A tread dimensional accuracy prediction method and system including an exit viscoelastic relaxation process

Abstract

This invention relates to the field of automated control technology in tire production, and more particularly to a method and system for predicting tread dimensional accuracy, including an exit viscoelastic relaxation process. This method establishes a composite geometric model, combining flow simulation within the die with viscoelastic relaxation deformation simulation in the free relaxation domain after exiting the die, to accurately simulate the entire process of stress release and deformation of the rubber compound from its confined flow within the die to free space. Through mapping and domain switching of the full-field stress tensor, the stable dimensions and morphology of the rubber compound during extrusion are predicted, and the die design is optimized based on the deviation between the target size and the actual predicted size. This method can be widely applied to the design and manufacturing of high-precision rubber products, reducing the number of trial runs and lowering production costs.

Description

Technical Field

[0001] This invention relates to the field of automated control technology in tire production, and in particular to a method and system for predicting tread size accuracy, including an exit viscoelastic relaxation process. Background Technology

[0002] Tire treads are typically formed by continuously extruding uncured rubber through a die after plasticization in an extruder. Because the tread compound is a non-Newtonian viscoelastic material with a significant memory effect, it is subjected to shear and tension within the die runner, resulting in a non-uniform velocity field, temperature field, and viscoelastic stress field. When the rubber leaves the die and loses its wall constraint, the elastic deformation energy stored within the compound is released, causing changes in cross-sectional morphology such as die expansion, shoulder upturn, bottom depression, and edge curling. Simultaneously, the tread continues to undergo viscoelastic relaxation under the influence of gravity, traction, and temperature reduction after die extrusion. Therefore, there is usually a significant difference between the final stable tread cross-section on the production line and the instantaneous cross-section at the die exit.

[0003] Traditional tread die design relies heavily on experience-based trial production. This involves initially designing the die based on the target tread cross-section, conducting extrusion tests after processing, and then repeatedly refining the die based on measured dimensions. This method not only has a long die processing and trial production cycle but also makes it difficult to accurately distinguish the dimensional deviations caused by uneven flow within the die, instantaneous die expansion, and long-term viscoelastic relaxation after die extrusion. Consequently, local die corrections lack a clear basis.

[0004] Existing technologies have attempted to improve the design efficiency of rubber extrusion dies through numerical simulation. For example, Chinese patent document CN1595404A discloses a digital design method for rubber product extrusion dies. This method involves selecting the melt constitutive equation, fitting material parameters based on rheological tests, establishing a geometric model, meshing, setting boundary conditions, and performing finite element analysis. The calculation results are then subjected to deformation and stress analysis, and the die design is revised based on the analysis results. This method can reduce reliance on human experience, but it mainly focuses on the flow analysis of the die and extruded material and the die shape correction. It lacks specific methods for handling the continuous transmission of residual stress and temperature conditions at the exit point when the rubber compound transitions from a flowing state to a freely deformable solid state after leaving the die. This makes it difficult to fully reflect the stress release and cross-sectional evolution of the tread over a long distance from the die.

[0005] US Patent Application US20090210189A1 discloses a polymer melt extrusion analysis system that can be used for die design. It uses an arbitrary Lagrange-Euler finite element method to solve the polymer viscoelastic constitutive equations, simulates free surface evolution and transient die-off expansion through mesh correction, and can adjust die parameters based on the calculated die-off expansion results. This scheme can predict the transient free surface changes of the polymer after leaving the die, but its analysis is still mainly based on a viscoelastic fluid free surface model. When the tread rubber has left the die a considerable distance, the flow velocity gradient gradually decreases, and it mainly exhibits finite deformation solid relaxation, continuing to use a fluid free surface model makes it difficult to simultaneously account for long-term stress decay, gravitational deformation, and traction deformation. Determining the calculation endpoint solely based on whether the expansion morphology changes tend to stabilize may result in situations where the cross-sectional dimensions remain essentially unchanged, but the internal residual stress has not been fully released, thus affecting the prediction accuracy of the final stable cross-section.

[0006] Furthermore, existing simulation results and die corrections typically only establish a correspondence at the overall dimensional level. When local contour deviations occur in the shoulder, center, or edge of the stable tread, it is difficult to accurately trace these deviations to the corresponding position at the die exit where there is uneven velocity or abnormal viscoelastic stress distribution. Die correction still relies on the designer's experience, which can easily lead to repeated iterations.

[0007] Therefore, existing technologies still need to address technical issues such as how to continuously retain the speed, temperature, and viscoelastic stress information at the die exit, how to balance instantaneous die expansion and long-term viscoelastic relaxation after die separation, how to reliably determine that the tread cross-section has reached a stable state, and how to map the local dimensional deviations of the stable cross-section to the specific correction positions of the die flow channel, in order to improve the accuracy of tread final dimension prediction and die optimization. Summary of the Invention

[0008] This invention aims to provide a method and system for predicting the accuracy of tread dimensions, including the exit viscoelastic relaxation process. By continuously transmitting the velocity, temperature, and viscoelastic stress state of the rubber material at the die exit, it simulates the long-term stress release and cross-sectional morphology evolution of the tread after demolding, accurately obtains a stable tread cross-section, and optimizes the die flow channel based on the deviation of the stable cross-section. This solves the problems of existing methods, such as difficulty in taking into account instantaneous demolding expansion and long-term viscoelastic relaxation, insufficient accuracy in predicting stable dimensions, and reliance on manual experience for die correction.

[0009] Firstly, in order to achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0010] A method for predicting tread dimensional accuracy that includes an exit viscoelastic stress relaxation process includes the following steps:

[0011] S1. Obtain the die structure parameters, extrusion process parameters, and rubber material parameters, establish the flow channel domain inside the die and the free relaxation domain outside the candidate die, and determine the length of the candidate free relaxation domain based on the preset average extrusion line speed, the main relaxation time of the rubber material, and the minimum characteristic size of the die exit.

[0012] S2, perform non-isothermal viscoelastic transient flow solution in the flow channel domain inside the die to obtain the velocity field, temperature field and viscoelastic stress tensor field of the die exit section at the same time scale. Calculate the actual average extrusion linear velocity at the die exit based on the velocity field and correct the length of the free relaxation region accordingly.

[0013] S3. Establish a local orthogonal basis at the die exit section, map the velocity field with mass flow rate consistency as constraint, and the temperature field with convective enthalpy consistency as constraint, and map the viscoelastic stress tensor field under the condition that the principal stress direction of the viscoelastic stress tensor is continuous and the error of the tensor invariant does not exceed the preset threshold, so as to form the initial motion state, initial temperature field and initial stress state of the rubber in the free relaxation domain.

[0014] S4. The rubber material in the free relaxation domain is equivalent to a finite strain hyperelastic-viscoelastic continuum. Under the action of gravity and the initial motion state, initial temperature field and initial stress state, large deformation transient relaxation solution is performed until the equivalent stress attenuation criterion and the cross-sectional shape convergence criterion of the whole region and the preset key region are simultaneously satisfied to obtain a stable cross section.

[0015] S5 registers the stable section with the target design section, and traces the local deviation back to the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die outlet according to the unified arc length coordinate. Based on this, it updates the local throttling size and outlet transition fillet at the corresponding position and returns to S1 until the size deviation and flow uniformity both meet the preset threshold. Then, it outputs the predicted size and die parameters.

[0016] As a further improvement, in steps S1 and S2, the length of the free relaxation region is determined as follows: during the first iteration, the candidate free relaxation region length is determined based on the preset average extrusion line speed of the extrusion process; after step S2 is completed, the actual average extrusion line speed at the die exit is determined based on the die exit speed field, and the length of the free relaxation region is corrected.

[0017] As a further improvement, in step S2, the PTT viscoelastic constitutive model or the Giesekus viscoelastic constitutive model is used to solve the non-isothermal transient flow of the rubber compound. The steady-state solution is first used as the initial state of the transient solution. When the relative changes of the outlet mass flow rate, the outlet average temperature, and the outlet average viscoelastic stress within a continuous 2-second time window are all no greater than 0.5%, the corresponding time window is determined as the extrusion stabilization time window. The coordinates, velocity vector, temperature, and six independent components of the second-order viscoelastic stress tensor of each sampling position at the die outlet section are obtained at the same output time.

[0018] As a further improvement, step S3 includes:

[0019] A reference surface with uniform parameterized coordinates is established at the die exit section, and a local orthogonal basis consisting of two tangential unit vectors and one normal unit vector along the extrusion direction is established at each sampling position of the reference surface.

[0020] The viscoelastic stress tensor output from the flow channel domain inside the die is transformed to the local orthogonal basis and eigenvalue decomposition is performed to obtain the principal stress values ​​and principal stress directions. The principal stress values ​​are interpolated to the inlet section of the free relaxation domain according to the unified parameterized coordinates, and the principal stress directions of adjacent sampling positions are kept continuous by minimum rotation projection. Tensor invariant correction is performed on the reconstructed viscoelastic stress tensor to ensure that the relative errors of the first, second, and third invariants before and after mapping are all no greater than 0.01%.

