A Finite Element Form Finding Method for Multi-span Transmission Conductors Based on BFGS Optimization Algorithm
The finite element form-finding method for multi-span transmission conductors based on the BFGS optimization algorithm solves the problems of high computational cost and poor convergence in the existing technology, and realizes efficient and stable form-finding in complex terrain, which is suitable for power grid transmission line design.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- 이너 몽골리아 일렉트릭 파워 그룹 컴퍼니 리미티드 이너 몽골리아 일렉트릭 파워 리서치 인스티튜트 브랜치
- Filing Date
- 2026-02-05
- Publication Date
- 2026-06-02
AI Technical Summary
Existing technologies are costly, have poor convergence, and are complex to implement when calculating the shape of power grid transmission lines. They are difficult to effectively handle multi-span continuous transmission lines, and their applicability is limited, especially in complex terrain.
A finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm is adopted. The transmission lines are discretized by finite element method, the total potential energy function is constructed and solved iteratively using the BFGS quasi-Newton algorithm, and the Wolfe condition and Armijo condition are combined to perform line search. The approximate matrix of the inverse Hessian matrix is updated to ensure convergence.
It achieves efficient and stable shape finding under complex terrain and harsh working conditions, avoids the iteration failure problem in traditional methods, and improves computational efficiency and engineering applicability.
Smart Images
Figure CN122133379A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of conductor shape finding technology, and in particular to a finite element shape finding method for multi-span transmission conductors based on the BFGS optimization algorithm. Background Technology
[0002] my country's power grid is massive, and national strategic projects highlight the uneven distribution of energy resources and load centers. The implementation of these strategic projects relies on the support of large-scale, long-distance transmission lines. These lines often need to traverse complex terrain, adapt to multi-span continuous layouts, and require insulator strings to work together to bear the load. Their design rationality and operational safety are directly related to the stability of energy transmission.
[0003] Accurate conductor alignment is a core foundational step in transmission line design and safety analysis. It is not only crucial for determining the conductor's equilibrium state but also a prerequisite for conducting subsequent dynamic response simulations such as wind deflection, icing, and de-icing jumps. Its accuracy and efficiency directly impact the reliability and economy of the entire transmission project. In traditional engineering practice, the catenary theory is widely used, but it struggles to effectively handle multi-span continuous transmission lines with fittings and insulator strings, particularly limiting its applicability in mountainous areas with complex terrain and significant elevation differences. With the development of computer technology, numerical simulation based on the finite element method has gradually become the mainstream approach to solving this problem. In this method, the conductor is discretized into a series of rod elements that only bear tension, and its equilibrium state is determined by solving a large-scale nonlinear equation system.
[0004] However, existing technical solutions, especially those using the classic Newton-Raphson iterative method, still face the following key challenges: a) High computational cost: It is necessary to construct and invert the tangent stiffness matrix (Hessian matrix) of the system. For long-distance lines containing a large number of nodes and elements, the computational and storage burden is significant; b) Poor convergence: The algorithm is sensitive to the initial configuration. As a typical flexible cable structure, the initial stiffness matrix of the transmission line is prone to singularity or ill-conditioned behavior, leading to iteration interruption or divergence; c) Complex implementation: The Newton-Raphson method involves complex matrix differentiation and assembly logic, which is difficult to implement in code and is not conducive to engineering applications and subsequent maintenance.
[0005] With the deepening application of artificial intelligence technology in power grid operation, as well as the construction of high-voltage interactive lines and the external transmission needs of large hydropower stations such as the Yalong River, the shortcomings of existing technologies in terms of computational efficiency, convergence stability and engineering applicability have become increasingly prominent. There is an urgent need for an efficient, stable and easy-to-engineer transmission conductor shape-finding method to meet the engineering design needs under complex working conditions. Summary of the Invention
[0006] In view of the above-mentioned prior art, the present invention provides a finite element form finding method for multi-span transmission conductors based on the BFGS optimization algorithm, which mainly solves the technical problems existing in the background art.
