Equation simulation processing method, device and equipment in multi-body dynamics system
By analyzing the dependencies of algebraic equations in multibody dynamics systems, reducing algebraic variables, and reorganizing the model, the problem of high computational complexity in multibody dynamics system simulation is solved, and simulation efficiency is significantly improved.
Patent Information
- Application Number
- CN202511698648.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-19
- Publication Date
- 2026-02-17
AI Technical Summary
In the DAE solution of multibody dynamics systems, the simultaneous processing of algebraic equations and differential equations leads to high computational complexity, low simulation efficiency, and an inability to effectively reduce the system solution scale and computational overhead.
By analyzing the dependencies in algebraic equations, algebraic variable reduction is performed, a directed dependency graph is constructed, and strongly connected component analysis is conducted to identify reducible nodes, which are then replaced with function expressions. The mathematical model is then reorganized to reduce computational complexity.
It significantly reduces the computational complexity and solution scale of multibody dynamics system simulations, improves simulation efficiency, and ensures the accuracy and efficiency of simulation results.
Smart Images
Figure CN121542541A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of multibody dynamics system simulation, and in particular to a method, apparatus and equipment for simulating equations in multibody dynamics systems. Background Technology
[0002] Multibody dynamics system simulation technology is widely used in key fields such as vehicle engineering, robotic systems, aerospace vehicle design, and industrial machinery. Its core is to simulate the dynamic behavior of a system by solving a system of differential-algebraic equations (DAEs), providing support for design optimization, performance verification, and fault prediction. The mathematical models of such systems typically contain a large number of second-order differential equations, first-order differential equations, and algebraic equations. The number of algebraic variables is enormous and deeply coupled with the differential equations, resulting in a massive solution scale and a dramatic increase in computational overhead for the entire DAE system.
[0003] In related technologies, DAE (Direct Algorithm) solutions for multibody dynamics systems typically employ direct solution methods, which involve simultaneously solving algebraic and differential equations. However, this method incorporates a large number of algebraic variables that can be directly solved computationally into the iterative solution process, causing the system solution scale to remain within the dimension of all variables, resulting in high computational complexity and low simulation efficiency. Summary of the Invention
[0004] This application provides a method, apparatus, and device for simulating equations in a multibody dynamics system, which reduces the computational complexity of differential-algebraic equation systems and improves simulation efficiency.
[0005] In a first aspect, this application provides a method for simulating equations in a multibody dynamics system, the method comprising:
[0006] Obtain the mathematical model of the multibody dynamics system to be simulated;
[0007] At least one algebraic equation in the mathematical model is reduced by algebraic variable reduction based on the dependencies between algebraic variables in the at least one algebraic equation, to obtain at least one target algebraic equation after reduction.
[0008] Based on the at least one target algebraic equation, the mathematical model is reorganized to obtain the reorganized mathematical model;
[0009] The multibody dynamics system is simulated and calculated based on the recombined mathematical model.
[0010] In one possible implementation, the step of reducing at least one algebraic equation in the mathematical model by algebraic variable reduction based on the dependencies between algebraic variables in the at least one algebraic equation to obtain at least one reduced target algebraic equation includes:
[0011] Obtain the at least one algebraic equation from the mathematical model;
[0012] Based on the at least one algebraic equation, a directed dependency graph of the algebraic equations in a multibody dynamics system is constructed. The directed dependency graph includes algebraic variable nodes and directed edges representing the dependencies between different algebraic variables.
[0013] Based on the directed dependency graph and preset rules, strong connected component analysis is performed to identify at least one reducible node. The preset rules include the dependency characteristics of the reducible node and / or the dependency characteristics of the non-reducible node.
[0014] The at least one reducible variable in the at least one algebraic equation is reduced to obtain the at least one target algebraic equation; the reducible variable is the algebraic variable corresponding to the reducible node.
[0015] In one possible implementation, the reduction process on at least one reducible variable in the at least one algebraic equation to obtain the at least one target algebraic equation includes:
[0016] Determine the function expression corresponding to the at least one reducible variable;
[0017] In the at least one algebraic equation, the at least one reducible variable is replaced with the corresponding function expression to obtain the at least one target algebraic equation.
[0018] In one possible implementation, the preset rules include:
[0019] If the node is the source node, then the node is determined to be a scalable node;
[0020] If a node is only depended on by other nodes but does not depend on other nodes to form a loop, then the node is determined to be a scalable node.
[0021] If the algebraic variable corresponding to a node depends on itself, then the node is determined to be an unreducible node.
[0022] If a strongly connected component has multiple nodes in its dependency relationship, and these multiple nodes form an algebraic ring, and one of these multiple nodes is a non-reducible node, then the other nodes among these multiple nodes are determined to be reducible nodes.
[0023] If a strongly connected component has multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and it is not determined whether any of these nodes are reducible, then one node is selected from these nodes and identified as a non-reducible node, while the other nodes are identified as reducible nodes.
[0024] In one possible implementation, selecting one node from the plurality of nodes as a non-reducible node includes:
[0025] The node with the fewest dependencies among the multiple nodes is identified as the non-reducible node;
[0026] or,
[0027] The first node among the plurality of nodes is determined to be a non-reducible node.
[0028] In one possible implementation, the step of reorganizing the mathematical model based on the at least one target algebraic equation to obtain a reorganized mathematical model includes:
[0029] If the mathematical model includes the reducible variable in other equations besides algebraic equations, then in the other equations, the reducible variable is replaced with the corresponding functional expression to obtain the reduced other equations. The other equations include at least one of second-order differential equations, first-order differential equations, and constraint equations.