[0021] Mass flow rate correction is applied to the mapped velocity field to ensure that the relative error of mass flow rate between the outlet side of the flow channel domain and the inlet side of the free relaxation domain inside the die is no greater than 0.5%; convective enthalpy correction is applied to the mapped temperature field to ensure that the relative error of convective enthalpy between the two sides is no greater than 0.5%.

[0022] The mapped velocity field is used to establish the initial motion state of the rubber compound in the free relaxation region, the mapped temperature field is used as the initial temperature field, and the mapped viscoelastic stress tensor is assigned to the material points near the inlet section of the free relaxation region to form the initial stress state. During continuous extrusion, the correspondence between the die exit time and the distance along the extrusion direction is established according to the average extrusion linear velocity at the actual die exit, and the initial motion state, initial temperature field and initial stress state are continuously assigned as the material enters the free relaxation region.

[0023] As a further improvement, in step S4, the finite strain hyperelastic-viscoelastic continuum adopts a combined model of the Ogden hyperelastic model and the generalized Maxwell viscoelastic network, or a combined model of the Ogden hyperelastic model and the Prony series viscoelastic term; the outer surface of the free relaxation domain is set as a free surface, a gravity load is applied to the rubber material, and an end traction load is applied or a conveyor belt support contact is set according to the actual production state; when the mesh quality is lower than the preset mesh quality threshold, adaptive remeshing is implemented, and the stress field, temperature field and displacement field are transferred to the updated mesh.

[0024] As a further improvement, in step S4, the criterion for the attenuation of equivalent stress is that the average equivalent stress of the entire region and the pre-specified shoulder and edge key regions are attenuated to less than 5% of their respective initial average equivalent stress.

[0025] As a further improvement, in step S4, the convergence criterion for the cross-sectional shape is: the change in the total width, the maximum thickness, and the coordinates of each preset contour point at adjacent sampling times is less than 0.05 mm, and is maintained continuously for no less than 1 second.

[0026] As a further improvement, in step S4, when the rubber material moves to the end of the free relaxation domain and still does not simultaneously satisfy the equivalent stress attenuation criterion and the cross-sectional shape convergence criterion, the moving calculation window is used to continue advancing the mold departure time coordinate until the two criteria are simultaneously satisfied and then a stable cross-section is output.

[0027] As a further improvement, in step S5, the center point of the tread bottom is used as the reference point and the tread bottom reference line is used as the lateral reference line. The stable section and the target design section are rigidly registered to obtain a set of local deviations, including total width deviation, maximum thickness deviation, shoulder upturn deviation, bottom indentation deviation and edge curling angle deviation. According to the same normalized arc length coordinates between the die exit section and the stable section, the set of local deviations is correlated with the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die exit.

[0028] As a further improvement, in step S5, when the local contour deviation and the corresponding position velocity deviation have a relationship of changing in the same direction, the local velocity deviation is reduced by adjusting the local throttling gap, throttling width, or throttling length at the corresponding position; when the correlation between the local contour deviation and the velocity deviation is lower than the preset correlation threshold and the viscoelastic stress principal gradient is higher than the adjacent region, the viscoelastic stress principal gradient is reduced by increasing the outlet transition fillet at the corresponding position or by making the local throttling structure smoothly transition towards the outlet direction.

[0029] As a further improvement, in step S5, for the parameterized die, a sensitivity relationship between the stable cross-sectional deviation and the die parameters is established by applying a disturbance to the local throttling size and the outlet transition fillet, and the die parameter update amount is determined based on the sensitivity relationship.

[0030] Secondly, the present invention also provides a tread size accuracy prediction system incorporating an exit viscoelastic stress relaxation process for implementing the method, comprising a processor and a memory, wherein the memory stores the following modules executed by the processor:

[0031] The model building module is used to obtain die structure parameters, extrusion process parameters and rubber material parameters, establish the flow channel domain inside the die and the free relaxation domain outside the candidate die, and determine the length of the candidate free relaxation domain based on the preset average extrusion line speed, the main relaxation time of the rubber material and the minimum feature size of the die exit.

[0032] The in-die flow solution module is used to solve non-isothermal viscoelastic transient flows, obtain the velocity field, temperature field and viscoelastic stress tensor field of the die exit section at the same time scale, and determine the actual die exit average extrusion linear velocity based on the velocity field to correct the length of the free relaxation region.

[0033] The data mapping and domain switching module is used to map the velocity field, temperature field and viscoelastic stress tensor field with mass flow rate, convective enthalpy flow and principal stress direction and tensor invariants of viscoelastic stress tensor as constraints, respectively, to construct the initial motion state, initial temperature field and initial stress state of the rubber compound in the free relaxation domain.

[0034] The free relaxation domain solid solution module is used to solve the large deformation transient relaxation of the rubber material as an equivalent finite strain hyperelastic-viscoelastic continuum. It outputs a stable cross section when the equivalent stress attenuation criterion and the cross section shape convergence criterion are simultaneously satisfied in the overall region and the preset key region.

[0035] The deviation tracing and die optimization module is used to trace back the local deviation of the stable section to the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die exit according to the unified arc length coordinate, update the local throttling size and exit transition fillet at the corresponding position, and trigger the model building module to recalculate iteratively.

[0036] The data mapping and domain switching module includes a reference surface construction unit, a coordinate transformation unit, a tensor eigenvalue decomposition unit, a constraint interpolation unit, a principal direction projection unit, a tensor invariant correction unit, and an initial state construction unit.

[0037] The reference surface construction unit is used to establish a unified parameterized coordinate between the die exit section and the free relaxation domain inlet section;

[0038] The coordinate transformation unit is used to transform the viscoelastic stress tensor into a local orthogonal basis consisting of two tangential directions and one normal direction;

[0039] The tensor eigenvalue decomposition unit is used to obtain the principal stress values ​​and principal stress directions of the viscoelastic stress tensor.

[0040] The constrained interpolation unit is used to interpolate the velocity field, temperature field, and principal stress values, and to correct the velocity field and temperature field based on the mass flow rate and convective enthalpy, respectively.

[0041] The main direction projection unit is used to maintain the continuity of the principal stress direction at adjacent positions through minimum rotational projection.

[0042] The tensor invariant correction unit is used to correct the reconstructed viscoelastic stress tensor so that the error between its tensor invariant and the tensor invariant of the corresponding tensor before mapping does not exceed a preset threshold.

[0043] The initial state construction unit is used to construct the initial motion state, initial temperature field, and initial stress state of the rubber compound within the free relaxation domain based on the mapping results.

[0044] The free relaxation domain solid solution module includes a combined constitutive element, an adaptive remesh element, a moving window element, and a dual-criteria termination element;

[0045] The combined constitutive unit is used to configure a combined model of the Ogden hyperelastic model and the generalized Maxwell viscoelastic network, or to configure a combined model of the Ogden hyperelastic model and the Prony series viscoelastic term.

[0046] The adaptive remeshing element is used to re-mesh when the mesh quality is lower than a preset mesh quality threshold, and to transfer the stress field, temperature field and displacement field to the updated mesh.

[0047] The moving window unit is used to continue advancing the demolding time coordinate of the corresponding material section when the rubber reaches the end of the free relaxation region but has not yet met the stability condition;

[0048] The dual-criteria termination unit is used to output a stable cross section when the average equivalent stress in the overall area and the preset shoulder and edge key areas is less than 5% of their respective initial values, and the changes in the total width, maximum thickness and coordinates of each preset contour point are all less than 0.05mm and are maintained continuously for no less than 1s.

[0049] Thirdly, the present invention also provides a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implement the steps of the method.

[0050] Fourthly, the present invention also provides a computer program product, including a computer program or instructions that, when executed by a processor, implement the steps of the method.

[0051] This invention reduces the loss of mass, heat, and residual stress information caused by switching solution domains by performing a conservation mapping between the fluid solution domain and the solid relaxation domain, and by converting the velocity field, temperature field, and viscoelastic stress tensor field at the die exit into the initial state of the rubber compound within the free relaxation domain while maintaining the principal stress directions and tensor invariants of the viscoelastic stress tensor. Treating the post-demolding rubber compound as a finite-strain hyperelastic-viscoelastic solid for large deformation transient relaxation allows for a comprehensive reflection of the combined effects of residual stress release, temperature changes, gravity, and production line traction on the tread cross-section, thus more accurately distinguishing between instantaneous demolding expansion and dimensional changes caused by long-term viscoelastic relaxation. By simultaneously employing… Determining the stable section using the equivalent stress decay criterion and the cross-sectional morphology convergence criterion can avoid prematurely terminating the calculation based solely on changes in external morphology, thereby improving the predictive reliability of parameters such as final width, thickness, shoulder upturn, bottom concavity, and edge curl angle. Furthermore, by tracing the local deviations of the stable section back to the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die exit, and adjusting the local throttling size and exit transition radius accordingly, a closed-loop optimization process of "state transfer—relaxation prediction—deviation tracing—die correction" can be formed, reducing the number of repeated die trials and manual die repairs, shortening the tread product development cycle, and improving the consistency of tread extrusion dimensions and die design efficiency. Attached Figure Description

[0052] Figure 1 This is a flowchart of the overall process for predicting the tread size accuracy of the present invention.

