Welding finite element rapid stiffness matrix assembly solving method based on temperature-stress

Abstract

The invention provides a temperature-stress-based welding finite element rapid stiffness matrix assembly solving method, and relates to the technical field of welding process finite element simulation, and the method optimizes a stiffness matrix updating strategy through double variables of temperature and stress based on local heat input characteristics of a welding process. In the elastic area, the calculation strategy is flexibly adjusted according to the temperature change amplitude, and in the plastic area, iterative correction is carried out aiming at the nonlinear behavior of the material. By intelligently identifying the key area and avoiding unnecessary global updating, the algorithm greatly reduces the calculation amount and significantly improves the simulation speed. Besides, in combination with a sequential solution method of thermal-mechanical coupling, the algorithm can provide rapid and accurate stress solution for a complex welding process on the premise of ensuring precision, and is particularly suitable for efficient simulation analysis of large weldments. The problem that in the welding process of large weldments, the rigidity matrix assembling efficiency is low is solved.

Description

Technical Field

[0001] This invention relates to the field of finite element simulation technology for welding processes, specifically to a rapid stiffness matrix assembly and solution method for welding finite elements based on temperature-stress. Background Technology

[0002] In modern engineering and manufacturing, welding technology is widely used in the manufacturing of various large structural components. The welding process involves complex physical phenomena, including localized heat input, rapid changes in the temperature field, and the resulting thermal stress and deformation. Accurate simulation and prediction of the thermo-mechanical coupling effects during welding are of great significance for optimizing welding processes, improving product quality, and reducing production costs.

[0003] Traditional welding simulation methods primarily rely on finite element analysis (FEA), using numerical calculations to simulate temperature distribution and stress-strain states during the welding process. However, the complexity of the welding process, especially the nonlinear behavior of localized high-temperature regions, poses significant challenges to the efficiency and accuracy of finite element analysis. On one hand, the high concentration of heat during welding leads to a rapid temperature increase in the weld and its surrounding area, creating a significant temperature gradient that affects the mechanical properties and microstructure of the material. On the other hand, there is a complex interaction between temperature changes and the stress field; the generation of thermal stress and the accumulation of residual stress have a crucial impact on the integrity and performance of the structure.

[0004] In finite element analysis, the assembly and solution of the stiffness matrix is ​​a core step in the computational process, and its efficiency directly affects the feasibility and practicality of the entire simulation. For large welded structures, traditional stiffness matrix assembly methods have significant limitations. For example, updating the stiffness matrix typically requires handling the mapping relationship between the local and global degrees of freedom of each element, which leads to a sharp increase in computational load in large-scale nonlinear problems. Furthermore, existing methods often require frequent updates to the stiffness matrix when dealing with high-temperature regions and plastic deformation during welding, further reducing computational efficiency.

[0005] In recent years, although researchers have proposed some optimization strategies, such as parallel computing, adaptive mesh generation, and sparse matrix techniques, these methods have improved computational efficiency to some extent. However, they still fall short of meeting practical engineering needs when dealing with the complex welding processes of large weldments. In particular, existing methods fail to fully utilize the local characteristics of the welding process in the high-temperature and plastic deformation regions, leading to wasted computational resources and efficiency bottlenecks.

[0006] Therefore, developing a finite element analysis method capable of efficiently handling temperature and stress changes during welding is of significant practical importance for improving the efficiency and accuracy of welding simulation. This method should fully utilize the local characteristics of the welding process, optimize the stiffness matrix update strategy, reduce unnecessary calculations, and maintain sufficient accuracy to meet the needs of large-scale welded structure analysis.

[0007] In view of the above, this application is hereby submitted. Summary of the Invention

[0008] This invention provides a rapid stiffness matrix assembly solution method for welding based on temperature-stress finite element method, which can at least partially improve the above-mentioned problems.

[0009] To achieve the above objectives, the present invention adopts the following technical solution: A rapid stiffness matrix assembly solution method for welding based on temperature-stress finite element method, comprising: The initial temperature and stress distribution are obtained, and matrix assembly preprocessing is performed based on the temperature and stress distribution to obtain the element stiffness matrix, which is then saved. Based on the variational principle and Galerkin method, the heat conduction formula of the transient temperature field and the convective heat transfer formula on the boundary are established and integrated by parts in sequence to obtain the transient finite element of the welding process. Based on the transient finite element method and the temperature field under time series, sequential coupled elastoplastic calculations are performed to obtain the plastic strain increment and elastic stress increment. Based on the difference between the stress yield state at the current moment and the temperature value cached at the previous moment, the element state judgment process is performed on the plastic strain increment and elastic stress increment to determine the element state. The yield state and temperature difference of the element stiffness matrix are judged, the judgment result is generated, and the element stiffness matrix is ​​updated or reused according to the judgment result and the element state.