[0030] The at least one target algebraic equation and the other equations after reduction are reorganized to obtain the reorganized mathematical model.
[0031] In one possible implementation, constructing a directed dependency graph of the algebraic equations in the multibody dynamics system based on the at least one algebraic equation includes:
[0032] Obtain all variables in the at least one algebraic equation to obtain multiple algebraic variables;
[0033] For each of the multiple algebraic variables, a corresponding node is established;
[0034] Based on the dependencies between algebraic variables in at least one algebraic equation, directed edges are connected between the nodes corresponding to the algebraic variables to obtain the directed dependency graph.
[0035] Secondly, this application provides an equation simulation processing device for a multibody dynamics system, the device comprising:
[0036] The acquisition module is used to acquire the mathematical model of the multibody dynamics system to be simulated.
[0037] The processing module is used to perform algebraic variable reduction processing on at least one algebraic equation in the mathematical model according to the dependency relationship between the algebraic variables in the at least one algebraic equation, so as to obtain at least one target algebraic equation after reduction.
[0038] The recombination module is used to recombine the mathematical model based on the at least one target algebraic equation to obtain a recombined mathematical model.
[0039] The simulation module is used to perform simulation calculations on the multibody dynamics system based on the recombined mathematical model.
[0040] Thirdly, this application provides an electronic device, including: a memory and a processor;
[0041] The memory stores computer-executed instructions;
[0042] The processor executes computer execution instructions stored in the memory, causing the processor to perform the first aspect and / or various possible implementations of the first aspect as described above.
[0043] Fourthly, this application provides a computer-readable storage medium storing computer-executable instructions, which, when executed by a processor, are used to implement the first aspect and / or various possible embodiments of the first aspect.
[0044] Fifthly, this application provides a computer program product, including a computer program that, when executed by a processor, implements the first aspect and / or various possible implementations of the first aspect.
[0045] This application provides a method, apparatus, and device for simulating equations in a multibody dynamics system. The method includes acquiring a mathematical model of the multibody dynamics system to be simulated, reducing at least one algebraic equation in the mathematical model by analyzing the dependencies between algebraic variables to obtain at least one reduced target algebraic equation, and then reorganizing the original mathematical model around this target algebraic equation to obtain a reorganized mathematical model. The simulation of the multibody dynamics system is then performed based on this reorganized mathematical model. In this process, by uncovering the dependencies between algebraic variables and reducing them to obtain the target algebraic equation, the system's solution scale and computational complexity can be significantly reduced, greatly improving the simulation efficiency of the multibody dynamics system. Attached Figure Description
[0046] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application.
[0047] Figure 1 A schematic diagram illustrating the application scenarios provided in the embodiments of this application;
[0048] Figure 2 A flowchart illustrating an embodiment of the method for simulating equations in a multibody dynamics system provided in the application.
[0049] Figure 3 This is a schematic diagram of the structure of the point spring model provided in the embodiments of this application;
[0050] Figure 4 This is a schematic diagram of the structure of a directed dependency graph provided in an embodiment of this application;
[0051] Figure 5 A flowchart illustrating Embodiment 2 of the method for simulating equations in a multibody dynamics system provided in the application;
[0052] Figure 6 A schematic diagram of the structure of the equation simulation processing device in the multibody dynamics system provided in the embodiments of this application;
[0053] Figure 7 This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application.
[0054] The accompanying drawings illustrate specific embodiments of this application, which will be described in more detail below. These drawings and descriptions are not intended to limit the scope of the concept in any way, but rather to illustrate the concept of this application to those skilled in the art through reference to particular embodiments. Detailed Implementation
[0055] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims.
[0056] Multibody dynamics simulation technology has become a core supporting technology in fields such as vehicle engineering, robotic systems, aerospace design, and industrial machinery. By constructing a mathematical model of the system and solving a system of mixed differential-algebraic equations (DAE), it accurately simulates the dynamic response, constraint interaction, and effects of external forces on rigid or flexible bodies, providing quantitative basis for key aspects such as product design optimization, performance verification under extreme conditions, and prediction of potential faults.
[0057] In vehicle handling stability analysis, the model needs to cover the dynamic relationships of components such as the vehicle body, suspension, and tires, and includes a large number of algebraic equations describing geometric constraints. In robot motion control, algebraic equations are used to characterize non-dynamic relationships such as joint constraints of robotic arms and sensor feedback mapping. In such models, the number of algebraic variables can typically reach one-third to one-half of the differential variables, and they are deeply coupled with first- and second-order differential equations, forming a complex coupled system. This directly leads to an exponential increase in the solution scale of the full DAE system and a dramatic increase in computational resource consumption.
[0058] In related technologies, the common approach for solving multibody dynamics systems using direct equation solving (DAE) is the direct solution method, which combines algebraic equations with differential equations. However, this method fails to distinguish the solution properties of algebraic variables, incorporating a large number of algebraic variables that can be directly derived through recursive dependencies between variables into the iterative solution process. This results in a large solution scale, high computational complexity, and low simulation efficiency.