[0053] Figure 2 This is a block diagram of the main modules of the tire tread size accuracy prediction system of the present invention.

[0054] Figure 3 This is a geometric schematic diagram of the composite simulation model of the present invention.

[0055] Figure 4 This is a schematic diagram showing the morphological evolution of the rubber compound after it enters the free relaxation region outside the die from the flow channel region inside the die. Detailed Implementation

[0056] The following is combined with Figures 1 to 4 The present invention will be further described below. The following embodiments use continuous extrusion of uncured tire tread rubber as an example to illustrate the physical connections, data relationships, and numerical implementation methods between the various modules and steps of the present invention. Those skilled in the art can adjust the mesh density, material parameters, and process parameters according to the actual tread size, rubber compound formulation, extruder specifications, and die structure. Without changing the technical concept of "exit complete field constraint mapping—fluid solution state to finite strain hyperelastic-viscoelastic solution state conversion—dual criteria to determine stable cross section—local deviation backtracking to the corresponding die position," corresponding adjustments can be made to realize the present invention.

[0057] It should be noted that the so-called "switch from fluid domain to solid relaxation domain" in this invention refers to a switch in numerical description, rather than a phase transition of the rubber compound. Uncured tread rubber exhibits significant shear flow and viscoelastic memory behavior inside the die, making it suitable for description as a viscoelastic fluid. After leaving the die, within the die-off timescale of interest in this invention, in order to continuously track the stress memory, gravitational deformation, and local contour evolution of material points, it is equivalent to an approximately incompressible finite-strain hyperelastic-viscoelastic continuum for body-based solution.

[0058] I. Terminology Explanation

[0059] Internal flow channel region of the die: This refers to the space within the die that is jointly defined by the feed section, flow branching section, compression section, shaping section, local throttling section, and outlet transition section for the flow of the rubber compound. Within this region, the rubber compound is constrained by the die wall and primarily exhibits non-isothermal, non-Newtonian viscoelastic flow.

[0060] The free relaxation region outside the die: refers to the calculated area extending outward from the die exit section along the average extrusion direction. In this region, the rubber compound is freed from the constraints of the runner wall and undergoes large deformation with limited strain under the influence of residual viscoelastic stress, gravity, and production line traction.

[0061] Domain switching: refers to converting the solution results of the viscoelastic fluid inside the die into the initial motion state, initial temperature field and initial stress state required for the finite strain hyperelastic-viscoelastic solution in the free relaxation domain at the die exit section, and then solving the subsequent material motion by the body large deformation method.

[0062] Viscoelastic stress tensor: refers to the second-order tensor that describes the elastic memory stress accumulated by the rubber compound due to shearing, stretching and deformation history, and contains at least three normal stress components and three independent shear stress components.

[0063] Principal relaxation time: refers to the time scale that represents the main stress relaxation process of the rubber compound at a given reference temperature. It can be obtained by fitting stress relaxation tests or determined by the principal time constant in the PTT model, Giesekus model or generalized Maxwell network.

[0064] Local orthogonal basis: refers to a coordinate basis established at each sampling position on the die exit reference surface, consisting of two tangential unit vectors and a normal unit vector along the average extrusion direction, used to eliminate the influence of different mesh orientations on tensor mapping.

[0065] Tensor invariants: These refer to the first, second, and third invariants of the viscoelastic stress tensor that remain unchanged under coordinate rotation. They are used to check whether non-physical principal value distortion occurs after the stress tensor is mapped.

[0066] Minimum rotation projection: refers to the direction corresponding to the smallest rotation angle when transforming the principal stress direction of the source interface to the local base of the target interface, under the conditions of satisfying the normal reference relationship and the continuity of the principal direction.

[0067] Stable section: refers to the section of the tread compound that simultaneously satisfies the equivalent stress decay criterion and the section shape convergence criterion after demolding. The stable section is not simply defined by a fixed demolding distance, but rather by both the stress state and the morphological state.

[0068] Deviation tracing: refers to establishing a correspondence between the local normal deviation and critical dimension deviation of the stable section relative to the target design section and the velocity deviation and viscoelastic stress principal gradient at the same position of the die exit, according to a unified arc length coordinate system, and determining the local throttling position and exit transition position that need to be corrected accordingly.

[0069] II. System Structure

[0070] like Figure 2 As shown, the system of this invention includes a processor, a memory, and a model building module, a die in-mold flow solving module, a data mapping and domain switching module, a free relaxation domain solid-state solving module, and a deviation tracing and die optimization module executed by the processor. These modules can run on the same computing workstation or on a multi-node computing platform connected via a network. To ensure data consistency within the same iteration, all modules share the same die parameter version number, material parameter version number, and process parameter version number.

[0071] Model building module reading Figure 2 The input parameters are shown on the left. The die structure parameters include at least the inlet size, the flow splitter size, the shaping section length, the local throttling gap, the throttling width, the throttling length, the outlet profile, and the outlet transition radius; the extrusion process parameters include at least the compound inlet temperature, the die wall temperature, the volumetric flow rate or inlet pressure, the preset average extrusion linear speed, the traction speed, and the environmental conditions; the compound material parameters include at least the density, specific heat capacity, thermal conductivity, PTT model parameters or Giesekus model parameters, Ogden model parameters, and Prony series or generalized Maxwell parameters.

[0072] The corresponding module for solving the flow inside the die Figure 2 The "Simulation of Viscoelastic Flow Inside the Die" section is used to calculate the velocity field, temperature field, and viscoelastic stress tensor field of the flow channel domain inside the die. The corresponding data mapping and domain switching module... Figure 2The "Exit Field Extraction and Mapping" section includes at least reference surface construction units, coordinate transformation units, tensor eigenvalue decomposition units, constraint interpolation units, principal direction projection units, tensor invariant correction units, and initial state construction units. This module does not simply copy each stress component at the exit point by point; instead, it constrains the mapping quality of velocity, temperature, and viscoelastic stress tensors under a unified parameterized coordinate system.

[0073] The corresponding module for solving solids in the free relaxation region Figure 2 The sections on "Transient Simulation of Hyperelastic-Viscoelastic Large Deformation in the Free Relaxation Domain" and "Stable Section Morphology and Size Extraction" include at least composite constitutive elements, adaptive remesh elements, moving window elements, and dual-criteria termination elements. Composite constitutive elements describe the finite strain and time-dependent stress release of the material after demolding; adaptive remesh elements control element distortion during large deformation; moving window elements continue advancing the demolding time if the stability criterion is not met after the material reaches the end of the preset free relaxation domain; and dual-criteria termination elements simultaneously check stress decay and changes in section morphology.

[0074] Deviation tracing and die optimization module correspondence Figure 2 The right-hand side displays the results analysis, accuracy assessment, and optimization suggestions. This module does not scale the entire module based solely on the total width or thickness, but rather according to... Figure 3 The normalized arc length coordinates shared by the exit reference surface and the stable section are used to trace back the local topographic deviations of the shoulder, center, bottom and edge to the corresponding die positions, and the local throttling size or exit transition fillet is selected as the correction object for different deviation sources.

[0075] III. Specific Technical Route for Implementing the Method of the Invention

[0076] like Figure 1 As shown, the method of the present invention includes S1 to S5.

[0077] S1. Obtain the die structure parameters, extrusion process parameters, and rubber material parameters, establish the flow channel domain inside the die and the free relaxation domain outside the candidate die, and determine the length of the candidate free relaxation domain based on the preset average extrusion line speed, the main relaxation time of the rubber material, and the minimum characteristic size of the die exit.

[0078] S2, perform non-isothermal viscoelastic transient flow solution in the flow channel domain inside the die to obtain the velocity field, temperature field and viscoelastic stress tensor field of the die exit section at the same time scale. Calculate the actual average extrusion linear velocity at the die exit based on the velocity field and correct the length of the free relaxation region accordingly.

[0079] S3. Establish a local orthogonal basis at the die exit section, map the velocity field with mass flow rate consistency as constraint, and the temperature field with convective enthalpy consistency as constraint, and map the viscoelastic stress tensor field under the condition that the principal stress direction of the viscoelastic stress tensor is continuous and the error of the tensor invariant does not exceed the preset threshold, so as to form the initial motion state, initial temperature field and initial stress state of the rubber in the free relaxation domain.

[0080] S4. The rubber material in the free relaxation domain is equivalent to a finite strain hyperelastic-viscoelastic continuum. Under the action of gravity and the initial motion state, initial temperature field and initial stress state, large deformation transient relaxation solution is performed until the equivalent stress attenuation criterion and the cross-sectional shape convergence criterion of the whole region and the preset key region are simultaneously satisfied to obtain a stable cross section.

[0081] S5 registers the stable section with the target design section, and traces the local deviation back to the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die outlet according to the unified arc length coordinate. Based on this, it updates the local throttling size and outlet transition fillet at the corresponding position and returns to S1 until the size deviation and flow uniformity both meet the preset threshold. Then, it outputs the predicted size and die parameters.