[0010] In summary, the temperature-stress-based rapid stiffness matrix assembly and solution method for welded finite element analysis focuses on improving the efficiency of stiffness matrix assembly and solution in the simulation of large welded structures. The core of this method lies in addressing the characteristics of localized high temperatures and nonlinear behavior during welding. By dividing the structure into different regions (such as elastic, plastic, high-temperature difference, and low-temperature difference regions), the algorithm can flexibly decide whether to update the stiffness matrix based on the element state and temperature changes. In the elastic region with small temperature changes, the algorithm uses a caching technique to retain the previous stiffness matrix, avoiding redundant calculations; while in the plastic region or with large temperature changes, the stiffness matrix is ​​dynamically updated based on the latest material parameters to adapt to changes in material properties.

[0011] Furthermore, this method incorporates a thermo-mechanical coupled sequential solution approach, iteratively correcting the nonlinear behavior of the plastic region to ensure the accuracy of the calculation results. Compared with traditional finite element simulation methods, this algorithm significantly reduces unnecessary computation, especially when dealing with large and complex welded structures, greatly shortening the computation time while maintaining high simulation accuracy. This method not only provides an efficient analytical tool for welding process optimization but also offers a new solution to the stress problem in hot forming processes, demonstrating broad application prospects and significant engineering implications. Attached Figure Description

[0012] Figure 1 This is a flowchart illustrating the rapid stiffness matrix assembly and solution method for welding based on temperature-stress finite element analysis provided in this embodiment of the invention. Figure 2 This is a schematic diagram of the framework of the rapid stiffness matrix assembly and solution method for welding based on temperature-stress finite element method provided in the embodiment of the present invention; Figure 3 This is a schematic diagram of the sequential coupling solution process for the elastoplastic model provided in the embodiments of the present invention; Figure 4 This is a schematic diagram of the theoretical process of the stiffness matrix update algorithm provided in the embodiment of the present invention. Detailed Implementation

[0013] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0014] refer to Figure 1 , Figure 2 As shown, the first embodiment of the present invention discloses a method for rapid assembly and solution of temperature-stress welding finite element method, which can be executed by a temperature-stress welding finite element rapid stiffness matrix assembly and solution device (hereinafter referred to as the solution device), specifically, by one or more processors within the solution device, to implement the following method: S1: Obtain the temperature and stress distribution under the initial state, and perform matrix assembly preprocessing based on the temperature and stress distribution to obtain the element stiffness matrix, and save it; Specifically, step S1 includes: obtaining the room temperature state and stress distribution under the initial state, and assembling multiple initial element stiffness matrices according to the temperature and stress distribution, wherein the element stiffness matrix is ​​used to describe the deformation characteristics of the material under loading. In this context, the initial room temperature and stress distribution are both zero, and the mathematical expression for the initial element stiffness matrix of the three-dimensional solid is: , The initial element stiffness matrix, The strain-displacement matrix, Here is the elastic stiffness matrix. For volume, For transpose; Based on the initial element stiffness matrix, a temperature effect term is added to obtain the element stiffness matrix, as shown in the formula: , ,in, The element stiffness matrix, , Let represent the thermal effect matrix caused by temperature change, and b be the gradient matrix of the shape function. The coefficient of thermal expansion is... Here is the stiffness matrix of the material. This represents the temperature change value. All element stiffness matrix data are stored in a preset storage space.

[0015] In this embodiment, a comprehensive acquisition and analysis of the temperature and stress distribution in the initial state is performed. This process is the foundation of the entire algorithm, providing initial conditions for subsequent matrix assembly and calculation. Specifically, in the initial state, it is typically assumed that the welded structure is at room temperature and the stress distribution is zero. This is because before the welding process begins, the structure has not been affected by the heat source, so the initial state can be regarded as a stress-free reference state. Based on this assumption, the finite element mesh of the entire welded structure is first divided, and each element is assigned an initial stiffness matrix. This initial stiffness matrix is ​​an important tool for describing the deformation characteristics of a material under loading. It is defined by the constitutive relation of the material and reflects the elastic response of the material under stress. The stiffness matrix is ​​used to describe the deformation characteristics of a material under loading and is usually defined by the constitutive relation of the material. In thermo-mechanical coupling problems, changes in temperature and stress affect the stiffness matrix of the material. Temperature affects the elastic modulus, thermal expansion, and other properties of the material, while stress directly affects the yield criterion and plastic deformation of the material.