[0007] To achieve the above objectives, the technical solution of this invention is implemented as follows: This invention discloses a finite element form-finding method for multi-span transmission conductors based on the BFGS optimization algorithm. The aforementioned form-finding method includes the following steps: Finite element discretization is performed on the transmission conductor with insulator strings to obtain multiple two-node rod elements; The endpoints of the two-node link element are defined as discrete nodes, and a node set is formed based on multiple discrete nodes; Based on the set of nodes, the total potential energy function of the system is constructed. Using the total potential energy function of the system as the objective function, the problem of finding the shape of the transmission line is transformed into an unconstrained optimization problem. The objective function is iteratively solved to output the set of node coordinates that minimizes the total potential energy. Combining the material and cross-sectional parameters of the transmission line and insulator string, the axial stress of each two-node rod element is calculated. The axial stress and the final set of node coordinates are used as the finite element form-finding result of the transmission line.
[0008] Optionally, the transmission conductor and insulator string are discretized using finite element method to obtain multiple two-node rod elements, specifically including: The geometric parameters, element parameters, material parameters, cross-sectional parameters, boundary constraint parameters, and external conservative force parameters of multiple transmission lines can be read from a text file in a preset format. The read geometric parameters are instantiated, and the instantiation result contains rod element class objects; Based on the tensile characteristics of the flexible cable of the transmission conductor and the rigid axial force transmission characteristics of the insulator string, the rod element object that only transmits axial mechanical response is selected as the discrete basic unit. The continuous transmission conductor along its length and the insulator string along its axis are divided into several segments to form a complete discrete structure covering the entire transmission conductor and insulator string. Each segment corresponds to a rod element object, and the rod element object is regarded as a two-node rod element.
[0009] Optionally, the objective function is solved iteratively using the BFGS quasi-Newton algorithm, specifically including: Initialize the initial coordinate vectors of the discrete nodes, set the initial approximation matrix of the inverse Hessian matrix, and initialize the iteration number parameter; The gradient vector corresponding to the objective function is calculated based on the node coordinates of the current iteration node. The gradient vector is then multiplied by a matrix-vector operation with the approximate matrix of the inverse Hessian matrix to obtain the search direction of the objective function. Perform a Wolfe conditional search along the search direction to determine the optimal step size, and update the node coordinate vector based on the optimal step size and the search direction to obtain the node coordinates of the new iterative node; Based on the new iteration node and the current iteration node, calculate its displacement increment vector and gradient increment vector, and update the approximate matrix of the inverse Hessian matrix based on the displacement increment vector and gradient increment vector. At the same time, determine whether the node coordinates of the new iteration node meet the convergence condition. If not, use the updated approximate matrix of the inverse Hessian matrix to recalculate the node coordinates of the new iteration node. If they meet the condition, output the final node coordinates of the node. Iterate through all nodes and repeat the above steps to output the set of node coordinates that minimizes the total potential energy.
[0010] Optionally, a Wolfe conditional search is performed along the search direction to determine the optimal step size, specifically including: Set the trial step size, and combine the node coordinates of the current iteration node with the search direction to calculate the node coordinates of the trial iteration point, the total potential energy of the trial iteration point, and the gradient vector. The total potential energy of the trial iteration point is checked using the Armijo condition. If the Armijo condition check passes, the curvature condition check is performed. Otherwise, the trial step size is adjusted, and the node coordinates of the trial iteration point, as well as the total potential energy and gradient vector of the trial iteration point, are recalculated. If the curvature condition verification passes, the trial step size is used as the optimal step size for output; otherwise, the trial step size is adjusted, the node coordinates of the trial iteration point, the total potential energy and gradient vector of the trial iteration point are recalculated, and the Armijo condition verification is performed again.
[0011] Optionally, the Armijo condition is:
[0012] In the formula, The coordinates of the current iteration node. Step size, For the search direction, As a preset constant, Let be the transpose of the gradient vector of the current iteration node. Let be the objective function for the total potential energy.
[0013] Optionally, the curvature condition is:
[0014] In the formula, For another preset constant, This is the gradient vector.
[0015] Optionally, an approximate matrix of the inverse Hessian matrix is updated based on the displacement increment vector and the gradient increment vector, and its update formula is as follows:
[0016]
[0017]
[0018] In the formula, This is an approximation of the updated inverse Hessian matrix. It is the identity matrix. It is the displacement increment vector. The gradient increment vector, To explore the coordinates of the iteration point, This is the gradient vector for probing the iteration point.