[0082] (a) S1. Establish the internal flow channel domain and candidate free relaxation domain of the die.

[0083] First, obtain the three-dimensional geometry of the die corresponding to the tread to be evaluated and the target stable cross-sectional profile. Perform a sealing check on the die geometry, and geometrically clean any minor machining burrs that do not affect flow, while retaining the flow islands, shaping sections, local throttling structures, minimum outlet clearance, and outlet transition fillet. Define the space inside the die, enclosed by the walls and actually occupied by the rubber material, as the internal flow channel domain of the die.

[0084] like Figure 3 As shown, the die exit face serves as the spatial interface between the internal flow channel region and the external free relaxation region of the die. In the first iteration, the preset average extrusion linear velocity in the extrusion process parameters is used. Master relaxation time of rubber compound Minimum feature size of the die exit and geometric length coefficient Calculate the length of the candidate free relaxation region :

[0085] ;

[0086] In the formula, The length of the candidate free relaxation region used in the first iteration; The preset average extrusion line speed given to the extrusion equipment or process card; This refers to the main relaxation time of the rubber compound; This refers to the minimum thickness or minimum clearance of the die exit section. This is the geometric length coefficient. The value can be 5 to 20, preferably 10; when the local thickness gradient at the outlet is large or the edge features are small, it can be 15 to 20; when the outlet cross section is relatively flat, it can be 5 to 10.

[0087] After S2 is completed, the actual average extrusion line speed at the die exit is calculated based on the exit velocity field. :

[0088] ;

[0089] In the formula, This represents the actual average extrusion line speed at the die exit. The cross-sectional area of ​​the die exit; This refers to the local velocity vector at the exit section. The unit normal vector of the cross section pointing outwards from the die; Let be the infinitesimal element of the outlet cross-sectional area.

[0090] Then correct the length of the free relaxation region as follows:

[0091] ;

[0092] In the formula, This represents the corrected length of the free relaxation region; the other parameters have the same meaning as before. Using this method, the initial modeling does not rely on the yet-to-be-obtained S2 exit velocity field, while a length correction is performed using the actual exit velocity after S2 is completed, thus avoiding inconsistencies between the free relaxation region length and the actual operating conditions.

[0093] When the principal relaxation time is determined through stress relaxation tests, a step strain can be applied to the uncured rubber compound at a reference temperature, and the result can be fitted using a multi-exponential function:

[0094] ;

[0095] In the formula, For a moment The relaxation stress; For long-term equilibrium stress; The number of relaxation components; For the first The initial stress contribution of each relaxation component; For the first The time constants of each relaxation component are used. The time constant that contributes the most to the overall relaxation process is taken as... In one implementation, three relaxation components are used at a reference temperature of 95°C, with time constants of 0.15s, 1.6s, and 12.5s, respectively, corresponding to normalized modulus coefficients of 0.18, 0.27, and 0.43, and a long-term equilibrium modulus coefficient of 0.12. Therefore, 12.5s is determined as the principal relaxation time.

[0096] Regarding the mesh, a body-fitted mesh is used in the flow channel region inside the die. The thickness of the first mesh layer near the wall can be 0.2–0.5 mm, the characteristic dimension of the outlet transition fillet and local throttling position can be 0.3–0.8 mm, and the thickness of other flow channel regions can be 1–3 mm. The free relaxation region uses a hexahedral mesh or a tetrahedral-hexahedral hybrid mesh suitable for large deformations. The characteristic dimension near the outlet can be 0.5–1 mm, gradually increasing to 2–5 mm along the extrusion direction. The maximum geometric projection error of the interface surfaces on both sides of the die outlet is preferably no greater than 0.05 mm.

[0097] (ii) S2, Solving the non-isothermal viscoelastic transient flow inside the die.

[0098] The purpose of S2 is not merely to obtain the pressure drop or total flow rate, but to obtain the same time-scaled outlet state that S3 can directly inherit. The rubber compound can be treated as an approximately incompressible non-Newtonian viscoelastic fluid. A preferred embodiment employs an exponential PTT model, the basic form of which is:

[0099] ;

[0100] in:

[0101] ;

[0102] In the formula, It is the viscoelastic stress tensor; The trace of the viscoelastic stress tensor; The relaxation time is at the current temperature; is the upper volumetric derivative of the viscoelastic stress tensor; The viscosity coefficient of the polymer; For strain rate tensor; For PTT elongation parameters.

[0103] In a specific parameter set, the reference temperature is 95℃, the viscosity of the viscous matrix is ​​2000 Pa·s, the polymer viscosity coefficient is 16500 Pa·s, the PTT reference relaxation time is 12.5 s, the PTT elongation parameter is 0.12, the compound density is 1120 kg / m³, the specific heat capacity is 1900 J / (kg·K), and the thermal conductivity is 0.28 W / (m·K). The effect of temperature on viscosity and relaxation time is expressed using the time-temperature shift factor.

[0104] ;

[0105] In the formula, It is the temperature shift factor; The current temperature; For reference temperature; , This is the temperature shift coefficient. In one embodiment, Take 95℃, Take 8.86, Take 101.6 K. The relaxation time and polymer viscosity coefficient at the current temperature are multiplied by the reference values. get.

[0106] Alternatively, the Giesekus viscoelastic constitutive model can be used. In one implementation, the Giesekus mobility coefficient is set to 0.18, and the reference relaxation time adopts the principal time constant consistent with the stress relaxation test of the same rubber compound. When selecting the PTT or Giesekus model, it should be ensured that the principal relaxation time of S1, the flow constitutive parameter of S2, and the Prony or generalized Maxwell parameter of S4 come from the same rubber compound test system, so that the "stress accumulated in the die" and the "stress released outside the die" have a consistent time scale.

[0107] The inlet can be a volumetric flow rate boundary or a velocity boundary, and the die wall can be set as a non-slip, isothermal wall. In one embodiment, the inlet temperature of the rubber compound is 105°C, the die wall temperature is 95°C, the preset average extrusion linear velocity is 0.12 m / s, and the outlet pressure is gauge pressure 0 Pa. If the actual rubber compound exhibits significant wall slip, a Navier slip boundary can be used, and the slip parameters are determined through capillary rheological testing.

[0108] During the solution process, steady-state results are first used as the initial state for transient calculations, followed by transient calculations. When the relative changes in outlet mass flow rate, outlet average temperature, and outlet average viscoelastic stress within a continuous 2-second time window are all no greater than 0.5%, this time window is defined as the stable output time window. At the same output moment, six independent components of the spatial coordinates, velocity vector, temperature, and second-order viscoelastic stress tensor of each sampling position at the outlet cross-section are derived, and the aforementioned actual average extrusion linear velocity is calculated from this velocity field. The length of the free relaxation region is then corrected accordingly. If the corrected length changes, only the free relaxation region mesh is regenerated, and the flow results inside the die remain the exit source field of this iteration.

[0109] (III) S3, Exit Complete Field Constraint Mapping and Fluid-Finite Strain Domain Switching

[0110] S3 is the corresponding Figure 3 The boundary position of the two computational domains, and corresponding to Figure 4 The location where the die transitions from a constrained state to a free deformation state. The technical objective is not simply to use a certain average pressure or average velocity as the downstream boundary, but to preserve the differences in velocity, temperature, principal stress values, and principal stress directions between different locations at the die exit, so that the shoulder, center, bottom, and edge retain their respective material memory states after entering the free relaxation region.

[0111] The internal flow mesh and the solid mesh of the free relaxation region serve different solvers, and their nodes typically cannot coincide one-to-one. Directly copying the six stress components based on the nearest node could disrupt the overall mass flow rate and energy state, and could also cause errors in the shear stress components due to different local coordinate directions on both sides. Therefore, this invention first establishes a unified parametric reference surface at the die exit section. For a typical tread section, it can be described using width and thickness parameters respectively; for sections with curved shoulders, grooves, or complex edges, normalized arc length coordinates are preferred. Normalized arc length coordinates Defined as:

[0112] ;

[0113] In the formula, The normalized arc length coordinates of the current sampling point; This represents the cumulative arc length from the starting point of the corresponding contour segment to the current sampling point. This is the total arc length of the contour segment. The range is 0 to 1. Subsequent step S5 also uses this coordinate system, thus providing a unified location identifier for both the "exit position" and the "stable section position".

[0114] Two tangential unit vectors are established at each sampling position on the reference plane. , and a normal unit vector This forms a locally orthogonal basis. The coordinate transformation matrix is ​​written as:

[0115] ;

[0116] In the formula, This is the direction cosine matrix between the global coordinate system and the local orthogonal basis; It is the first tangential unit vector along the direction of increasing cross-sectional width or arc length; Located within the cross section and The orthogonal second tangential unit vector; It is the normal unit vector along the average extrusion direction.

[0117] The exit viscoelastic stress tensor in the global coordinate system Transform to a local orthogonal basis:

[0118] ;

[0119] In the formula, The viscoelastic stress tensor under a locally orthogonal basis; This is the transpose of the coordinate transformation matrix. Perform eigenvalue decomposition:

[0120] ;

[0121] In the formula, It is an orthogonal matrix composed of the three principal stress directions; It is a diagonal matrix composed of the three principal stress values.