[0016] Initially, since both temperature and stress are zero, the calculation of the element stiffness matrix can be simplified to a basic form. This basic stiffness matrix reflects the elastic behavior of the material in the absence of thermal effects and serves as the starting point for subsequent calculations. However, temperature changes during welding significantly affect the mechanical properties of the material, thus requiring the introduction of temperature effects into the initial stiffness matrix. To this end, a temperature effect term is defined to account for the influence of temperature changes on the stiffness matrix; that is, a temperature effect term is added to the original stiffness matrix. Adding this temperature effect term to the initial stiffness matrix yields an element stiffness matrix that considers the temperature effect. This improved stiffness matrix not only considers the elastic properties of the material but also incorporates the influence of temperature changes on material properties, enabling subsequent finite element analysis to more accurately reflect the thermo-mechanical coupling effects during welding.

[0017] After calculating the initial element stiffness matrix, the stiffness matrix data of all elements is stored in a pre-defined storage space. This step is crucial because it provides the foundational data for subsequent iterative calculations. By storing this data in a cache, the algorithm can quickly access this information during subsequent calculations, significantly improving computational efficiency. This data caching strategy not only reduces the need for redundant calculations but also facilitates subsequent dynamic updates, enabling the algorithm to flexibly adjust the stiffness matrix according to changes in temperature and stress during the welding process.

[0018] Please see Figure 3 S2, Based on the variational principle and Galerkin method, the heat conduction formula of the transient temperature field and the convective heat transfer formula on the boundary are established and integrated by parts in sequence to obtain the transient finite element of the welding process. Specifically, step S2 includes: during the welding process, the heat conduction formula for the transient temperature field inside the weldment is: ,in, , , These represent the thermal conductivity of the welding material in the x-axis, y-axis, and z-axis directions, respectively. The specific heat capacity of the material, For the density of the material, It is a heat source acting inside the workpiece; During welding, heat transfer consists of convective heat transfer and radiative heat transfer. The formula for convective heat transfer at the boundary of the weldment is: , , , All are the direction cosines of the boundary outward normal vector. The convective heat transfer coefficient is... Ambient temperature; The formula for radiative heat transfer is: ,in, The Stefan-Boltzmann constant has a value of , The blackness coefficient of the object; Based on the variational principle, the heat conduction formula and the convective heat transfer formula on the boundary of the transient temperature field are established and processed. Their equivalent integral forms are then established, and integration by parts is performed using the Galerkin method. The temperature inside each element is obtained through interpolation functions and element nodal temperatures. The general format of the transient finite element method for the welding process is thus obtained as follows: ,in, For the element heat capacity matrix, For the unit heat conduction matrix, The load vector at the right end of the element. This is a temperature vector.

[0019] Based on the node numbers within the temperature field analysis domain, the nodes and vectors within the temperature field analysis domain are assembled to obtain the global matrix in the entire computational space. The element-node correspondence within the finite element mesh is as follows: , For the heat capacity matrix of all units The sum, For all element heat conduction matrices The sum; It is the vector of all element loads. The sum; In terms of time, the element-node correspondence within the finite element mesh is discretized using a difference method, as shown in the formula: The temperature field distribution at each time step is obtained, where, For time step, The temperature vector at time n+1, Let n be the temperature vector. The vector of all element loads at time n+1 The sum of .

[0020] In this embodiment, the transient temperature field during the welding process is accurately simulated. This step involves processing the heat conduction equation and its boundary conditions, transforming them into a computable form using the finite element method. First, a heat conduction model of the transient temperature field during welding needs to be established. During welding, heat is mainly transferred through three mechanisms: heat conduction, convection, and radiation. The transient temperature field inside the weldment is calculated to satisfy the heat conduction equation, which describes the heat transfer process within the weldment and forms the basis of the temperature field simulation. At the boundaries of the weldment, heat transfer is mainly achieved through convection and radiation. These boundary conditions ensure the integrity and accuracy of heat transfer during the welding process.

[0021] Furthermore, to transform the aforementioned heat conduction equations and boundary conditions into finite element form, variational principles and the Galerkin method are required. Using variational principles, the heat conduction equations and boundary conditions are transformed into equivalent integral forms, and then integrated by parts is performed using the Galerkin method. In this way, the temperature inside each element can be obtained through interpolation functions and element nodal temperatures. Ultimately, a general format for the transient finite element method of the welding process can be obtained.

[0022] By assembling the nodes and vectors within the analysis domain according to their node numbers, a global matrix over the entire computational space can be obtained, further determining the element-node correspondence within the finite element mesh. Next, to solve for the time-dependent temperature field distribution, the above equations need to be discretized over time; by solving these discretized equations, the temperature field distribution at each time step can be obtained.