[0059] To address the aforementioned problems, the inventors considered using the dependencies between algebraic variables to selectively reduce the algebraic equations, thereby decreasing the solution scale of the mathematical model for multibody dynamics systems. Based on this, after numerous experiments, the inventors discovered that a mathematical model of the multibody dynamics system to be simulated can be obtained first. Then, at least one algebraic equation can be extracted from this model. Based on the dependencies between algebraic variables in these equations, the algebraic variables are reduced to obtain at least one reduced target algebraic equation. Subsequently, the original mathematical model is reorganized based on this target algebraic equation to obtain a reorganized mathematical model. Finally, the multibody dynamics system is simulated using the reorganized mathematical model. In this process, by accurately reducing the algebraic variables that are not independently solvable, the large amount of redundant computation caused by solving all equations simultaneously can be avoided, significantly reducing the complexity of numerical computation and thus greatly improving simulation efficiency. Based on this, this application proposes a method for processing equations in multibody dynamics systems to improve the simulation efficiency of multibody dynamics systems.
[0060] Figure 1 This is a schematic diagram illustrating an application scenario provided in an embodiment of this application. Please refer to [link / reference]. Figure 1 The mathematical model of a multibody dynamics system includes algebraic equation 1, algebraic equation 2, and other equations. Algebraic equation 1 and algebraic equation 2 include parameter variables f1 and f2. Based on the dependency relationship between parameter variables f1 and f2, parameter variable f2 can be reduced to obtain the reduced target algebraic equation A. The mathematical model is then reorganized based on the target algebraic equation A and other equations to obtain the reorganized mathematical model.
[0061] Optionally, other equations may include at least one of second-order differential equations, first-order differential equations, and constraint equations.
[0062] The technical solution of this application and how it solves the above-mentioned technical problems will be described in detail below with specific embodiments. These specific embodiments can be combined with each other, and the same or similar concepts or processes may not be described again in some embodiments. The embodiments of this application will be described below with reference to the accompanying drawings.
[0063] Figure 2 This is a flowchart illustrating an embodiment of the method for simulating equations in a multibody dynamics system provided in the application. Please refer to [link / reference]. Figure 2 The method may include:
[0064] S201. Obtain the mathematical model of the multibody dynamics system to be simulated.
[0065] The execution entity of this application embodiment can be an electronic device or an equation simulation processing device in a multibody dynamics system disposed in an electronic device. The equation simulation processing device in a multibody dynamics system can be implemented by software or by a combination of software and hardware. The equation simulation processing device in a multibody dynamics system can be a processor in an electronic device. For ease of understanding, the technical solution of this application will be described below using an electronic device as an example.
[0066] In this step, the electronic device can acquire the mathematical model of the multibody dynamics system to be simulated. This mathematical model can be an abstract mathematical expression of the physical properties, laws of motion, and constraints of the multibody system. Its sources may include files exported from multibody dynamics simulation software, analytical models built by developers based on physical laws, or semi-empirical models obtained by fitting experimental data.
[0067] Optionally, the mathematical model is typically a system of mixed differential-algebraic equations (DAEs) consisting of second-order differential equations, first-order differential equations, algebraic equations, and constraint equations. Its expression can be shown in equation (1):
[0068] Formula (1)
[0069] Second-order differential equations are primarily derived from the Newton-Euler equations, describing the translational, rotational, and modal dynamics of rigid and flexible bodies. Here, M represents the mass matrix. The acceleration representing the generalized coordinate q, This represents the transpose of the constrained Jacobian matrix. Represents the Lagrange multipliers. This represents a generalized force vector, with independent variables including the generalized coordinate q and the generalized velocity. Generalized acceleration Algebraic variables First-order differential variable , first derivative And time t.
[0070] The first-order differential equations mainly originate from the state-space equations of the control system, the heat conduction equations, the displacement integral equations driven by acceleration / velocity, and user-defined dynamic processes. Among these, This represents the right-hand side function of a first-order differential equation.
[0071] Algebraic equations can come from a wide range of sources, including output equations of control systems, displacement equations of displacement-driven systems, user-defined algebraic relations, and equations introduced to achieve complex modeling (such as defining auxiliary variables related to external loads and constraint reactions).
[0072] The constraint equations mainly come from the geometric constraint equations of mechanical systems (such as ball joints, column joints, universal joints, etc.), drive constraint equations, transmission coupling equations, gear transmission equations, etc.
[0073] To facilitate understanding, a simplified point mass spring model will be used as an example to illustrate the mathematical model of the multibody dynamics system to be simulated.
[0074] Figure 3 This is a schematic diagram of the structure of a point mass spring model provided in an embodiment of this application. Please refer to [link / reference]. Figure 3 The masses are respectively and Two point masses are undergoing single-degree-of-freedom translational motion in the x-direction. The requirements are... The speed follows a given function sports. and All are external loads, satisfying , The mathematical model of the multibody dynamics system to be simulated can be shown in formula (2):
[0075] Formula (2)
[0076] Formula (2) includes two second-order differential equations, one first-order differential equation (velocity integral equation), one constraint equation, and two algebraic equations. Here, k represents the stiffness coefficient of the spring connecting the two particles.
[0077] S202. Based on the dependencies between algebraic variables in at least one algebraic equation in the mathematical model, perform algebraic variable reduction processing to obtain at least one reduced target algebraic equation.
[0078] In this step, at least one algebraic equation can be obtained from the mathematical model. Based on the dependencies between the algebraic variables in the at least one algebraic equation, the algebraic variables are reduced to obtain at least one target algebraic equation after reduction.
[0079] In one optional implementation, after obtaining at least one algebraic equation from the mathematical model, a directed dependency graph (DG) of the algebraic equations in the multibody dynamics system can be constructed based on the at least one algebraic equation. The directed dependency graph includes algebraic variable nodes and directed edges representing the dependencies between different algebraic variables. Then, based on the directed dependency graph and preset rules, strongly connected component analysis can be performed to determine at least one reducible node. At least one reducible variable in the at least one algebraic equation is then reduced to obtain at least one target algebraic equation. Here, the reducible variable is the algebraic variable corresponding to the reducible node.