[0122] Principal stress values ​​are interpolated using area-weighted methods based on uniform parametric coordinates and source element shape functions. Instead of linear component interpolation, the principal stress directions are determined using the exit normal as a common reference direction, with the minimum rotation matrix calculated within the target local basis. If two principal stress values ​​at adjacent locations are close and their eigenvector order may be interchanged, the principal direction from the previous sampling location is used as a continuity reference, and the arrangement with the smallest direction angle is selected to avoid 180° flips or unreasonable exchanges of principal directions at adjacent grid points.

[0123] After reconstructing the principal value and principal direction, the tensor invariants are checked. The first, second, and third invariants are as follows:

[0124] ;

[0125] In the formula, , , These are the first, second, and third invariants of the viscoelastic stress tensor, respectively. Let be the determinant of the viscoelastic stress tensor. In one implementation, the relative error of each invariant before and after mapping is controlled within 0.01%; when the error of any invariant exceeds 0.01%, the principal stress values ​​obtained by interpolation are slightly constrained and the tensor is reconstructed until the error limit is met.

[0126] A mass flow rate constraint is applied to the velocity field. The interface mass flow rate is:

[0127] ;

[0128] In the formula, For interface quality flow rate; Density of the rubber compound; For the interface section; It is a local velocity vector; The unit normal vector is used for the interface. First, the velocity is interpolated according to the reference surface, then the mass flow rates at the exit side of the aperture and the inlet side of the free relaxation domain are compared. When the relative error of the initial mapped mass flow rate is no greater than 2%, a fine-tuning of the target inlet normal velocity is allowed based on the ratio of the mass flow rates on both sides; when the error is greater than 2%, instead of masking the mesh mapping error by large scaling, the reference surface sampling density is increased and re-interpolated. In a preferred embodiment, the final relative error of the mass flow rate is controlled within 0.5%.

[0129] The temperature field employs convective enthalpy flow constraint, rather than mistakenly equating "equal temperature" with heat flux conservation. The interface convective enthalpy flow is defined as:

[0130] ;

[0131] In the formula, For interface convection enthalpy flow; This refers to the specific heat capacity of the rubber compound. This represents the local temperature; the meanings of the other parameters are the same as before. In one implementation, the relative error of the convective enthalpy before and after mapping is controlled within 0.5%; a uniform, minute correction of no more than 0.5°C is allowed to be applied to the target temperature field. If the error limit is still not met even when 0.5°C is reached, the temperature field interpolation is re-executed.

[0132] After completing the constraint mapping of velocity, temperature, and stress tensors, the initial state of the free relaxation domain is constructed. The mapped velocity field is used to form the initial motion state of the inlet material point; the mapped temperature field is used as the initial temperature of the material point, and the relaxation time and viscoelastic parameters at this position are updated according to the aforementioned time-temperature shift factor; the mapped viscoelastic stress tensor is used as the initial stress memory state of the inlet material point.

[0133] In the implementation where the finite strain solver accepts the current configuration Cauchy stress as the initial stress input, the exit viscoelastic stress tensor, after local basis transformation and invariant correction, is directly assigned to the inlet material point according to the current configuration. The hydrostatic pressure component is then redefined in subsequent solid solution operations using approximately incompressible constraints. When the solid solver requires a reference configuration stress metric, the mapped current configuration stress is converted to the corresponding reference configuration stress according to the solver's initial deformation gradient definition. Thus, S3 performs not only coordinate component transformation but also stress metric processing consistent with the initial stress interface of the actual solid solver.

[0134] To provide an initial estimate of the free relaxation region, the initial stress in the inlet-adjacent region can be configured along the extrusion direction with a constant extrapolation segment and a decay extrapolation segment. In one embodiment, the length of the constant extrapolation segment is 5% of the length of the free relaxation region, and the decay characteristic length is 25% of the length of the free relaxation region. The decay segment can be represented as:

[0135] ;

[0136] In the formula, Distance from the entrance to the free relaxation area Initial stress guess at the location; This refers to the inlet viscoelastic stress tensor after interface mapping. The length of the extrapolated segment is a constant value; This is the attenuation characteristic length. For In the region, the initial stress is taken as This extrapolation is only used to construct the initial numerical state and does not replace the stress relaxation calculated by S4 based on the actual material constitutive model.

[0137] In continuous extrusion, the nominal relationship between die departure time and distance along the extrusion direction is:

[0138] ;

[0139] In the formula, The nominal time elapsed for a certain cross section from leaving the die exit point; This is the distance from the cross-section to the die exit; S2 represents the actual average extrusion line velocity at the die exit. After each injection time step, a new layer of material is generated at the entrance of the free relaxation region and assigned the corresponding initial motion state, initial temperature field, and initial stress state. The injection time step can be 0.01–0.05 s; in one specific embodiment, it is 0.02 s.

[0140] Through the above S3 process. Figure 4 Although different numerical descriptions are used on either side of the boundary, the local flow rate, temperature level, principal stress magnitude, and principal stress direction at the outlet can be continuously transmitted. Therefore, Figure 4 The non-uniform morphological changes observed on the right shoulder and edge are driven by the actual spatial distribution of residual stress within the die, rather than by a pre-set uniform expansion ratio or a human-experienced coefficient. This constitutes the key technical basis for distinguishing the die from corrections based solely on instantaneous die-off expansion.

[0141] (iv) S4. Transient solution of large deformation of finite strain hyperelastic-viscoelastic in free relaxation region and determination of stable section by dual criteria

[0142] The S4 core does not simply "extend" the free relaxation region; instead, it uses the stress memory of material points as the initial state and continuously calculates the large deformation and time-dependent stress release of the rubber compound after it leaves the die. It then uses both the internal stress state and the external morphology to determine whether stability has been achieved. This avoids prematurely terminating the calculation when only observing small short-term changes in the cross-section.

[0143] The rubber compound in the free relaxation region is equivalent to an approximately incompressible finite-strain hyperelastic-viscoelastic continuum. The deformation gradient at a point in the material can be written as:

[0144] ;

[0145] In the formula, For deformation gradient; It is a second-order unit tensor; The displacement vector of the material point; This is a deformation gradient determinant used to characterize local volume changes. For uncured tread rubber, a higher bulk modulus is used within the computational timescale of this invention. Keep it close to 1 to reflect the approximately incompressible characteristic.

[0146] A preferred embodiment employs a second-order Ogden hyperelastic model, whose strain energy density function is:

[0147] ;

[0148] In the formula, Strain energy per unit reference volume; This represents the number of terms in the Ogden model. For the first Shear modulus coefficient; For the first strain hardening coefficient; , , These are the three principal elongation ratios after removing volumetric deformation; Bulk modulus; The meaning is the same as before. In one specific implementation, The first shear modulus is set to 2, the first strain hardening coefficient is set to 0.62 MPa, the first strain hardening coefficient is set to 1.35, the second shear modulus is set to 0.18 MPa, the second strain hardening coefficient is set to 5.0, and the bulk modulus is set to 120 MPa.

[0149] The time-dependent viscoelastic response is represented using a Prony series or an equivalent generalized Maxwell network. In the Prony representation:

[0150] ;

[0151] In the formula, For temperature Next experience time The subsequent shear relaxation modulus; Instantaneous shear modulus; This is the long-term equilibrium modulus coefficient; The number of Prony terms; For the first Normalized modulus coefficients of each Prony term; This corresponds to the time constant; The temperature shift factor defined for S2. In one specific implementation, The value is 3. The three normalized modulus coefficients are 0.18, 0.27 and 0.43 respectively, corresponding to time constants of 0.15s, 1.6s and 12.5s respectively. The long-term equilibrium modulus coefficient is 0.12. The sum of each normalized modulus coefficient and the long-term equilibrium modulus coefficient is 1.

[0152] The outer surface of the free relaxation region is set as a free surface, without any displacement constraints that hinder local normal deformation. The entire rubber compound is subjected to a gravitational acceleration of 9.81 m / s². In actual production lines, if a conveyor belt supports the compound, a non-penetrating contact can be established between the bottom surface of the rubber compound and the conveyor belt, with a friction coefficient of 0.15–0.30, preferably 0.20. If a suspended extrusion experiment is used, no bottom contact is provided. When a traction device is present downstream, it is preferable to apply a traction speed according to the actual process; in one embodiment, the ratio of the traction speed to the actual average extrusion line speed is 1.02; alternatively, an equivalent end traction stress can be used, in one embodiment, it is 0.015 MPa. For verification conditions without traction, the end is set as a free end, and overall rigid body drift is eliminated only through weak constraints.

[0153] S4 can be solved using implicit transient, explicit dynamic, or quasi-static time integration. One specific implementation uses implicit transient solution, with an initial time step of 0.002 s. After five consecutive time steps converge normally, the next time step increases by a factor of 1.25, with a maximum time step not exceeding 0.1 s. The first 1 s is the rapid stress release phase, and the time step is preferably no greater than 0.02 s. After entering the slow relaxation phase, it can be increased to 0.05–0.1 s. To suppress high-frequency numerical oscillations unrelated to the relaxation of the actual material, an equivalent critical damping ratio of 0.03 can be used. This numerical damping is only used to improve numerical stability and does not participate in the determination of the main relaxation time of the rubber compound.