[0019] Optionally, the constant The value range of this constant is from 1e-4 to 0.1. The value range is from 0.1 to 0.9.
[0020] A second aspect of the present invention discloses an electronic device, including a memory, a processor, and a computing program stored in the memory and executable on the processor, characterized in that the processor, when executing the program, implements the method as described in any of the preceding claims.
[0021] A third aspect of the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method as described in any of the preceding claims.
[0022] The beneficial effects of this invention are as follows: It employs a BFGS quasi-Newton algorithm, exhibiting superlinear convergence characteristics. By introducing a line search strategy that satisfies the Wolfe condition, the positive definiteness of the approximate Hessian matrix during iterative updates is strictly guaranteed, fundamentally avoiding the iteration failure problem caused by matrix singularity or non-positive definiteness in the traditional Newton method. This design enables the algorithm to demonstrate excellent adaptability and stability under harsh conditions such as complex terrain and unfavorable initial configurations, solving the pain points of traditional methods such as sensitivity to initial values and difficulty in convergence. Attached Figure Description
[0023] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only preferred embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0024] Figure 1 This is a flowchart illustrating the finite element form-finding method for multi-segment transmission lines based on the BFGS optimization algorithm in the embodiments of this application. Figure 2 This is a flowchart illustrating the BFGS optimization algorithm in an embodiment of this application; Figure 3 This is a schematic diagram of the Wolfe conditional search process in the embodiments of this application; Figure 4 This is a schematic diagram of the shape finding results of a single-span transmission conductor in an embodiment of this application; Figure 5 This is a schematic diagram of the shape finding results for the three-tower, two-span transmission conductor in an embodiment of this application; Figure 6 This is a schematic diagram of the shape finding results of a three-tower, two-span transmission conductor with a rotation angle β in an embodiment of this application. Detailed Implementation
[0025] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used in this specification is for the purpose of describing particular embodiments only and is not intended to limit the invention. In the following description, the expression "some embodiments" refers to a subset of all possible embodiments; however, it should be understood that "some embodiments" can be the same subset or different subsets of all possible embodiments and can be combined with each other without conflict.
[0026] In the following description, numerous specific details are set forth in order to provide a more thorough understanding of the invention. However, it will be apparent to those skilled in the art that the invention can be practiced without one or more of these details. In other instances, certain technical features well-known in the art have not been described in order to avoid obscuring the invention.
[0027] It should be understood that the present invention can be embodied in various forms and should not be construed as being limited to the embodiments set forth herein. Rather, providing these embodiments will make the disclosure thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Furthermore, the terminology used herein is intended only to describe particular embodiments and is not intended to limit the invention. When used herein, the singular forms “a,” “an,” and “the” are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the terms “compose” and / or “comprising,” when used in this specification, identify the presence of the stated features, integers, steps, operations, elements, and / or components, but do not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups. When used herein, the term “and / or” includes any and all combinations of the associated listed items.
[0028] It should also be noted that when an element is referred to as being "fixed to" another element, it can be directly attached to the other element or there may be an intervening element. When an element is referred to as being "connected to" another element, it can be directly connected to the other element or there may be an intervening element. The terms "vertical," "horizontal," "inner," "outer," "left," "right," and similar expressions used herein are for illustrative purposes only and do not represent the only possible implementation.
[0029] To fully understand this invention, a detailed structure will be presented in the following description to illustrate the technical solution proposed by this invention. Optional embodiments of the invention are described in detail below; however, in addition to these detailed descriptions, the invention may have other embodiments.
[0030] Please refer to the attached document. Figures 1 to 6 The first aspect of this application provides a finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm. The aforementioned method includes the following steps: S1. The transmission line with insulator strings is discretized by finite element method to obtain multiple two-node rod elements; Specifically, data is read from a specified text file (e.g., model.txt). This file uses a keyword-driven format to define the relevant parameters of multiple transmission conductors, including geometric parameters describing node locations, element parameters specifying element connection relationships, material parameters characterizing material mechanical properties (e.g., elastic modulus, density), cross-sectional parameters reflecting cross-sectional geometry (e.g., cross-sectional area), boundary constraint parameters defining constraint types and associated nodes, and external conservative force parameters defining core external forces (mainly parameters related to the self-gravity of the transmission conductors and insulator strings).