[0080] Figure 4 This is a schematic diagram of the structure of a directed dependency graph provided in an embodiment of this application. Please refer to [link / reference]. Figure 4 A directed dependency graph includes nodes with algebraic variables. and and representing algebraic variables and The directed edges P1 and P2 that define the dependency relationship between them.
[0081] In one specific implementation, the preset rules include:
[0082] If a strongly connected component has multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and one of these nodes is a non-reducible node, then the other nodes in the multiple nodes are determined to be reducible nodes.
[0083] For example, a strongly connected component's dependency contains nodes. and nodes And nodes and nodes Forming an algebraic ring, where there exists a node between two nodes. If the node is non-reducible, then the node will be... The node was identified as a scalable node. Among them, the node... A non-reducible node can be a node. The driving force term, which directly participates in the differential equation, is the core input to the system's dynamic response.
[0084] If a strongly connected component has multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and it is not determined whether any of these nodes are reducible, then select one node from the multiple nodes to determine it as a non-reducible node, and determine the other nodes as reducible nodes.
[0085] Optionally, a node can be selected from multiple nodes and designated as a non-reducible node using any of the following methods.
[0086] Method (1): The node with the fewest dependencies among multiple nodes is determined as the non-reducible node.
[0087] Method (2): Determine the first node among multiple nodes as the non-reducible node.
[0088] For example, a strongly connected component's dependency contains nodes. and nodes And nodes and nodes Forming an algebraic ring, and the nodes and nodes If it is not determined whether reduction is possible, then the node can be determined from the two nodes. As non-reducible nodes, and set the nodes The node was identified as a scalable node. Among them, the node... For nodes and nodes The first node in the process.
[0089] For example, after obtaining algebraic equation 1 and algebraic equation 2 from the mathematical model shown in formula (2), a directed dependency graph of the algebraic equations in the multibody dynamics system can be constructed based on algebraic equation 1 and algebraic equation 2 (e.g., ...). Figure 4 As shown in the figure, based on the directed dependency graph and preset rules, strongly connected component analysis can be performed to determine the nodes that can be reduced to... For the reducible variables in algebraic equation 1 After reduction, the target algebraic equation is obtained. .
[0090] S203. Based on at least one objective algebraic equation, reorganize the mathematical model to obtain the reorganized mathematical model.
[0091] In this step, the mathematical model can be reorganized based on at least one target algebraic equation obtained after algebraic variable reduction, resulting in a reorganized mathematical model. Specifically, if other equations in the mathematical model besides the algebraic equations include reducible variables, these reducible variables are replaced with corresponding functional expressions to obtain reduced equations. The at least one target algebraic equation and the reduced equations are then reorganized to obtain the reorganized mathematical model. These other equations include at least one of second-order differential equations, first-order differential equations, and constraint equations.
[0092] For example, the second-order differential equation 2 in formula (1) includes reducible variables. Then, in the second-order differential equation 2, the reducible variables can be... Replace with the corresponding expression The reduced second-order differential equation 2 is obtained. By recombining at least one target algebraic equation and the reduced second-order differential equation 2, a mathematical model as shown in formula (3) can be obtained.
[0093] Formula (3)
[0094] In formula (3), the second-order differential equation 1, the first-order differential equation, and the constraint equation do not include reducible variables. Therefore, the original form can be retained in the recombined mathematical model to ensure that the recombined model can still accurately describe the core dynamic characteristics and constraint relationships of the multibody system, providing a complete and non-redundant equation basis for subsequent simulation calculations.
[0095] S204. Perform simulation calculations on the multibody dynamics system based on the recombined mathematical model.
[0096] For example, electronic devices can use numerical calculation methods based on the recombined mathematical model shown in formula (2) to simulate and calculate the point spring of a multibody dynamic system.
[0097] In the embodiments of this application, a mathematical model of the multibody dynamics system to be simulated can be obtained. At least one algebraic equation in the mathematical model is then subjected to algebraic variable reduction processing based on the dependencies between algebraic variables, resulting in at least one reduced target algebraic equation. Subsequently, the original mathematical model is reorganized around the target algebraic equation to obtain a reorganized mathematical model. Finally, the simulation calculation of the multibody dynamics system is completed based on the reorganized mathematical model. By analyzing the dependencies between algebraic variables to achieve precise reduction, retaining only the core target algebraic equation and reorganizing the mathematical model, the number of non-independent algebraic variables that need to be included in the iterative solution process is significantly reduced, which can significantly reduce the system solution scale and computational complexity, thereby greatly improving the simulation efficiency of the multibody dynamics system.
[0098] exist Figure 2 Based on the illustrated embodiment, the following, in conjunction with Figure 5 The simulation method for the equations in the above-mentioned multibody dynamics system will be further explained in detail.
[0099] Figure 5 This is a flowchart illustrating Example 2 of the method for simulating equations in a multibody dynamics system provided in the application. Please refer to [link / reference]. Figure 5 The method may include:
[0100] S501. Obtain the mathematical model of the multibody dynamics system to be simulated.
[0101] Optionally, a mathematical model of the multibody dynamics system to be simulated can be obtained. This mathematical model may include algebraic equations and other equations, which may include at least one of second-order differential equations, first-order differential equations, and constraint equations.