[0154] Because the shoulders and edges may experience significant local rotation and warping, mesh quality needs to be checked at each time step. In one implementation, adaptive remeshing is triggered when any element's normalized Jacobian value is below 0.25 or its aspect ratio is greater than 8. After remeshing, stress, temperature, and displacement fields are projected onto the new mesh, and field transfer errors are checked. The total mass transfer error is controlled within 0.1%, and the total internal energy transfer error and the region-average equivalent stress transfer error are each controlled within 1%; if these thresholds are exceeded, the local mesh size is reduced and the transfer is repeated.

[0155] The first criterion for a stable cross section is the equivalent stress attenuation criterion. The equivalent stress is expressed in the second invariant form of the deviatoric stress tensor:

[0156] ;

[0157] In the formula, Equivalent stress; This is the deviatoric stress tensor obtained by subtracting hydrostatic stress from the total stress tensor; "" indicates the second-order tensor double dot product. To avoid the overall average value masking the local residual stress in the shoulder or edge, this invention calculates the volume-weighted average equivalent stress for the overall region and the pre-specified key regions of the left and right shoulders and left and right edges, and defines the regional residual ratio:

[0158] ;

[0159] In the formula, For the region At any moment The equivalent residual stress ratio; The current volume-weighted average equivalent stress in this region; This corresponds to the initial value. A preferred embodiment requires that the entire area and all preset shoulder and edge key areas satisfy the following. That is, the average equivalent stress decreases to less than 5% of the corresponding initial value.

[0160] The second criterion is the cross-sectional shape convergence criterion. Periodic sampling is performed on the total width, maximum thickness, and coordinates of multiple preset contour points. The maximum shape change between adjacent sampling times is defined as:

[0161] ;

[0162] In the formula, For a moment The maximum morphological change; , These are the current total width and the maximum thickness, respectively. The morphology sampling interval; For the first The positions of preset contour points at the current time. In one specific implementation, the topography sampling interval is 0.2s, requiring... The diameter should be less than 0.05 mm and maintained continuously for at least 1 second.

[0163] A section is considered stable only when both the aforementioned equivalent stress attenuation criterion and section morphology convergence criterion are met. The reason for using dual criteria is that while the external dimensions of the tread may change little over a short period, high residual viscoelastic stress may still exist at the shoulder and edges; using only morphological changes would prematurely terminate the calculation. Conversely, the overall average stress may be low, but a local point at a certain edge may still be slowly curling; using only overall stress cannot represent the actual stable morphology. Therefore, simultaneously examining the local stress state and geometric changes improves the physical reliability of the stable section determination.

[0164] The purpose of the free relaxation region length is to provide sufficient space for large deformation calculations, rather than fixing the termination time to a single master relaxation time. If the material does not meet the dual criteria when it reaches the end of the free relaxation region, a moving window element is activated to continue advancing the demolding time coordinates. One possible implementation is to retain the demolding time and internal state of the material section after it reaches the end and continue time integration, while looping the upstream mesh, which has entered an approximately stable translational state, back to the inlet. Another possible implementation is to fix the body label of the end material section and continue solving for the internal stress and local morphology of that section. By using a moving window, candidate free relaxation regions do not need to be physically modeled until all materials reach the maximum transport distance of 5% residual stress, and stable results with long demolding times can still be obtained continuously.

[0165] For example, when the actual average extrusion line speed is 0.12 m / s, the main relaxation time is 12.5 s, the minimum exit feature size is 4 mm, and the geometric length coefficient is 10, the free relaxation domain length is taken as 1.50 m according to S1. The material reaches the 1.50 m position, corresponding to a nominal die departure time of 12.5 s; if the shoulder or edge still does not meet the 5% stress decay condition at this time, the window is moved to continue the calculation. In a verification case, at 41.6 s, the overall structure, the left and right shoulders, and the left and right edges all meet the stress decay threshold, and the maximum change in the profile within 1 second is 0.032 mm, at which point a stable section is output. This processing is consistent with... Figure 4 This is consistent with the phenomenon shown that "morphological changes continue to occur after export".

[0166] (v) S5. Source tracing of local deviations in stable sections and closed-loop updating of die parameters

[0167] S5 does not simply address "re-testing the mold after predicting failure," but rather determines which deviation at the stable section corresponds to which velocity or stress anomaly at the mold exit, and selects local mold parameters with a clear physical effect for point-to-point correction. This step enables... Figure 1 The feedback loop in the process is transformed from empirical trial and error into a parameter closed loop with positional correspondence and sensitivity constraints.

[0168] First, the stable section obtained from S4 is rigidly registered with the target design section. Preferably, the center point of the tread bottom is used as the reference point, and the tread bottom reference line is used as the lateral reference line. Only overall translation and rigid body rotation are eliminated; non-rigid registration is avoided to prevent the artificial elimination of real local topographic errors during the registration process. Both the stable section and the target section are sampled using the same normalized arc length coordinates as the S3 reference plane. In one embodiment, 401 contour sampling points are used, with secondary densification in the left and right shoulder and left and right edge regions, ensuring that the distance between adjacent sampling points in key areas is no greater than 0.5 mm.

[0169] No. The normal deviation of each contour point is expressed as:

[0170] ;

[0171] In the formula, For the first Normal deviation of each contour point; To predict the coordinates of the corresponding points on the stable section; The coordinates of the corresponding points of the target cross-section; The unit normal vector of the target contour at that point. In addition to the contour point deviation, the total width deviation, maximum thickness deviation, left and right shoulder upturn deviation, bottom indentation deviation, and left and right edge curl angle deviation are also extracted to form a set of local deviations.

[0172] The velocity deviation at the same arc length position at the die exit is defined as:

[0173] ;

[0174] In the formula, Position of arc length Normal velocity deviation; This is the local velocity vector at that location; This is the normal direction for the exit at this location; Let be the average extrusion line speed at the outlet. The principal gradient of viscoelastic stress is defined as:

[0175] ;

[0176] In the formula, The principal gradient of viscoelastic stress; This is the principal stress value with the largest absolute value at that location; The coordinates are the arc length coordinates of the exit point.

[0177] According to the unified normalized arc length coordinates, , and Corresponding to the same location. To reduce the noise impact of individual grid points, local window averaging is used for velocity deviation and stress principal value gradient. In one implementation, the local window width is 3% of the total exit arc length, the window uses Gaussian weighting, the weight coefficient at the window center is 1, and the weight at the window boundary is decayed to 0.20. The normalized arc lengths 0–0.10 and 0.90–1.00 can be defined as the edge region, 0.10–0.30 and 0.70–0.90 as the shoulder region, and 0.30–0.70 as the center region.

[0178] When a region exhibits a positive profile deviation and the corresponding normal velocity is higher than the average velocity, it indicates that the local flow rate is likely too high. The primary solution is to reduce the die outlet gap at that location, decrease the effective flow area corresponding to the throttling width, or increase the local throttling length. Conversely, when a region exhibits a negative profile deviation and the outlet velocity is lower than the average velocity, the local throttling size should be adjusted in the opposite direction. The change in local throttling size should preferably be limited to within 0.30 mm to avoid causing a sudden change in the flow path with a single correction.

[0179] When the correlation between profile deviation and velocity deviation in a certain region is weak, but the principal gradient of viscoelastic stress in that region is significantly higher than that in adjacent regions, it is preferentially determined that the uneven residual stress is caused by local high shear or abrupt change in flow direction. In this case, the outlet gap should not be simply changed. Instead, the outlet transition fillet at the corresponding position should be increased or the local throttling structure should be smoothly transitioned towards the outlet direction. In one embodiment, the amount of single outlet transition fillet correction is limited to within 0.50 mm.

[0180] To further obtain the "correction amount" from the "correction direction," a finite difference method can be used to establish the sensitivity matrix for the parameterized aperture. For the ... Individual Mouth Mold Parameters Applying positive and negative perturbations respectively, we obtain the first... Sensitivity of the stability section deviation index to this parameter:

[0181] ;

[0182] In the formula, For the sensitivity matrix, the first Line 1 Column elements; For the first One stable section deviation index; For the first Each die parameter; The corresponding disturbance amounts are as follows. In one embodiment, the local throttling size disturbance amount is 0.20 mm, and the outlet transition fillet disturbance amount is 0.50 mm.

[0183] The parameter update amount can be determined using regularized weighted least squares:

[0184] ;

[0185] In the formula, Update the vector for the protrusion parameters; This is the iteration step size coefficient; This is the sensitivity matrix; This is the deviation weight matrix; The regularization coefficient is used. It is the identity matrix; This is the stable section deviation vector. In one specific implementation, Take 0.60, The weights are set to 0.05; the total width weight is 0.20, the maximum thickness weight is 0.20, the combined weight of the left and right shoulder upturn is 0.25, the weight of the bottom indentation is 0.15, and the combined weight of the left and right edge curl angles is 0.20. If the calculated update amount of a certain parameter exceeds the aforementioned single limit value, then the update amounts of all parameters are reduced by the same proportion to maintain the update direction relationship between different die parameters.

