them, F int K is the internal force vector of the node. e Here is the stiffness matrix of the prestressed beam element. For prestressed beam elements in Displacement vector in the coordinate system.
5. The prestressed beam simulation method based on vector finite element method according to claim 3, characterized in that, The governing equations for the nodes are expressed as follows: Where, m i Let be the quality matrix of the i-th node. Let F be the acceleration vector of the i-th node. i ext Let F be the external force vector at node i. i int Let be the internal force vector of the i-th node.
6. A prestressed beam simulation system based on vector finite element method, characterized in that, The system is applied to the prestressed beam simulation method based on vector finite element method as described in any one of claims 1-5, and the system comprises: The model building module is used to establish a numerical model of a prestressed beam using the vector finite element method. The discretization module is used to discretize the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements. Stiffness matrix calculation unit, used to calculate the stiffness matrix of each prestressed beam element; The control equation determination module is used to determine the control equations for each node based on the stiffness matrix. The solution and update module is used to solve the governing equations of each node using the central difference method, and to update the state of each node based on the solution results, thus completing the simulation of the prestressed beam.
7. A computer device, comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that the processor executes the computer program to implement the prestressed beam simulation method based on vector finite element method as described in any one of claims 1-5.