[0102] For example, the mathematical model of the multibody dynamics system to be simulated can be obtained, specifically including second-order differential equation 1, second-order differential equation 2, algebraic equation 1, algebraic equation 2, first-order differential equation and constraint equation as shown in formula (2).
[0103] S502. Obtain at least one algebraic equation from the mathematical model.
[0104] For example, algebraic equation 1 and algebraic equation 2 can be obtained from the mathematical model shown in formula (2).
[0105] S503. Based on at least one algebraic equation, construct a directed dependency graph of the algebraic equations in a multibody dynamics system.
[0106] In this step, a directed dependency graph of the algebraic equations in a multibody dynamics system can be constructed based on at least one algebraic equation and according to all algebraic nodes and their dependencies.
[0107] In one specific implementation, the algebraic equations involved in the multibody dynamics system are rearranged to achieve the form of formula (1). Here, the left side of the equation can be called the left-hand term, i.e., the algebraic variable to be solved, and the right side of the equation can be called the right-hand term. The left-hand term of the algebraic equation can be defined as an algebraic node in the topological graph. Analyzing the algebraic variables contained in the right-hand term of the equation, these algebraic variables constitute the local dependencies of the left-hand term node.
[0108] In one alternative implementation, all variables in at least one algebraic equation can be obtained to obtain multiple algebraic variables. For each of the multiple algebraic variables, a corresponding node is established. Based on the dependency relationship between the algebraic variables in at least one algebraic equation, directed edges are connected between the nodes corresponding to the algebraic variables to obtain a directed dependency graph.
[0109] S504. Based on the directed dependency graph and preset rules, perform strong connected component analysis to determine at least one node that can be reduced.
[0110] In this step, based on preset rules, strongly connected component analysis can be performed on the directed dependency graph to identify at least one reducible node. The preset rules include the dependency characteristics of reducible nodes and / or the dependency characteristics of non-reducible nodes.
[0111] Alternatively, strongly connected component (SCC) algorithms, such as the classic Tarjan algorithm or Kosaraju algorithm, can be used to perform strongly connected component analysis on directed dependency graphs to identify the mutual dependencies between nodes and clusters of algebraic variables forming loops, thereby obtaining strongly connected components.
[0112] Taking the Tajan algorithm as an example, the process of constructing strongly connected components of a directed dependency graph of algebraic variables needs to be completed by combining a single depth-first search (DFS) with a stack structure and timestamp records. The core is to efficiently identify clusters of algebraic variables (strongly connected components) that are mutually dependent and form loops.
[0113] First, at the beginning of the algorithm execution, the basic data structure can be initialized. This includes initializing the global counter index to 0, which is used to allocate a unique timestamp for the first visit of a node; creating the dfn and low arrays and setting them both to -1, which respectively record the node access order and the earliest node timestamp that can be traced back; and initializing an empty stack to temporarily store the nodes of the current DFS path, with a boolean array inStack[] marking whether a node is in the stack to avoid duplicate processing or misjudgment.
[0114] After initialization, the algorithm begins to traverse the nodes in the directed dependency graph. If an unvisited algebraic variable node u is encountered (i.e., dfn[u] = -1), the processing flow for that node is immediately initiated: the value of the global counter index is synchronously assigned to dfn[u] and low[u], and the allocation of the first access timestamp is completed. At this time, low[u] is initialized to its own timestamp, indicating that the earliest node that node u can trace back to is itself; then node u is pushed onto the stack, and inStack[u] is set to true, indicating that it enters the temporary queue of the current DFS path, preparing for subsequent dependency tracing and loop identification.
[0115] After node u completes its initial state record, it needs to further traverse all its dependent neighbor nodes v (i.e., there exists a directed edge from u to v, corresponding to the algebraic variable u depending on v), and process them differently based on the access state and stack state of v: if v has not been visited, v is processed recursively. After the recursion backtracks, low[u] is updated with low[v] to ensure that low[u] can reflect the earliest node that u can reach through v; if v has been visited and is in the stack, it means that the two form a circular dependency, and low[u] needs to be updated with dfn[v] to mark the starting reference of the cycle; if v has been visited but is not in the stack, it means that it has belonged to another SCC and no processing is required.
[0116] After all the neighbors of node u have been processed, the backtracking judgment phase begins: If dfn[u] equals low[u], it means that u is the root node of a certain SCC. At this time, all nodes from u to the top of the stack form a complete strongly connected component, corresponding to a cluster of mutually dependent algebraic variables. These nodes need to be popped in sequence to form an SCC set, and the inStack status flag is updated to indicate that it has been processed; If dfn[u] is not equal to low[u], it means that the SCC to which u belongs needs to be rooted at the earlier node corresponding to low[u]. It is necessary to continue backtracking to the previous node and wait for the root node to meet the conditions for unified extraction.
[0117] Optionally, the preset rules may include:
[0118] 1) If the node is the source node, then the node is determined to be a shrinkable node;
[0119] 2) If a node is only depended on by other nodes but does not depend on other nodes to form a loop, then the node is determined to be a reducible node;
[0120] For example, nodes Only by node (for calculation) If a node is provided that depends on other nodes but does not depend on any other nodes, and does not form a loop with other nodes, then the node is determined. It is a node that can be reduced.
[0121] 3) If the algebraic variable corresponding to a node depends on itself, then the node is determined to be an irreducible node;
[0122] 4) If a strongly connected component contains multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and one of these nodes is a non-reducible node, then the other nodes in the multiple nodes are determined to be reducible nodes.
[0123] 5) If a strongly connected component has multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and it is not determined whether any of the nodes are reducible, then select one node from the multiple nodes to determine it as a non-reducible node, and determine the other nodes as reducible nodes.