[0031] After parsing the above parameters, a global data container called StructData is constructed based on the C++ language. This container contains multiple sub-containers for storing pointers to different types of objects. Then, the parsed geometric parameters are instantiated as Node class objects, and the element parameters, material parameters, and section parameters are associated and instantiated as Element_nonTruss3D rod element class objects. At the same time, the boundary constraint parameters are instantiated as Boundary boundary class objects, and the external conservative force parameters are instantiated as LoadNode load class objects. The pointers of the above types of objects are stored in the corresponding sub-containers of the StructData global data container to achieve structured management of parameters.
[0032] Considering the force characteristics of transmission conductors as flexible cables bearing only axial tension, and the rigid force transmission characteristics of insulator strings primarily transmitting mechanical loads axially, the Element_nonTruss3D rod element class object, which only transmits axial mechanical response and does not consider bending and torsional effects, was selected as the discrete basic unit. Based on the preset mechanical calculation accuracy requirements, and taking into account the span length of the transmission conductor, the complexity of the terrain, and the structural dimensions of the insulator string, the continuous transmission conductor along its length and the insulator string along its axis were reasonably divided into several segments. Each segment corresponds to an Element_nonTruss3D rod element class object. These rod element class objects are connected in series through shared Node class objects at both ends, ultimately forming a complete discrete structure covering the entire transmission conductor and insulator string. At this point, each Element_nonTruss3D rod element class object serves as a discretized two-node rod element. S2. Define the endpoints of the two-node link element as discrete nodes, and form a node set based on multiple discrete nodes; Specifically, the process iterates through all Element_nonTruss3D rod element objects stored in the StructDat global data container, extracts the two endpoints of each rod element object, defines each endpoint as a discrete node, and assigns a unique identifier to each discrete node to avoid node confusion. Simultaneously, based on the geometric parameters read from a pre-formatted text file, each discrete node is assigned corresponding initial 3D coordinates, and a displacement vector storage field is reserved to record the dynamic changes of node coordinates during subsequent iterative solutions. Based on this, the boundary conditions and external conservative forces corresponding to the discretized structure are determined. The boundary conditions include constraint directions (such as x / y / z axis directions) and nodes associated with the constraints (such as the end nodes of the insulator strings connecting the towers). The external conservative forces include the gravity of the conductors and the insulator strings themselves.
[0033] For each discrete node, its corresponding Element_nonTruss3D rod element object, Boundary boundary object, and LoadNode load object information are associated to clarify the correspondence between nodes, elements, constraints, and loads, ensuring that nodes can carry core data such as geometric coordinates, constraint association identifiers, and load distribution records. All defined discrete nodes are instantiated as Node class objects, and pointers to each Node object are sequentially stored in the node sub-container of the StructData global data container. This sub-container enables centralized management and unified access to all discrete nodes, ultimately forming a node set covering the overall discrete structure of the transmission conductor and insulator string, carrying complete core data.
[0034] S3. Construct the total potential energy function of the system based on the set of nodes, and use the total potential energy function of the system as the objective function to transform the shape finding problem of the transmission line into an unconstrained optimization problem; Specifically, the expression for the total potential energy function of the constructed system is:
[0035] In the formula, Let be the total potential energy function of the system. For total strain energy, It is the potential energy of external forces.
[0036] When calculating the total strain energy, since the discrete elements of the transmission conductor and insulator string are two-node rod elements that only transmit axial mechanical response, it is only necessary to calculate and sum the tensile strain energy of each rod element. During the calculation, by traversing all Element_nonTruss3D rod element objects stored in the StructData global data container, based on the elastic modulus, section parameters, material parameters, initial length, and axial deformation in the current iteration state associated with the rod element, the tensile strain energy of a single rod element is calculated using the following formula:
[0037]
[0038] The total strain energy of the system is obtained by summing the tensile strain energies of all the bar elements. .