[0023] S3, based on the transient finite element method and the temperature field under the time series, perform sequentially coupled elastoplastic calculations to obtain the plastic strain increment and elastic stress increment; Specifically, step S3 includes: using the V. Mises yield criterion to determine the stress yield state of the material. Determining the stress yield state of a material, among other things, For equivalent stress, , Yield strength; When the equivalent stress is determined to reach the preset yield condition, the strain increment follows the V. Mises flow law, yielding the relationship between the plastic strain increment and the stress increment. The mathematical expression for the plastic strain increment is: , A positive, undetermined finite quantity. This is along the normal direction of the yield surface F = 0 in stress space; Obtain the elastic stress increment during the welding process Based on the set relations in the element stiffness matrix, the relationship between displacement and strain is obtained. ,in, Here is the stiffness matrix. For the strain increment, This is the temperature-stress matrix. For temperature increment, The displacement-strain matrix, This represents the displacement increment of the element node. , The element stiffness matrix, For unit thermal strain load.

[0024] In this embodiment, sequentially coupled elastoplastic calculations are performed based on transient finite element analysis and the temperature field under a time series to obtain the plastic strain increment and elastic stress increment. This process not only involves complex judgment of material mechanical behavior but also requires accurate calculation of strain and stress increments to reflect the deformation and stress state of the material during welding.

[0025] First, based on the temperature field distribution obtained from transient finite element analysis, and combined with the physical and mechanical properties of the material, sequential coupled elastoplastic calculations are performed. The core of this calculation process is determining the stress-yield state of the material, specifically using the Von Mises yield criterion to assess whether the material has entered the plastic deformation stage. The Von Mises yield criterion is a widely used yield criterion for isotropic materials, effectively determining the yield behavior of materials under multiaxial stress states. Specifically, when the equivalent stress reaches or exceeds the yield strength, the material begins to undergo plastic deformation.

[0026] After determining that the material has entered the plastic stage, the strain increment follows Von Mises' flow law. This law describes the relationship between the plastic strain increment and the stress increment during the plastic deformation stage; this relationship reflects the deformation behavior of the material in the plastic stage and is an important component of elastoplastic calculations. Simultaneously, the elastic stress increment during the welding process also needs to be calculated. Based on the geometric relationships in the element stiffness matrix, the relationship between displacement and strain can be obtained. The temperature-stress matrix reflects the influence of temperature changes on stress. Furthermore, the strain increment can be calculated using the displacement-strain matrix and the element nodal displacement increment. Finally, combining the element stiffness matrix and the element thermal strain load, the element nodal displacement increment is obtained.

[0027] Based on the above calculation formulas, not only can the increment of plastic strain in the material during welding be obtained, but the increment of elastic stress can also be accurately calculated. This sequentially coupled elastoplastic calculation method fully considers the influence of temperature changes on the mechanical properties of the material during welding, as well as the deformation behavior of the material under different stress states. Compared with traditional elastoplastic analysis methods, this method can more accurately reflect the complex mechanical behavior during welding, especially when dealing with plastic deformation in high-temperature regions, significantly improving calculation accuracy. Furthermore, by coupling the calculation sequence of the temperature field and stress field, the complexity and computational cost of simultaneously solving thermo-mechanical coupling problems are avoided. This step-by-step solution strategy not only improves computational efficiency but also makes the entire analysis process more stable and reliable. In practical applications, this method can provide important technical support for the optimization of welding processes, the prediction of residual stress, and the reliability assessment of welded structures.

[0028] Please see Figure 4S4, based on the difference between the stress yield state at the current moment and the temperature value cached at the previous moment, perform element state judgment processing on the plastic strain increment and elastic stress increment to determine the element state. Specifically, step S4 includes: when the stress yields... When the element is in a state of plastic deformation, the stress yields. At that time, the element is in an elastic deformation state; Obtain the preset temperature change threshold ,when When, the unit state is classified as a high temperature difference region, when At that time, the unit state is classified as a low temperature difference region.

[0029] In this embodiment, the key step in determining the element state is based on the difference between the current stress-yield state and the cached temperature value from the previous moment. This allows for the differentiation of different mechanical behaviors and temperature variation ranges of the element during the welding process, thereby achieving efficient utilization of computational resources.

[0030] The stress-yield state of an element is determined using the Von Mises yield criterion. If the equivalent stress reaches or exceeds the material's yield strength, the element is considered to be in a plastic deformation state; conversely, if the equivalent stress is lower than the yield strength, the element is in an elastic deformation state. This method based on stress-yield state can accurately distinguish the mechanical behavior of the element during the welding process, providing a basis for subsequent stiffness matrix updates.