[0124] Optionally, a node can be selected from multiple nodes as a non-reducible node based on heuristic rules, such as the node with the fewest dependencies or the first node in the sequence. For example, the node with the fewest dependencies among multiple nodes can be determined as a non-reducible node; or, the first node among multiple nodes can be determined as a non-reducible node.
[0125] For example, strongly connected component analysis can be performed based on directed dependency graphs and preset rules to determine... It is a node that can be reduced.
[0126] S505. Reduce at least one reducible variable in at least one algebraic equation to obtain at least one target algebraic equation.
[0127] In this step, for the reducible variables contained in at least one algebraic equation, a reduction process can be performed using substitution: the reducible variables are replaced by their dependent variables, eliminating the reducible variables from the equation, resulting in at least one target algebraic equation. Here, the reducible variables are the algebraic variables corresponding to the reducible nodes.
[0128] In one alternative implementation, a functional expression corresponding to at least one reducible variable can be determined, and in at least one algebraic equation, at least one reducible variable is replaced with the corresponding functional expression to obtain at least one target algebraic equation.
[0129] For example, in determining Once a node is identified as scalable, further reduction nodes can be determined. The corresponding function expression is Furthermore, the algebraic equation 1 of formula (2) can be reduced to a smaller number of variables. Reduce the equation to the target algebraic equation. At the same time, algebraic equation 2 in formula (2) can be deleted, that is, the number of nodes can be reduced. The corresponding function expression.
[0130] S506. If the equations in the mathematical model other than the algebraic equations include reducible variables, then in the other equations, replace the reducible variables with the corresponding function expressions to obtain the other equations after reduction.
[0131] In this step, you can query other equations in the mathematical model besides algebraic equations (such as second-order differential equations, first-order differential equations, geometric constraint equations, etc.). If these include reducible variables, you can substitute the function expressions corresponding to the reducible variables into the other equations to eliminate the reducible variables, ensuring that the other equations only retain non-reducible variables, and obtain the other equations after reduction.
[0132] For example, the second-order differential equation 2 in formula (2) includes reducible variables. Then the function expression corresponding to the reducible variable can be... Substituting into the second-order differential equation 2, we obtain the reduced second-order differential equation 2: .
[0133] S507. Reorganize at least one objective algebraic equation and the other equations after reduction to obtain the reorganized mathematical model.
[0134] For example, the target algebraic equation, the second-order differential equation 1, the first-order differential equation, the constraint equation, and the reduced second-order differential equation 2 in formula (2) can be reorganized to obtain the reorganized mathematical model, the expression of which can be shown in formula (3).
[0135] S508. Perform simulation calculations on the multibody dynamics system based on the recombined mathematical model.
[0136] In this step, an appropriate numerical solution method (such as Newton's iteration method commonly used in algebraic equations) can be used to solve the recombined mathematical model together. The dynamic change law of the core variables of the system can be obtained through calculation, and then the key simulation indicators such as the motion characteristics and force state of the multibody dynamics system can be analyzed.
[0137] For example, by performing simulation calculations on the mathematical model shown in formula (3), the particle can be output. , Animated motion trajectories are used to evaluate the dynamic characteristics of a point spring.
[0138] In the embodiments of this application, a mathematical model of a multibody dynamics system can be obtained, algebraic equations can be extracted from it, and a directed dependency graph can be constructed. Based on the directed dependency graph and preset rules, strongly connected component analysis is performed to determine at least one reducible node. The reducible variables (i.e., the algebraic variables corresponding to the reducible nodes) in at least one algebraic equation are reduced to obtain at least one target algebraic equation. If other equations in the mathematical model besides algebraic equations (such as differential equations, constraint equations, etc.) include reducible variables, they are replaced with corresponding functional expressions to obtain other equations after reduction. The target algebraic equation and the other equations after reduction are recombined to obtain a simplified mathematical model that retains the core characteristics. Finally, the multibody dynamics system is simulated based on the recombined mathematical model. In the above process, by deeply mining the dependencies of algebraic variables and applying the strongly connected component algorithm, the reducible non-independent algebraic variables (which can be obtained by directly calculating their dependencies) can be accurately identified. This can significantly reduce the number of such variables included in the iterative solution process, thereby significantly reducing the solution scale and computational complexity of the multibody dynamics system. While ensuring the accuracy of the simulation results, it effectively improves the efficiency of simulation calculation, providing technical support for the efficient analysis and optimization of complex multibody systems.
[0139] Figure 6 This is a schematic diagram of the equation simulation processing device in a multibody dynamics system provided in an embodiment of this application. Please refer to... Figure 6 The equation simulation processing device 10 for multibody dynamics systems includes:
[0140] Module 11 is used to acquire the mathematical model of the multibody dynamics system to be simulated.
[0141] Processing module 12 is used to perform algebraic variable reduction processing on at least one algebraic equation in the mathematical model according to the dependency relationship between the algebraic variables in the at least one algebraic equation, so as to obtain at least one target algebraic equation after reduction.
[0142] Reorganization module 13 is used to reorganize the mathematical model based on at least one target algebraic equation to obtain the reorganized mathematical model;
[0143] Simulation module 14 is used to perform simulation calculations on the multibody dynamics system based on the recombined mathematical model.
[0144] The equation simulation processing device for multibody dynamics systems provided in this application embodiment can execute the technical solution shown in the above method embodiment. Its implementation principle and beneficial effects are similar, and will not be repeated here.