[0039] The calculation of external potential energy is based on the gravity of the transmission conductor and the insulator string. By traversing all Node objects in the node sub-container of the StructData container, the gravity load allocated to each node is obtained. The gravity load is obtained by the mass distribution of the rod element. Combined with the vertical coordinate of the node in the current three-dimensional coordinate system, i.e. the z-axis coordinate, the total external potential energy is obtained by logically accumulating the gravity potential energy calculation.
[0040] S4. Iteratively solve the objective function and output the set of node coordinates that minimizes the total potential energy. Combine the material parameters and cross-sectional parameters of the transmission line and insulator string to calculate the axial stress of each two-node rod element. Use the axial stress and the final set of node coordinates as the finite element form finding result of the transmission line.
[0041] See Figure 2 Preferably, in this embodiment, the BFGS quasi-Newton algorithm is used to iteratively solve the objective function. The termination condition for the iteration is that the L2 norm of the gradient vector is less than a preset convergence threshold or the maximum number of iterations is reached. Each iteration includes the following core steps: Initialize the initial coordinate vectors of the discrete nodes, set the initial approximation matrix of the inverse Hessian matrix, and initialize the iteration number parameter; The gradient vector corresponding to the objective function is calculated based on the node coordinates of the current iteration node. The gradient vector is then multiplied by a matrix-vector operation with the approximate matrix of the inverse Hessian matrix to obtain the search direction of the objective function. Perform a Wolfe conditional search along the search direction to determine the optimal step size, and update the node coordinate vector based on the optimal step size and the search direction to obtain the node coordinates of the new iterative node; Based on the new iteration node and the current iteration node, calculate its displacement increment vector and gradient increment vector, and update the approximate matrix of the inverse Hessian matrix based on the displacement increment vector and gradient increment vector. At the same time, determine whether the node coordinates of the new iteration node meet the convergence condition. If not, use the updated approximate matrix of the inverse Hessian matrix to recalculate the node coordinates of the new iteration node. If they meet the condition, output the final node coordinates of the node. Iterate through all nodes and repeat the above steps to output the set of node coordinates that minimizes the total potential energy.
[0042] Specifically, when using the BFGS quasi-Newton algorithm to iteratively solve the objective function, the initial coordinate vectors of the discrete nodes are first initialized. The initial coordinate vector is determined based on geometric parameters read from a pre-formatted text file, and then an initial approximation matrix of the inverse Hessian matrix is set. and the initial approximate matrix The Hessian matrix is considered as the identity matrix, and the iteration number parameter k is initialized to 0. It should be noted that the Hessian matrix, in the traditional Newton-Raphson method, is the matrix formed by taking the partial derivatives of the objective function. It is a fundamental application of the traditional Newton-Raphson method. The true inverse Hessian matrix, on the other hand, is the inverse of the matrix formed by taking the partial derivatives of the objective function. It can accurately provide the optimal search direction for the optimization problem. However, the optimization variables (the three-dimensional coordinates of all discrete nodes) of the multi-span transmission line finite element model have extremely high dimensions. Directly calculating the true inverse Hessian matrix requires solving a massive number of second-order partial derivatives and performing large-scale matrix inversion operations, resulting in extremely high computational costs and storage burdens. Therefore, this embodiment adopts an approximation matrix design using the inverse Hessian matrix. It does not require prior knowledge of the true inverse Hessian matrix; it only requires dynamic correction through local feedback generated by iteration. This allows for low-cost approximation of the true inverse Hessian matrix, balancing efficiency and accuracy.
[0043] Furthermore, based on the node coordinates of the current iteration node (At the very beginning) ), calculate the gradient vector corresponding to the objective function , where the gradient vector The formula for calculating is:
[0044] in, For internal force vectors, This is the vector of external forces.
[0045] It should be noted that the external force vector The value of remains constant throughout the BFGS iteration. Its initial calculation relies on external conservative force information read from a pre-formatted text file, including the gravity parameters of the transmission conductor and insulator string, node load distribution rules, and the type and magnitude of external loads. During calculation, the computer program parses the load parameters, node geometric positions, and element mass distribution information from the text file. Based on the finite element load equivalence principle, the distributed gravity of the transmission conductor and insulator string is transformed into concentrated loads at each discrete node. At the same time, other external conservative forces defined in the text file are integrated and assembled into an initial external force vector according to the node number order. In all subsequent BFGS iterations, regardless of how the node coordinates are updated, The initial calculation result is always used and is not dynamically adjusted during the iteration process; it is only used as a fixed reference vector in gradient calculation.