[0186] Flow uniformity is evaluated using the outlet velocity variation coefficient:

[0187] ;

[0188] In the formula, As an indicator of uneven export speed; The cross-sectional area of ​​the die exit; This represents the component of the local velocity in the exit normal direction; This refers to the actual average extrusion line speed at the outlet. In one specific embodiment, when the total width error is no greater than 1.0 mm, the maximum thickness error is no greater than 0.5 mm, the shoulder upturn error is no greater than 0.3 mm, the bottom indentation error is no greater than 0.3 mm, and the edge curling angle error is no greater than 1.0°, and... If the value is not greater than 0.05, both dimensional deviation and flow uniformity are deemed to meet the requirements; otherwise, they are treated as... Figure 1 The feedback loop updates the local throttling dimensions of the die and the outlet transition fillet, refreshes the S1 model, and re-executes the calculation.

[0189] Through the above S5, the present invention can distinguish between "cross-sectional dimension expansion caused by local high flow rate" and "exit post-relaxation morphology distortion caused by local high stress gradient". The former type of problem is mainly solved by adjusting the local throttling size, while the latter type of problem is solved primarily by adjusting the outlet transition fillet and smooth transition structure. Therefore, it can reduce the uncertainty of die design caused by repeated trial and error based solely on the final size.

[0190] IV. Specific Application and Implementation Methods

[0191] The target of this application is a mold for a passenger car tire tread. The target stable section has a total width of 278.0 mm and a maximum thickness of 18.0 mm. The upward curve of the left and right shoulders is 1.0 mm, the bottom is 0.5 mm indented, and the curl angle of the left and right edges is 4.0°.

[0192] The inlet temperature of the rubber compound is set to 105℃, the die wall temperature to 95℃, the preset average extrusion line speed to 0.12 m / s, and the ratio of traction speed to actual average extrusion line speed to 1.02. The principal relaxation time of the rubber compound at the 95℃ reference temperature is 12.5 s, the minimum exit feature size is 4 mm, and the geometric length coefficient is 10. The candidate length S1 is determined by the greater of the product of 0.12 m / s and 12.5 s, and the lower geometric limit of 40 mm, and is therefore chosen as 1.50 m. After the solution for S2 stabilizes, the actual average extrusion line speed is calculated by integrating the exit speed field. When there is a difference between this actual value and the preset process value, the length of the free relaxation region is recalculated and updated using the actual value.

[0193] The die uses an exponential PTT model, with a viscous matrix viscosity of 2000 Pa·s, a polymer viscosity coefficient of 16500 Pa·s, a PTT reference relaxation time of 12.5 s, a PTT elongation parameter of 0.12, and a compound density of 1120 kg / m³. 3 It has a specific heat capacity of 1900 J / (kg·K) and a thermal conductivity of 0.28 W / (m·K). The external die adopts a second-order Ogden model and a three-term Prony series. The Ogden parameters are 0.62 MPa, 1.35, 0.18 MPa and 5.0, respectively, with a bulk modulus of 120 MPa. The Prony normalized modulus coefficients are 0.18, 0.27 and 0.43, corresponding to time constants of 0.15 s, 1.6 s and 12.5 s, respectively, and a long-term equilibrium modulus coefficient of 0.12.

[0194] S3 completed Figure 3 After mapping at the outlet interface, the relative error of the final mass flow rate is controlled to be no greater than 0.5%, the relative error of the convective enthalpy flow to be no greater than 0.5%, and the relative errors of the three invariants of the viscoelastic stress tensor to be no greater than 0.01%. The continuous material injection time step is set to 0.02s. The outer surface of the free relaxation domain is a free surface, the gravitational acceleration is set to 9.81m / s², and the traction velocity ratio is set to 1.02. S4 adopts implicit transient solution with an initial time step of 0.002s, a maximum time step of 0.1s, and a numerical damping ratio of 0.03; remeshing is performed when the normalized Jacobian is less than 0.25 or the element aspect ratio is greater than 8.

[0195] If the material still fails to meet the 5% stress attenuation threshold and the 0.05mm morphological change threshold for all overall and critical areas when it reaches the 1.50m end, the calculation is continued using a moving window. In a corresponding working condition, when the demolding time advances to 41.6s, the equivalent residual stress ratio of the overall structure, left and right shoulders, and left and right edges is all below 5%, and the maximum contour change within 1 second is 0.032mm. At this point, the stable section is determined.

[0196] S5 according to Figure 1 The feedback process traces the deviations at the stable cross-section. When the outlet velocity deviation in the right shoulder region is positive and the principal gradient of viscoelastic stress is 1.64 times that of the adjacent central region, it is determined that the local flow rate in the right shoulder is too high and there is significant stress concentration. The corresponding local throttling gap is reduced by 0.24 mm, and the outlet transition radius at that location is increased from 2.0 mm to 2.5 mm, and S1 to S5 are executed again. These correction amounts are all lower than the single correction limit set in this embodiment, which can avoid the formation of new abrupt flow channels.

[0197] As can be seen from the complete implementation described above, Figure 3 It provides the basic composite domains and interfaces required for S1, S2, and S3. Figure 4 This reflects the physical significance of S3 domain switching and S4 long-term relaxation. Figure 2 This reflects the data flow between various software modules. Figure 1 This forms a closed loop encompassing the modeling, solving, stability determination, and deviation tracing described above. Therefore, those skilled in the art can implement this invention according to the given material parameters, mesh parameters, mapping error limits, time steps, and die parameter update rules, without needing to create new computer theories or empirical model-correction rules.

[0198] The above embodiments are for illustrative purposes only and are not intended to limit the scope of protection of this invention. For different tire specifications, different tread compound formulations, or different extrusion equipment, the reference temperature, master relaxation time, mesh size, local window width, material parameters, and optimization weights can be adjusted while maintaining the technical concept of this invention; the above parameter adjustments should be determined through corresponding material tests and actual process conditions. Any implementation method that uses exit full-field constraint mapping to continuously enter the finite strain large deformation relaxation solution, uses stress-morphology dual criteria to determine the stable section, and then uses unified position coordinates to backtrack local deviations to the corresponding structural parameters of the die can be implemented according to the disclosure of this invention.

[0199] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0200] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0201] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0202] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0203] In a typical configuration, a computing device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.

[0204] Memory may include non-persistent memory in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.

[0205] Computer-readable media includes both permanent and non-permanent, removable and non-removable media that can store information using any method or technology. Information can be computer-readable instructions, data structures, modules of programs, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include transient computer-readable media, such as modulated data signals and carrier waves.

Claims

1. A method for predicting tread dimensional accuracy including an exit viscoelastic stress relaxation process, characterized in that, Includes the following steps: S1. Obtain the die structure parameters, extrusion process parameters, and rubber material parameters, establish the flow channel domain inside the die and the free relaxation domain outside the candidate die, and determine the length of the candidate free relaxation domain based on the preset average extrusion line speed, the main relaxation time of the rubber material, and the minimum characteristic size of the die exit. S2, perform non-isothermal viscoelastic transient flow solution in the flow channel domain inside the die to obtain the velocity field, temperature field and viscoelastic stress tensor field of the die exit section at the same time scale. Calculate the actual average extrusion linear velocity at the die exit based on the velocity field and correct the length of the free relaxation region accordingly. S3. Establish a local orthogonal basis at the die exit section, map the velocity field with mass flow rate consistency as constraint, and the temperature field with convective enthalpy consistency as constraint, and map the viscoelastic stress tensor field under the condition that the principal stress direction of the viscoelastic stress tensor is continuous and the error of the tensor invariant does not exceed the preset threshold, so as to form the initial motion state, initial temperature field and initial stress state of the rubber in the free relaxation domain. S4. The rubber material in the free relaxation domain is equivalent to a finite strain hyperelastic-viscoelastic continuum. Under the action of gravity and the initial motion state, initial temperature field and initial stress state, large deformation transient relaxation solution is performed until the equivalent stress attenuation criterion and the cross-sectional shape convergence criterion of the whole region and the preset key region are simultaneously satisfied to obtain a stable cross section. S5 registers the stable section with the target design section, and traces the local deviation back to the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die outlet according to the unified arc length coordinate. Based on this, it updates the local throttling size and outlet transition fillet at the corresponding position and returns to S1 until the size deviation and flow uniformity both meet the preset threshold. Then, it outputs the predicted size and die parameters.

2. The method for predicting tire tread size accuracy according to claim 1, characterized in that, In steps S1 and S2, the length of the free relaxation region is determined as follows: during the first iteration, the candidate free relaxation region length is determined based on the preset average extrusion line speed of the extrusion process. After step S2 is completed, the actual average extrusion line speed at the die exit is determined based on the die exit speed field, and the length of the free relaxation region is corrected. And / or, in step S2, the PTT viscoelastic constitutive model or the Giesekus viscoelastic constitutive model is used to solve the non-isothermal transient flow of the rubber compound; the steady-state solution is first used as the initial state of the transient solution. When the relative changes of the outlet mass flow rate, the outlet average temperature, and the outlet average viscoelastic stress within a continuous 2s time window are all no greater than 0.5%, the corresponding time window is determined as the extrusion stabilization time window, and the coordinates, velocity vector, temperature, and six independent components of the second-order viscoelastic stress tensor of each sampling position at the die outlet section are obtained at the same output time.