[0145] In one possible implementation, the processing module 12 is specifically used for:
[0146] Obtain at least one algebraic equation from the mathematical model;
[0147] Based on at least one algebraic equation, a directed dependency graph of the algebraic equations in a multibody dynamics system is constructed. The directed dependency graph includes algebraic variable nodes and directed edges representing the dependencies between different algebraic variables.
[0148] Strongly connected component analysis is performed based on directed dependency graphs and preset rules to identify at least one reducible node. The preset rules include the dependency characteristics of reducible nodes and / or the dependency characteristics of non-reducible nodes.
[0149] Reduce at least one reducible variable in at least one algebraic equation to obtain at least one target algebraic equation; the reducible variable is the algebraic variable corresponding to the reducible node.
[0150] In one possible implementation, the processing module 12 is specifically used for:
[0151] Determine the function expression corresponding to at least one reducible variable;
[0152] In at least one algebraic equation, at least one reducible variable is replaced with the corresponding functional expression to obtain at least one target algebraic equation.
[0153] In one possible implementation, the preset rules include:
[0154] If the node is the source node, then the node is determined to be a shrinkable node;
[0155] If a node is only depended on by other nodes but does not depend on other nodes to form a loop, then the node is determined to be a reducible node.
[0156] If the algebraic variable corresponding to a node depends on itself, then the node is determined to be an irreducible node.
[0157] If a strongly connected component has multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and one of these nodes is a non-reducible node, then the other nodes in the multiple nodes are determined to be reducible nodes.
[0158] If a strongly connected component has multiple nodes in its dependency relationship, and these nodes form an algebraic ring, and it is not determined whether any of these nodes are reducible, then select one node from the multiple nodes to determine it as a non-reducible node, and determine the other nodes as reducible nodes.
[0159] In one possible implementation, the preset rules include:
[0160] The node with the fewest dependencies among multiple nodes is identified as the non-reducible node;
[0161] or,
[0162] The first node among multiple nodes is determined as the non-reducible node.
[0163] In one possible implementation, the reassembly module 13 is specifically used for:
[0164] If the mathematical model contains reduceable variables in equations other than algebraic equations, then in the other equations, the reduceable variables are replaced with the corresponding functional expressions to obtain the reduced other equations. The other equations include at least one of second-order differential equations, first-order differential equations, and constraint equations.
[0165] Reorganize at least one objective algebraic equation and the other equations after reduction to obtain a reorganized mathematical model.
[0166] In one possible implementation, the processing module 12 is specifically used for:
[0167] Obtain all variables in at least one algebraic equation to obtain multiple algebraic variables;
[0168] For each algebraic variable among multiple algebraic variables, a corresponding node is created;
[0169] Based on the dependencies between algebraic variables in at least one algebraic equation, directed edges are connected between the nodes corresponding to the algebraic variables to obtain a directed dependency graph.
[0170] The equation simulation processing device for multibody dynamics systems provided in this application embodiment can execute the technical solution shown in the above method embodiment. Its implementation principle and beneficial effects are similar, and will not be repeated here.
[0171] Figure 7 This is a schematic diagram of the structure of an electronic device provided in an embodiment of this application. Please refer to... Figure 7 The electronic device 7 may include at least one processor 21 and a memory 22. Optionally, the electronic device 20 may also include a communication component 23. The processor 21, the memory 22, and the communication component 23 are connected via a bus 24.
[0172] In the specific implementation process, at least one processor 21 executes computer execution instructions stored in memory 22, causing at least one processor 21 to perform the above-described method.
[0173] The specific implementation process of processor 21 can be found in the above method embodiments, and its implementation principle and technical effect are similar. It will not be repeated here.
[0174] In one optional implementation, the electronic device provided in this application embodiment can run software for executing equation processing methods in multibody dynamics systems. In specific execution, this software can sequentially complete the acquisition of the mathematical model of the multibody dynamics system, the extraction of algebraic equations and the construction of a directed dependency graph, the identification of reducible nodes based on strongly connected component algorithms and preset rules, the substitution and replacement of reducible variables, and the recombination of equations. Finally, based on the recombined mathematical model, the simulation calculation of the multibody dynamics system is performed, thereby significantly reducing the solution scale and computational complexity, and achieving efficient and accurate analysis of complex multibody dynamics systems.
[0175] In the above embodiments, it should be understood that the processor can be a Central Processing Unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), etc. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the method disclosed in this invention can be directly implemented by a hardware processor, or implemented by a combination of hardware and software modules within the processor.
[0176] The memory may include random access memory (RAM) and may also include non-volatile memory (NVM), such as at least one disk storage device.
[0177] The bus can be an Industry Standard Architecture (ISA) bus, a Peripheral Component Interconnect (PCI) bus, or an Extended Industry Standard Architecture (EISA) bus, etc. Buses can be categorized as address buses, data buses, control buses, etc. For ease of illustration, the buses shown in the accompanying drawings are not limited to a single bus or a single type of bus.
[0178] This application also provides a computer program product, including a computer program that, when executed by a processor, implements the above-described method.
[0179] This application also provides a computer-readable storage medium storing computer-executable instructions, which, when executed by a processor, implement the above-described method.
[0180] The aforementioned readable storage medium can be implemented by any type of volatile or non-volatile storage device or a combination thereof, such as static random access memory (SRAM), electrically erasable programmable read-only memory (EEPROM), erasable programmable read-only memory (EPROM), programmable read-only memory (PROM), read-only memory (ROM), magnetic storage, flash memory, magnetic disk, or optical disk. The readable storage medium can be any available medium accessible to a general-purpose or special-purpose computer.