[0046] In addition, internal force vectors This is the core vector reflecting the internal mechanical response of the system. Its value is calculated in real time as the node coordinates are updated during the iteration process, and it is not read from a text file or fixed. In each iteration, the program performs the following calculations based on the node coordinates of the current iteration point: First, it traverses all Element_nonTruss3D rod element objects in the StructData global data container. For each rod element, it calculates the actual length, axial deformation, and axial strain of the element based on the current coordinates of its two end nodes. Then, it calculates the axial stress of the element by combining the material parameters and section parameters associated with the element. Based on the finite element stress, strain, and tangent stiffness matrices built into the finite element program, it calculates the internal force values of each element. Finally, it summarizes the internal forces of all elements in the order of node numbers to form the internal force vector corresponding to the current iteration point. The entire calculation process was completed in real time by the finite element program.
[0047] The gradient vector is essentially the unbalanced force vector of the system. Approximate matrix with the inverse Hessian matrix Perform matrix-vector multiplication to obtain the search direction of the objective function. Its search direction The calculation expression is:
[0048] This direction ensures that the objective function decreases along this direction, along the search direction. Perform a Wolfe conditional search to determine the optimal step size. During the search process, the preset Armijo condition and curvature condition must be met. Through multiple trials and step size adjustments, the optimal step size that simultaneously satisfies both conditions is found. Then, the node coordinate vector is updated according to the following formula to obtain the node coordinates of the new iterative node. .
[0049]
[0050] After obtaining the node coordinates of the new iteration node Subsequently, the convergence condition determination process is carried out simultaneously with the update process of the approximate matrix of the inverse Hessian matrix. During the convergence condition determination process, the coordinates of the new iteration nodes are used as the basis for the determination. Calculate the gradient vector corresponding to the new iteration node. Then calculate the gradient vector. norm Whether it is less than a preset tolerance, wherein the norm of the gradient vector is... , which is the length of the gradient vector, or the square root of the sum of the squares of all elements in the gradient. If it is less than , it means that the unbalanced force of the system is small enough to be negligible, and the new iteration point is... The corresponding total potential energy of the system The minimum value has been reached, and the transmission line is in a state of mechanical equilibrium, so there is no need to continue iterating.
[0051] In the process of updating the approximate matrix of the inverse Hessian matrix, it is based on the coordinates of the new iteration nodes. Coordinates of the current iteration node The displacement increment vector is calculated according to the following formula. :
[0052] Based on the coordinates of the new iteration node Calculate the gradient vector corresponding to the new iteration node. The gradient vector based on the new iteration node gradient vector of the current iteration node The gradient increment vector is calculated according to the following formula. :
[0053] These two vectors are the corrected approximation matrices. The core feedback information contains the "law of the influence of coordinate changes on unbalanced forces," and the true inverse Hessian matrix is essentially the global matrix describing this law. Therefore, correction is achieved through these two vectors. This allows it to gradually approximate the true inverse Hessian matrix. It checks if the inner product of the displacement increment vector and the gradient increment vector is greater than 0. If it is, the BFGS correction formula is used to update the approximate matrix of the inverse Hessian matrix. The update formula is as follows:
[0054] Where I is the identity matrix set in the aforementioned process. If the inner product of the displacement increment vector and the gradient increment vector is less than 0, the approximate matrix of the inverse Hessian matrix is restored to the initial identity matrix I.
[0055] It should be noted that the update result of the approximate matrix of the inverse Hessian matrix does not affect the judgment of the convergence condition. Only when the norm of the gradient vector is greater than the preset tolerance, the updated approximate matrix of the inverse Hessian matrix is used to repeat the above steps to recalculate and obtain the node coordinates of the new iteration node.