[0031] After determining the mechanical state of the element, it is necessary to further subdivide the element state based on temperature changes. To this end, a preset temperature change threshold is introduced, which is typically set based on the temperature characteristics of the heat-affected zone of the welding material. By comparing the temperature of the element at the current moment with the temperature value cached at the previous moment, the element can be divided into high-temperature difference regions and low-temperature difference regions. Specifically, when the temperature difference is greater than or equal to the threshold, the element is classified as a high-temperature difference region; while when the temperature difference is less than the threshold, the element is classified as a low-temperature difference region.

[0032] This temperature-varying region partitioning method has significant advantages. In high-temperature regions, the mechanical properties and thermal expansion behavior of the material change significantly, thus requiring an update of the stiffness matrix to reflect the material's deformation characteristics at high temperatures. In low-temperature regions, however, the material's properties change relatively little, allowing the use of the previously cached stiffness matrix, thereby avoiding unnecessary calculations and significantly improving computational efficiency.

[0033] By combining element state assessment with temperature changes, the stiffness matrix update strategy can be dynamically adjusted during welding. This approach considers not only the mechanical behavior of materials under different stress states but also the impact of temperature variations on material properties. Compared to traditional stiffness matrix update methods, it flexibly adjusts the calculation strategy based on the actual state of the elements, avoiding the computational burden of global updates while ensuring computational accuracy. When dealing with large welded structures, this optimized update strategy significantly reduces computational load, improves simulation efficiency, and provides a more efficient and accurate analytical tool for welding process optimization and structural performance evaluation.

[0034] S5 determines the yield state and temperature difference of the element stiffness matrix, generates a determination result, and updates or reuses the element stiffness matrix based on the determination result and the element state. Specifically, step S5 includes: when it is determined that the element stiffness matrix has reached the plastic deformation or high temperature difference region, the stiffness matrix is ​​recalculated and saved according to the material parameters corresponding to the element state and temperature; When it is determined that the element stiffness matrix has reached the elastic deformation or low temperature difference region, the most recently saved element stiffness matrix will continue to be used.

[0035] Preferably, when it is determined that the element stiffness matrix has reached the plastic deformation or high temperature difference region, the stiffness matrix is ​​recalculated and saved based on the material parameters corresponding to the element state and temperature. Specifically: According to the formula The constitutive matrices of the stiffness matrices in the yield region and the high temperature difference region are updated, and the new constitutive matrix parameters are used as the basis for the update. and the incremental stiffness matrix update formula The element stiffness matrix is ​​updated, where, Let i be the updated stiffness matrix of the i-th element. For the volume or region of a unit, This is the new constitutive matrix calculated based on the current temperature region conditions. This is the constitutive matrix obtained from the temperature region of the previous stage. A new stiffness matrix is ​​obtained by superimposing an incremental matrix generated by changes in structural parameters onto the original stiffness matrix. ,in, For the updated stiffness matrix, This is the original old stiffness matrix. This is the sum of the increment matrices caused by changes in structural parameters.

[0036] For the stiffness matrix in the plastic stage, a correction algorithm is used to correct it, and an approximate solution is obtained. ,in, For displacement vectors, It is the product of the stiffness matrix and the displacement vector. This is the thermal strain load vector; Based on the Newton-Raphson iterative solution algorithm, assuming the nth approximate solution is... For nonlinear equations that do not satisfy In this case, we perform the (n+1)th iteration to obtain the (n+1)th approximate solution. , , , ,in, This is the correction term for the solution obtained in the nth iteration. This is the tangent matrix in the nth iteration; Solving the nonlinear equation yields the solution equation for the t-th increment step: , , .

[0037] Preferably, the nonlinear equation is solved to obtain the solution equation in the t-th increment step, specifically as follows: Determine the initial linear solution system of equations: , , ,in, This is the stress value obtained after the previous iteration step converges; According to the formula , , Calculate the stress and strain values; Using the radial return stress algorithm, the relationship between the trial stress and the yield function is established, and the expression is as follows: Where G is the shear modulus. Let be the yield function; Based on the plastic strain increment, the plastic strain in each iteration step is obtained: ,in, This represents the equivalent plastic strain increment at the (m+1)th iteration step. This represents the equivalent plastic strain increment at step m. For yield strength, The stress is the equivalent stress, and h is the strengthening coefficient; In the iterative process, each incremental step is divided into two nonlinear problems, which are then solved.