[0181] An exemplary readable storage medium is coupled to a processor, enabling the processor to read information from and write information to the readable storage medium. Of course, the readable storage medium can also be a component of the processor. The processor and the readable storage medium can reside in an Application Specific Integrated Circuit (ASIC). Alternatively, the processor and the readable storage medium can exist as discrete components in the device.
[0182] The division of units is merely a logical functional division; in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be indirect coupling or communication connection through some interfaces, devices, or units, and may be electrical, mechanical, or other forms.
[0183] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.
[0184] In addition, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit.
[0185] If a function is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.
[0186] Those skilled in the art will understand that all or part of the steps of the above-described method embodiments can be implemented by hardware related to program instructions. The aforementioned program can be stored in a computer-readable storage medium. When executed, the program performs the steps of the above-described method embodiments; and the aforementioned storage medium includes various media capable of storing program code, such as ROM, RAM, magnetic disks, or optical disks.
[0187] Finally, it should be noted that other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This invention is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein, and is not limited to the precise structures described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is limited only by the appended claims.
Claims
1. A method of equation simulation processing in a multi-body dynamics system, characterized by, The method comprises: acquiring a mathematical model of a multibody dynamics system to be simulated; performing algebraic variable reduction processing on at least one algebraic equation in the mathematical model according to the dependency relationship between algebraic variables in the at least one algebraic equation to obtain at least one target algebraic equation after reduction; reorganizing the mathematical model based on the at least one target algebraic equation to obtain a reorganized mathematical model; performing simulation calculation on the multibody dynamics system according to the reorganized mathematical model.
2. The method of claim 1, wherein, The method comprises: acquiring the at least one algebraic equation from the mathematical model; constructing a directed dependency graph of algebraic equations in the multibody dynamics system based on the at least one algebraic equation, wherein the directed dependency graph comprises algebraic variable nodes and directed edges representing the dependency relationship between different algebraic variables; performing strong connected component analysis based on the directed dependency graph and a preset rule to determine at least one reducible node, wherein the preset rule comprises dependency characteristics of a reducible node and / or dependency characteristics of an irreducible node; performing reduction processing on at least one reducible variable in the at least one algebraic equation to obtain the at least one target algebraic equation, wherein the reducible variable is an algebraic variable corresponding to the reducible node.
3. The method of claim 2, wherein, The method comprises: determining a function expression corresponding to the at least one reducible variable; replacing the at least one reducible variable with the corresponding function expression in the at least one algebraic equation to obtain the at least one target algebraic equation.
4. The method of claim 2, wherein, The preset rule comprises: if a node is a source node, the node is determined to be a reducible node; if a node is only dependent on other nodes but does not form a loop with other nodes, the node is determined to be a reducible node; if an algebraic variable corresponding to a node is dependent on itself, the node is determined to be an irreducible node; if a plurality of nodes form an algebraic loop in the dependency relationship of a strong connected component, and one of the plurality of nodes is an irreducible node, the other nodes in the plurality of nodes are determined to be reducible nodes; if a plurality of nodes form an algebraic loop in the dependency relationship of a strong connected component, and none of the plurality of nodes is determined to be reducible or irreducible, one of the plurality of nodes is selected to be an irreducible node, and the other nodes are determined to be reducible nodes.
5. The method of claim 4, wherein, The method comprises: selecting a node with the least dependency from the plurality of nodes to be an irreducible node; or selecting a first node from the plurality of nodes to be an irreducible node. The method comprises:
6. The method according to any one of claims 2 to 5, characterized in that, If the reducible variable is included in other equations in the mathematical model except the algebraic equations, the reducible variable is replaced by a corresponding function expression in the other equations to obtain condensed other equations, the other equations including at least one of a second-order differential equation, a first-order differential equation, and a constraint equation; The at least one target algebraic equation and the condensed other equations are reorganized to obtain the reorganized mathematical model.
7. The method according to any one of claims 2 to 5, characterized in that, The directed dependency graph of the algebraic equations in the multi-body dynamics system is constructed based on the at least one algebraic equation, including: All variables in the at least one algebraic equation are obtained to obtain a plurality of algebraic variables; A corresponding node is established for each algebraic variable in the plurality of algebraic variables; According to the dependency relationship between the algebraic variables in the at least one algebraic equation, the nodes corresponding to the algebraic variables are connected by directed edges to obtain the directed dependency graph.
8. An equation simulation processing device in a multi-body dynamics system, characterized by comprising: The device includes: An acquisition module is configured to acquire a mathematical model of a multi-body dynamics system to be simulated; A processing module is configured to perform algebraic variable reduction processing on at least one algebraic equation in the mathematical model according to a dependency relationship between algebraic variables in the at least one algebraic equation to obtain at least one target algebraic equation after reduction; A reorganization module is configured to reorganize the mathematical model based on the at least one target algebraic equation to obtain a reorganized mathematical model; A simulation module is configured to perform simulation calculation on the multi-body dynamics system according to the reorganized mathematical model.
9. An electronic device, comprising: It includes: A memory and a processor; The memory stores computer execution instructions; The processor executes the computer execution instructions stored in the memory, so that the processor executes the method of any one of claims 1-7.
10. A computer-readable storage medium, characterized in that, The computer readable storage medium stores computer execution instructions, and the computer execution instructions are executed by the processor to implement the method of any one of claims 1-7.
11. A computer program product, characterised in that, It includes a computer program, which is executed by the processor to implement the method of any one of claims 1-7.