[0056] See Figure 3Preferably, in this embodiment, a Wolfe conditional search is performed along the search direction to determine the optimal step size, specifically including: Set an initial trial step size, calculate the search direction using the search direction calculation formula shown above, and then combine the node coordinates of the current iteration node and the calculated search direction to calculate the node coordinates of the trial iteration point, the total potential energy of the trial iteration point, and the gradient vector of the trial iteration point using the formula shown above. The total potential energy at the trial iteration point is checked against the Armijo condition, where the Armijo condition is:
[0057] In the formula, The coordinates of the current iteration node. Step size, For the search direction, As a preset constant, Let be the transpose of the gradient vector of the current iteration node. Let be the objective function for the total potential energy.
[0058] If the Armijo condition check fails, the trial step size is adjusted using an interpolation formula, which is as follows:
[0059] After adjusting the trial step size using the above interpolation formula, the search direction, node coordinates, total potential energy, and gradient vector are recalculated until the Armijo condition verification is passed. If the Armijo condition verification is passed, the curvature condition verification continues, where the curvature condition is:
[0060] In the formula, For another preset constant, This is the gradient vector.
[0061] If the curvature condition check fails, the trial step size is adjusted using the aforementioned interpolation formula. The search direction, node coordinates, total potential energy, and gradient vector are recalculated, and the Armijo condition check is performed again. This iterative process of adjusting the trial step size and repeating the check is continued until the trial step size satisfies both the Armijo and curvature conditions. Finally, the trial step size that satisfies both conditions is taken as the optimal step size for the current iteration. It should be noted that, and This is a preset constant that satisfies 0 < < <1, specifically, the constant The value range of this constant is from 1e-4 to 0.1. The value range is from 0.1 to 0.9.
[0062] After traversing all nodes and iteratively calculating, the set of node coordinates that minimizes the total potential energy is output. The displacement value corresponding to each degree of freedom is extracted from the final displacement vector obtained through iterative solving. Based on the pointers to all Node class objects stored in the StructData global data container, each node object is traversed, and the displacement value of the corresponding degree of freedom is accurately written into the displacement member u of that node object. This achieves the associated storage of displacement data and node objects, ensuring that the displacement information of each node can be directly accessed through its member variables. Pointers to all Element_nonTruss3D rod element class objects are obtained through the StructData container and traversed one by one. Since the node update calculation has been completed in the previous stage, the final coordinates of the nodes at both ends of each rod element have been obtained through the aforementioned calculation process, and the final geometric state of the element (such as actual length and axial deformation) is thus determined. The Element_nonTruss3D class object encapsulates stress calculation logic, and the stored stress value is the final stress calculated based on the final geometric state of the element and the associated material parameters (elastic modulus) and cross-sectional parameters (cross-sectional area). Simultaneously, the axial force result is obtained through the calculation formula: Axial Force = Final Stress of Element × Cross-sectional Area in the Cross-sectional Parameter.
[0063] After obtaining the node coordinates and axial force results, the final 3D coordinates of all Node class objects and the final stress or calculated axial force of all Element_nonTruss3D class objects are read from the StructData container and written to the specified text file for storage according to the formatted structure of "node number-final X coordinate-final Y coordinate-final Z coordinate" and "element number-final stress-axial force".
[0064] A second aspect of the present invention discloses an electronic device, including a memory, a processor, and a computing program stored in the memory and executable on the processor, characterized in that the processor, when executing the program, implements the method as described in any of the preceding claims.
[0065] A third aspect of the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method as described in any of the preceding claims.
[0066] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. The scope of protection of the present invention should be determined by the scope of the claims.
Claims
1. A finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm, characterized in that, The aforementioned shape-finding method includes the following steps: Finite element discretization is performed on the transmission conductor with insulator strings to obtain multiple two-node rod elements; The endpoints of the two-node link element are defined as discrete nodes, and a node set is formed based on multiple discrete nodes; Based on the set of nodes, the total potential energy function of the system is constructed. Using the total potential energy function of the system as the objective function, the problem of finding the shape of the transmission line is transformed into an unconstrained optimization problem. The objective function is iteratively solved to output the set of node coordinates that minimizes the total potential energy. Combining the material and cross-sectional parameters of the transmission line and insulator string, the axial stress of each two-node rod element is calculated. The axial stress and the final set of node coordinates are used as the finite element form-finding result of the transmission line.