[0038] Solve the nonlinear equation with displacement field as the unknown variable to obtain the displacement field increment in each Newton-Rapson iteration; The displacement field increment is used as a parameter input to solve for the strain increment and stress increment. Then, a nonlinear iteration is performed on the plastic strain increment according to the formula for plastic strain increment. After convergence, the plastic strain increment and stress increment are obtained. The nonlinear equation with displacement field as the unknown variable is solved until numerical convergence.

[0039] In this embodiment, when the element stiffness matrix reaches the plastic deformation stage or is in the high temperature difference region, it is necessary to recalculate the stiffness matrix based on the material parameters corresponding to the current state and temperature of the element. This operation is based on an important understanding: in the plastic deformation and high temperature difference regions, the mechanical properties of the material will change significantly, so the stiffness matrix needs to be updated to accurately reflect these changes. Specifically, firstly, the constitutive matrix of the stiffness matrix in the yield region and the high temperature difference region is updated. The constitutive matrix is ​​the core parameter describing the mechanical behavior of the material, and its update ensures that the stiffness matrix accurately reflects the mechanical properties of the material in the current state. Subsequently, the element stiffness matrix is ​​updated according to the new constitutive matrix parameters and the incremental stiffness matrix update formula. Simply put, a new stiffness matrix is ​​obtained by superimposing the incremental matrix generated by the change in structural parameters on the original stiffness matrix. This update method not only considers the changes in material parameters but also retains the information of the original stiffness matrix, thereby improving computational efficiency while ensuring computational accuracy.

[0040] For the stiffness matrix in the plastic stage, due to the nonlinear characteristics of material plastic deformation, a correction algorithm is needed to obtain a more accurate approximate solution. Through iterative solving, the true solution of the nonlinear equations can be gradually approximated, thus obtaining a more accurate distribution of the displacement, strain, and stress fields. In each increment step, the initial linear solution equations are first determined, with the initial stress values ​​typically taken from the convergence result of the previous iteration step. Subsequently, the stress and strain values ​​are calculated according to the formulas, and a radial return stress algorithm is used to establish the relationship between the actual stress and the trial stress. The core of this algorithm lies in determining whether the material has entered the plastic stage by using trial stress and yield function, and adjusting the stress and strain increments accordingly. This algorithm effectively handles the nonlinear behavior of materials in the plastic stage, ensuring the accuracy of the calculation results.

[0041] In the iterative process, each increment step involves solving the nonlinear problem twice. First, the nonlinear equation with displacement as the unknown variable is solved to obtain the displacement increment in each Newton-Raphson iteration. Then, the displacement increment is used as a parameter input to solve for the strain and stress increments, and a nonlinear iteration is performed for the plastic strain increment according to the formula. Finally, the nonlinear equation with displacement as the unknown variable is solved again until numerical convergence.

[0042] In summary, the temperature-stress-based rapid stiffness matrix assembly and solution method for welding finite element analysis significantly improves simulation efficiency while maintaining computational accuracy by dynamically monitoring temperature and stress changes during the welding process and intelligently updating the element stiffness matrix. It is specifically optimized to address the efficiency and accuracy issues in the simulation of large welded structures.

[0043] In practical applications, this method first performs real-time evaluation of the element state during the welding process. By distinguishing between elastic and plastic deformation regions, as well as high-temperature and low-temperature difference regions, it achieves precise management of the stiffness matrix. For elements in the elastic deformation or low-temperature difference regions, the algorithm directly uses the existing stiffness matrix, avoiding unnecessary recalculation and significantly reducing the waste of computational resources. For elements in the plastic deformation or high-temperature difference regions, the stiffness matrix is ​​recalculated based on the latest material parameters, ensuring an accurate reflection of changes in material properties. This selective update strategy not only improves computational efficiency but also avoids the computational burden of global updates.

[0044] Next, to address the nonlinear behavior of materials during welding, a modified algorithm and iterative solution mechanism were introduced. By combining the Newton-Raphson iterative method, the algorithm effectively handles the complex mechanical behavior of materials under high temperature and stress, ensuring high accuracy and reliability of the simulation results. Furthermore, this method further optimizes the computational process by introducing a temperature effect term and a sequential solution strategy for thermo-mechanical coupling. This sequential solution approach not only simplifies the complex thermo-mechanical coupling problem but also significantly reduces computational complexity, making the algorithm more efficient when handling large-scale welded structures.

[0045] In summary, the temperature-stress-based rapid stiffness matrix assembly solution method for welding finite element analysis provides an efficient and accurate solution by optimizing the stiffness matrix update strategy and iterative solution mechanism. Compared with traditional methods, this method not only significantly improves simulation efficiency but also substantially enhances computational accuracy, making it particularly suitable for complex simulation analysis of large welded structures. This innovative analysis tool provides strong support for welding process optimization, residual stress prediction, and reliability assessment of welded structures, demonstrating significant engineering application value and broad application prospects.