3. The method for predicting tire tread size accuracy according to claim 1, characterized in that, Step S3 includes: A reference surface with uniform parameterized coordinates is established at the die exit section, and a local orthogonal basis consisting of two tangential unit vectors and one normal unit vector along the extrusion direction is established at each sampling position of the reference surface. The viscoelastic stress tensor output from the flow channel domain inside the die is transformed to the local orthogonal basis and eigenvalue decomposition is performed to obtain the principal stress values ​​and principal stress directions. The principal stress values ​​are interpolated to the inlet section of the free relaxation domain according to the unified parameterized coordinates, and the principal stress directions of adjacent sampling positions are kept continuous by minimum rotation projection. Tensor invariant correction is performed on the reconstructed viscoelastic stress tensor to ensure that the relative errors of the first, second, and third invariants before and after mapping are all no greater than 0.01%. Mass flow rate correction is applied to the mapped velocity field to ensure that the relative error of mass flow rate between the outlet side of the flow channel domain and the inlet side of the free relaxation domain inside the die is no greater than 0.5%; convective enthalpy correction is applied to the mapped temperature field to ensure that the relative error of convective enthalpy between the two sides is no greater than 0.5%. The mapped velocity field is used to establish the initial motion state of the rubber compound in the free relaxation region, the mapped temperature field is used as the initial temperature field, and the mapped viscoelastic stress tensor is assigned to the material points near the inlet section of the free relaxation region to form the initial stress state. During continuous extrusion, the correspondence between the die exit time and the distance along the extrusion direction is established according to the average extrusion linear velocity at the actual die exit, and the initial motion state, initial temperature field and initial stress state are continuously assigned as the material enters the free relaxation region.

4. The method for predicting tire tread size accuracy according to claim 1, characterized in that, In step S4, the finite strain hyperelastic-viscoelastic continuum adopts a combined model of the Ogden hyperelastic model and the generalized Maxwell viscoelastic network, or a combined model of the Ogden hyperelastic model and the Prony series viscoelastic term; the outer surface of the free relaxation region is set as a free surface, a gravity load is applied to the rubber material, and an end traction load is applied or a conveyor belt support contact is set according to the actual production state; when the mesh quality is lower than the preset mesh quality threshold, adaptive remeshing is implemented, and the stress field, temperature field and displacement field are transferred to the updated mesh; And / or, in step S4, the equivalent stress attenuation criterion is: the average equivalent stress of the entire region and the pre-specified shoulder and edge key regions are all attenuated to less than 5% of their respective initial average equivalent stress; And / or, in step S4, the convergence criterion for the cross-sectional shape is: the change in the total width, the maximum thickness, and the coordinates of each preset contour point at adjacent sampling times is less than 0.05 mm, and is maintained continuously for no less than 1 second; And / or, in step S4, if the rubber material moves to the end of the free relaxation domain and still does not simultaneously satisfy the equivalent stress decay criterion and the cross-sectional shape convergence criterion, the moving calculation window is used to continue advancing the mold departure time coordinate until the two criteria are simultaneously satisfied and a stable cross-section is output.

5. The method for predicting tire tread size accuracy according to claim 1, characterized in that, In step S5, the center point of the tread bottom is used as the reference point and the tread bottom reference line is used as the lateral reference line. The stable section and the target design section are rigidly registered to obtain a set of local deviations, including total width deviation, maximum thickness deviation, shoulder upturn deviation, bottom indentation deviation, and edge curling angle deviation. According to the same normalized arc length coordinates between the die exit section and the stable section, the set of local deviations is correlated with the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die exit. And / or, in step S5, when the local profile deviation and the corresponding position velocity deviation have a cyclic relationship, the local velocity deviation is reduced by adjusting the local throttling gap, throttling width, or throttling length at the corresponding position; when the correlation between the local profile deviation and the velocity deviation is lower than the preset correlation threshold and the viscoelastic stress principal gradient is higher than the adjacent region, the viscoelastic stress principal gradient is reduced by increasing the outlet transition fillet at the corresponding position or by making the local throttling structure smoothly transition towards the outlet direction. And / or, in step S5, for the parameterized die, a sensitivity relationship between the stable cross-sectional deviation and the die parameters is established by applying a disturbance to the local throttling size and the outlet transition fillet, and the die parameter update amount is determined according to the sensitivity relationship.

6. A tread dimensional accuracy prediction system incorporating an exit viscoelastic stress relaxation process, characterized in that, A method for implementing the method of any one of claims 1 to 5, comprising a processor and a memory, wherein the memory stores the following modules executed by the processor: The model building module is used to obtain die structure parameters, extrusion process parameters and rubber material parameters, establish the flow channel domain inside the die and the free relaxation domain outside the candidate die, and determine the length of the candidate free relaxation domain based on the preset average extrusion line speed, the main relaxation time of the rubber material and the minimum feature size of the die exit. The in-die flow solution module is used to solve non-isothermal viscoelastic transient flows, obtain the velocity field, temperature field and viscoelastic stress tensor field of the die exit section at the same time scale, and determine the actual die exit average extrusion linear velocity based on the velocity field to correct the length of the free relaxation region. The data mapping and domain switching module is used to map the velocity field, temperature field and viscoelastic stress tensor field with mass flow rate, convective enthalpy flow and principal stress direction and tensor invariants of viscoelastic stress tensor as constraints, respectively, to construct the initial motion state, initial temperature field and initial stress state of the rubber compound in the free relaxation domain. The free relaxation domain solid solution module is used to solve the large deformation transient relaxation of the rubber material as an equivalent finite strain hyperelastic-viscoelastic continuum. It outputs a stable cross section when the equivalent stress attenuation criterion and the cross section shape convergence criterion are simultaneously satisfied in the overall region and the preset key region. The deviation tracing and die optimization module is used to trace back the local deviation of the stable section to the velocity deviation and viscoelastic stress principal gradient at the corresponding position of the die exit according to the unified arc length coordinate, update the local throttling size and exit transition fillet at the corresponding position, and trigger the model building module to recalculate iteratively.

7. The tread size accuracy prediction system according to claim 6, characterized in that, The data mapping and domain switching module includes a reference surface construction unit, a coordinate transformation unit, a tensor eigenvalue decomposition unit, a constraint interpolation unit, a principal direction projection unit, a tensor invariant correction unit, and an initial state construction unit. The reference surface construction unit is used to establish a unified parameterized coordinate between the die exit section and the free relaxation domain inlet section; The coordinate transformation unit is used to transform the viscoelastic stress tensor into a local orthogonal basis consisting of two tangential directions and one normal direction; The tensor eigenvalue decomposition unit is used to obtain the principal stress values ​​and principal stress directions of the viscoelastic stress tensor. The constrained interpolation unit is used to interpolate the velocity field, temperature field, and principal stress values, and to correct the velocity field and temperature field based on the mass flow rate and convective enthalpy, respectively. The main direction projection unit is used to maintain the continuity of the principal stress direction at adjacent positions through minimum rotational projection. The tensor invariant correction unit is used to correct the reconstructed viscoelastic stress tensor so that the error between its tensor invariant and the tensor invariant of the corresponding tensor before mapping does not exceed a preset threshold. The initial state construction unit is used to construct the initial motion state, initial temperature field, and initial stress state of the rubber compound within the free relaxation domain based on the mapping results.

8. The tread size accuracy prediction system according to claim 6, characterized in that, The free relaxation domain solid solution module includes a combined constitutive element, an adaptive remesh element, a moving window element, and a dual-criteria termination element; The combined constitutive unit is used to configure a combined model of the Ogden hyperelastic model and the generalized Maxwell viscoelastic network, or to configure a combined model of the Ogden hyperelastic model and the Prony series viscoelastic term. The adaptive remeshing element is used to re-mesh when the mesh quality is lower than a preset mesh quality threshold, and to transfer the stress field, temperature field and displacement field to the updated mesh. The moving window unit is used to continue advancing the demolding time coordinate of the corresponding material section when the rubber reaches the end of the free relaxation region but has not yet met the stability condition; The dual-criteria termination unit is used to output a stable cross section when the average equivalent stress in the overall area and the preset shoulder and edge key areas is less than 5% of their respective initial values, and the changes in the total width, maximum thickness and coordinates of each preset contour point are all less than 0.05mm and are maintained continuously for no less than 1s.

9. A computer-readable storage medium having a computer program or instructions stored thereon, characterized in that, When the computer program or instructions are executed by the processor, the processor performs the steps of the tread size accuracy prediction method according to any one of claims 1 to 5.

10. A computer program product, comprising a computer program or instructions, characterized in that, When the computer program or instructions are executed by the processor, the processor performs the steps of the tread size accuracy prediction method according to any one of claims 1 to 5.

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