2. The finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm according to claim 1, characterized in that, Finite element discretization is performed on the transmission conductor and insulator string to obtain multiple two-node rod elements, specifically including: The geometric parameters, element parameters, material parameters, cross-sectional parameters, boundary constraint parameters, and external conservative force parameters of multiple transmission lines can be read from a text file in a preset format. The read geometric parameters are instantiated, and the instantiation result contains rod element class objects; Based on the tensile characteristics of the flexible cable of the transmission conductor and the rigid axial force transmission characteristics of the insulator string, the rod element object that only transmits axial mechanical response is selected as the discrete basic unit. The continuous transmission conductor along its length and the insulator string along its axis are divided into several segments to form a complete discrete structure covering the entire transmission conductor and insulator string. Each segment corresponds to a rod element object, and the rod element object is regarded as a two-node rod element.
3. The finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm according to claim 2, characterized in that, The objective function is solved iteratively using the BFGS quasi-Newton algorithm, specifically including: Initialize the initial coordinate vectors of the discrete nodes, set the initial approximation matrix of the inverse Hessian matrix, and initialize the iteration number parameter; The gradient vector corresponding to the objective function is calculated based on the node coordinates of the current iteration node. The gradient vector is then multiplied by a matrix-vector operation with the approximate matrix of the inverse Hessian matrix to obtain the search direction of the objective function. Perform a Wolfe conditional search along the search direction to determine the optimal step size, and update the node coordinate vector based on the optimal step size and the search direction to obtain the node coordinates of the new iterative node; Based on the new iteration node and the current iteration node, calculate its displacement increment vector and gradient increment vector, and update the approximate matrix of the inverse Hessian matrix based on the displacement increment vector and gradient increment vector. At the same time, determine whether the node coordinates of the new iteration node meet the convergence condition. If not, use the updated approximate matrix of the inverse Hessian matrix to recalculate the node coordinates of the new iteration node. If they meet the condition, output the final node coordinates of the node. Iterate through all nodes and repeat the above steps to output the set of node coordinates that minimizes the total potential energy.
4. The finite element form-finding method for multi-span transmission lines based on the BFGS optimization algorithm according to claim 3, characterized in that, Perform a Wolfe conditional search along the search direction to determine the optimal step size, specifically including: Set the trial step size, and combine the node coordinates of the current iteration node with the search direction to calculate the node coordinates of the trial iteration point, the total potential energy of the trial iteration point, and the gradient vector. The total potential energy of the trial iteration point is checked using the Armijo condition. If the Armijo condition check passes, the curvature condition check is performed. Otherwise, the trial step size is adjusted, and the node coordinates of the trial iteration point, as well as the total potential energy and gradient vector of the trial iteration point, are recalculated. If the curvature condition verification passes, the trial step size is used as the optimal step size for output; otherwise, the trial step size is adjusted, the node coordinates of the trial iteration point, the total potential energy and gradient vector of the trial iteration point are recalculated, and the Armijo condition verification is performed again.
5. The finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm according to claim 4, characterized in that, The Armijo condition is: In the formula, The coordinates of the current iteration node. Step size, For the search direction, As a preset constant, Let be the transpose of the gradient vector of the current iteration node. Let be the objective function for the total potential energy.
6. The finite element method for finding the shape of multi-span transmission lines based on the BFGS optimization algorithm according to claim 5, characterized in that, The curvature condition is: In the formula, For another preset constant, This is the gradient vector.
7. The finite element form-finding method for multi-span transmission lines based on the BFGS optimization algorithm according to claim 6, characterized in that, The approximate matrix for updating the inverse Hessian matrix based on the displacement increment vector and gradient increment vector is updated using the following formula: In the formula, This is an approximation of the updated inverse Hessian matrix. It is the identity matrix. It is the displacement increment vector. The gradient increment vector, To explore the coordinates of the iteration point, This is the gradient vector for probing the iteration point.
8. The finite element form-finding method for multi-span transmission lines based on the BFGS optimization algorithm according to claim 7, characterized in that, The constant The value range of this constant is from 1e-4 to 0.
1. The value range is from 0.1 to 0.
9.
9. An electronic device, comprising a memory, a processor, and a computing program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the method as described in any one of claims 1 to 8.
10. A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method as described in any one of claims 1 to 8.