[0046] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A temperature-stress-based welding finite element fast stiffness matrix assembly solving method, characterized in that, Comprise: Obtain the temperature and stress distribution under the initial state, and perform matrix assembly preprocessing according to the temperature and stress distribution, obtain the unit stiffness matrix, and save it; According to the variational principle and the Galerkin method, the heat conduction formula of the transient temperature field and the convection heat transfer formula on the boundary are sequentially established and processed and integrated, and the transient finite element of the welding process is obtained; According to the transient finite element and the temperature field under the time sequence, the elastic-plastic calculation of sequential coupling is carried out, and the plastic strain increment and the elastic stress increment are obtained; Based on the difference between the stress yield state at the current time and the temperature value cached at the last time, the plastic strain increment and the elastic stress increment are processed for unit state judgment to determine the unit state; The yield state of the unit stiffness matrix and the temperature difference value are judged to generate a judgment result, and the unit stiffness matrix is updated or used according to the judgment result and the unit state.

2. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 1, characterized in that, Obtain the temperature and stress distribution under the initial state, and perform matrix assembly preprocessing according to the temperature and stress distribution, obtain the unit stiffness matrix, and save it, specifically: Obtain the room temperature state and stress distribution under the initial state, and assemble a plurality of initial unit stiffness matrices according to the temperature and stress distribution, wherein the unit stiffness matrix is used to describe the deformation characteristics of the material under loading; Wherein, the room temperature state and stress distribution at initial time are 0, the mathematical expression of initial element stiffness matrix of three-dimensional entity is: , is initial element stiffness matrix, is strain-displacement matrix, is elastic stiffness matrix, is volume, is transpose; On the basis of the initial element stiffness matrix, a temperature effect term is added to obtain the element stiffness matrix, the formula is: , , wherein, is the element stiffness matrix, , represents the thermal effect matrix caused by temperature change, b is the gradient matrix of the shape function, is the thermal expansion coefficient, is the stiffness matrix of the material, is the temperature change value; Save the data of all unit stiffness matrices in a preset storage space.

3. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 2, characterized in that, According to the variational principle and the Galerkin method, the heat conduction formula of the transient temperature field and the convection heat transfer formula on the boundary are sequentially established and processed and integrated, and the transient finite element of the welding process is obtained, specifically: In the welding process, the heat conduction formula of the transient temperature field inside the welding object is: wherein, , , and represent the thermal conductivity coefficients of the welding material in the x-axis, y-axis and z-axis directions respectively, is the specific heat capacity of the material, is the density of the material, is the heat source acting inside the workpiece; In the welding process, heat is composed of convective heat transfer and radiative heat transfer, and the formula of convective heat transfer on the boundary of the welding object is: , , , are the direction cosines of the boundary outer normal vector, is the convective heat transfer coefficient, is the ambient temperature; The formula for radiative heat exchange is: where, is the Stefan-Boltzmann constant, with a value of , is the blackness coefficient of the object; According to the variational principle, the heat conduction formula of the transient temperature field and the convection heat transfer formula on the boundary are established and processed, the equivalent integral form is established, and the partial integral processing is carried out through the Galerkin method, the temperature inside each unit body is obtained through the interpolation function and the unit node temperature, and the general format of the transient finite element of the welding process is obtained as follows: wherein, is the unit heat capacity matrix, is the unit heat conduction matrix, is the unit right end load vector, is the temperature vector. According to the numbering of the nodes in the temperature field analysis domain, the nodes and vectors in the temperature field analysis domain are assembled to obtain a global matrix on the entire calculation space, wherein the element-node correspondence in the finite element grid is: , is the sum of all element heat capacity matrices , is the sum of all element heat conduction matrices ; is the sum of all element load vectors ; In time, the element-node correspondence in the finite element mesh is discretely processed in the differential form, and the formula is: , to obtain the temperature field distribution at each time step, where, is the time step, is the temperature vector at time n+1, is the temperature vector at time n, is the sum of all element load vectors at time n+1 .

4. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 3, characterized by, Based on the transient finite element and the temperature field under the time sequence, the elastic-plastic calculation of sequential coupling is carried out, and the plastic strain increment and the elastic stress increment are obtained, specifically: Stress yield condition of a material using the V. Mises yield criterion A determination of a stress yield condition of a material is made, wherein is an equivalent stress, , is a yield strength; When it is judged that the equivalent stress reaches the preset yield condition, the strain increment follows the V. Mises flow rule to obtain the relationship between the plastic strain increment and the stress increment, wherein the mathematical expression of the plastic strain increment is: , is a positive undetermined finite amount, is along the normal direction of the yield surface F = 0 in the stress space; Acquiring elastic stress increment in welding process According to the set relation in the unit stiffness matrix, the relation between displacement and strain is obtained Wherein, The stiffness matrix is, The strain increment is, The temperature-stress matrix is, The temperature increment is, The displacement-strain matrix is, The unit node displacement increment is, , The unit stiffness matrix is, The unit thermal strain load is.

5. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 4, characterized in that, Based on the difference between the stress yield state at the current time and the temperature value cached at the last time, the plastic strain increment and the elastic stress increment are processed for unit state judgment to determine the unit state, specifically: When the stress yield condition is satisfied, the element state is plastic deformation, and when the stress yield condition is satisfied, the element state is elastic deformation; acquiring a preset temperature change threshold When the cell state is classified as a high temperature difference region, and when the cell state is classified as a low temperature difference region.

6. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 5, characterized in that, The yield state of the unit stiffness matrix and the temperature difference value are judged to generate a judgment result, and the unit stiffness matrix is updated or used according to the judgment result and the unit state, specifically: When it is judged that the unit stiffness matrix reaches the plastic deformation or high temperature difference area, the stiffness matrix is recalculated according to the material parameters corresponding to the unit state and the temperature, and is saved; When it is judged that the unit stiffness matrix reaches the elastic deformation or low temperature difference area, the last saved unit stiffness matrix is continued to be used.

7. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 6, characterized in that, When it is judged that the unit stiffness matrix reaches the plastic deformation or high temperature difference area, the stiffness matrix is recalculated according to the material parameters corresponding to the unit state and the temperature, and is saved, specifically: According to the formula The constitutive matrix of the stiffness matrix of the yield region and the high temperature difference region is updated, and the new constitutive matrix parameters are obtained according to the new constitutive matrix parameters And the incremental stiffness matrix update formula , the unit stiffness matrix is updated, wherein The updated stiffness matrix of the i-th unit is The volume or area of the unit is The new constitutive matrix calculated based on the current temperature region condition is The constitutive matrix obtained by the last stage temperature region; On the basis of the original stiffness matrix, the incremental matrix caused by the change of structural parameters is superimposed to obtain a new stiffness matrix wherein, is the updated new stiffness matrix, is the original old stiffness matrix, is the sum of the incremental matrices caused by the change of structural parameters.

8. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 7, characterized by, Also include: The stiffness matrix in the plastic stage is corrected by a correction algorithm to obtain an approximate solution wherein, is a displacement vector, is a product of the stiffness matrix and the displacement vector, is a thermal strain load vector; Based on the Newton-Raphson iterative solution algorithm, assuming the n-th approximate solution is , for the case that the nonlinear equation does not satisfy , the n+1-th iteration solution is obtained, and the n+1-th approximate solution is , , , , wherein is the correction term of the n-th iteration solution, is the tangent matrix under the n-th iteration; Solving the nonlinear equation, the solution equation in the tth increment step is obtained: , , .

9. The temperature-stress based welded finite element fast stiffness matrix assembly solution method according to claim 7, characterized by, Solve the nonlinear equation to obtain the solution equation in the tth increment step, specifically: Determine the initial linear system of equations: , , where is the stress value obtained after convergence of the previous iteration step; The stress strain values are calculated according to the formula , , ; The radial return algorithm with radial return stress is used to establish the relationship between the trial stress and the actual stress according to the trial stress and the yield function, and the expression is as follows: wherein G is the shear modulus, is the yield function. According to the plastic strain increment, the plastic strain in each iteration step is obtained as: wherein, is the equivalent plastic strain increment of the m+1 iteration step, is the equivalent plastic strain increment of the m step, is the yield strength, is the equivalent stress, and h is the hardening coefficient. In the process of loop iteration, each increment step is divided into two times of nonlinear problems, and is solved.

10. The temperature-stress based welded finite element fast stiffness matrix assembly solution method of claim 9, wherein, In the process of loop iteration, each increment step is divided into two times of nonlinear problems, and is solved, specifically: Solving the nonlinear equation with displacement field as the unknown quantity to obtain the displacement field increment in each Newton-Rapson iteration; Inputting the displacement field increment as a parameter to solve the strain increment and the stress increment, and performing nonlinear iteration on the plastic strain increment according to the plastic strain increment formula to obtain the plastic strain increment and the stress increment after convergence; Solving the nonlinear equation with displacement field as the unknown quantity until numerical convergence.

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