EVA foam foaming ratio uniformity closed-loop control method based on thickness feedback

Abstract

The application discloses a thickness feedback-based EVA foam uniformity closed-loop control method, and relates to the technical field of plastic foam products.The method synchronously collects the base material reference thickness distribution before foaming and the real-time thickness distribution of the foam after foaming, and generates an equivalent foaming ratio distribution; the equivalent foaming ratio distribution is analyzed in time and space gradient, an equipotential line skeleton map is constructed, and spatial non-uniformity is extracted; a foaming ratio strain rate field is constructed by tracking the time change rate of the area of the closed region of the equipotential line; the topological tension field is used to drive the deformation simulation of the skeleton, the predicted spatial non-uniformity after the future control period is previewed, and the pre-adjustment control instruction for each independent adjustment section is generated in advance; the topological tension field is cyclically self-calibrated through skeleton deformation residual feedback, a closed-loop control with continuously converging prediction accuracy is formed, and the application effectively reduces the product density fluctuation, and improves the mechanical property consistency and yield rate of the foam.

Description

Technical Field

[0001] This invention relates to the field of plastic foam technology, and in particular to a closed-loop control method for the uniformity of EVA foam expansion ratio based on thickness feedback. Background Technology

[0002] EVA foam, as a new type of environmentally friendly packaging material, is widely used in the protective packaging of electronic appliances, communication equipment, and precision instruments due to its excellent water and moisture resistance, shock absorption, and plasticity. EVA foam is typically manufactured using a continuous foaming process, in which EVA substrate containing a foaming agent is continuously fed into foaming equipment and foamed under specific temperature and pressure conditions. The uniformity of the foaming ratio is one of the core indicators for evaluating the quality of foam products. Uneven distribution of the foaming ratio along the width and travel directions directly affects the density consistency, mechanical property uniformity, and subsequent processing adaptability of the foam.

[0003] Currently, the industry mainly relies on a combination of offline sampling and manual adjustment to control the uniformity of foaming ratio. This approach suffers from shortcomings such as detection lag, untimely adjustment, and difficulty in covering continuous detection across the entire width. Some improved solutions introduce online thickness detection methods, calculating the foaming ratio by comparing the thickness changes before and after foaming, and adjusting the foaming process parameters accordingly. However, such solutions still face the following technical bottlenecks in practical applications: First, the thickness of the substrate itself fluctuates, and the thickness difference measured after foaming inevitably includes interference from the background difference in the thickness of the substrate, causing the calculated foaming ratio to fail to truly reflect the expansion capacity of the material imparted by the foaming process itself. Second, existing feedback regulation mostly adopts threshold triggering or PID deviation correction control, whose control actions are based only on the ratio deviation at the current or historical time, lacking the ability to predict the evolution trend of the ratio non-uniformity in the future time dimension, resulting in the regulation always being in a "lag-following" state, making it difficult to implement preventive intervention before the fluctuation expands. Third, the analysis of the spatial distribution data of the foaming ratio in existing solutions remains at the level of statistical dispersion, without deeply exploring the spatial structural characteristics of the ratio distribution and the evolution law of this spatial structure in the time series, causing the control strategy to be disconnected from the physical nature of the foaming process, and limiting the regulation accuracy and robustness. The aforementioned problems have long constrained the improvement of EVA foam continuous foaming production towards higher uniformity and consistency. There is an urgent need for a closed-loop control method that can remove the background interference of incoming materials, has the ability to predict non-uniform states, and whose control logic is consistent with the physical mechanism of the foaming process. Summary of the Invention

[0004] This invention overcomes the shortcomings of the prior art and provides a closed-loop control method for the uniformity of EVA foam expansion ratio based on thickness feedback.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows: This invention discloses a closed-loop control method for the uniformity of EVA foam expansion ratio based on thickness feedback, comprising the following steps: S1. During the continuous foaming process, the baseline thickness distribution of the unfoamed substrate before foaming and the real-time thickness distribution of the foam after foaming are collected simultaneously to form a thickness distribution data pair. S2. Perform a point-by-point correlation calculation between the real-time thickness distribution and the reference thickness distribution to generate an equivalent foaming ratio distribution; S3. Perform spatiotemporal gradient analysis on the equivalent foaming ratio distribution, extract the spatial non-uniformity of the equivalent foaming ratio distribution, and construct the ratio strain rate field in the foaming process based on the change of the spatial non-uniformity in the time series. S4. Calculate the predicted spatial non-uniformity that will occur after a preset control cycle using the aforementioned strain rate field. S5. When the predicted spatial non-uniformity does not fall within the preset uniformity allowable range, a pre-adjustment instruction to adjust the foaming process parameters in advance is generated. S6. Repeat the above steps to update the spatial non-uniformity and the expansion ratio field with the newly generated equivalent foaming ratio distribution after adjustment, and then update the predicted spatial non-uniformity and the pre-control command to form a closed-loop control based on the expansion ratio field calculation.

[0006] This invention addresses the technical deficiencies in the prior art and offers the following advantages: By synchronously collecting the thickness distribution before and after foaming and generating an equivalent foaming ratio distribution based on the difference in the thickness of the incoming material, this invention eliminates the interference of inherent substrate unevenness on foaming ratio detection, ensuring the ratio distribution truly reflects the expansion capacity of the foaming process itself. Based on this, an equipotential line skeleton map is constructed, and the directional derivative field of the expansion stress is extracted to characterize the spatial structure of the non-uniform ratio. Furthermore, a ratio strain rate field is constructed by tracking the time change rate of the area of ​​the equipotential line closed region, capturing the evolution trend of the non-uniform state. The topological tension field is used to drive deformation simulation, predicting the ratio distribution pattern after future control cycles, achieving forward-looking non-uniformity prediction. Pre-control commands are generated for each independent adjustment segment based on the prediction results, completing preventative intervention before the actual expansion of fluctuations. Finally, the topological tension field is cyclically self-calibrated through skeleton deformation residual feedback, ensuring the prediction accuracy continuously converges with production. By upgrading the foaming ratio control from a passive, delayed response to an active, predictive intervention, the control accuracy and robustness of the ratio uniformity during continuous foaming are improved, effectively reducing product density fluctuations and enhancing the consistency of foam mechanical properties and yield. Attached Figure Description

[0007] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other embodiments can be obtained from these drawings without creative effort.

[0008] Figure 1 This is a flowchart of the first method of the present invention; Figure 2 This is a flowchart of the second method of the present invention. Detailed Implementation

[0009] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other.

[0010] like Figure 1 , Figure 2 As shown, this invention discloses a closed-loop control method for the uniformity of EVA foam expansion ratio based on thickness feedback, comprising the following steps: S1. During continuous foaming, the baseline thickness distribution of the unfoamed substrate before foaming and the real-time thickness distribution of the foam after foaming are simultaneously collected to form a thickness distribution data pair. S2. The real-time thickness distribution and the baseline thickness distribution are correlated point-by-point to generate an equivalent foaming ratio distribution. S3. The equivalent foaming ratio distribution is analyzed using spatiotemporal gradient analysis to extract its spatial non-uniformity. Based on the change of the spatial non-uniformity over time, a ratio strain rate field is constructed during the foaming process. S4. The ratio strain rate field is used to calculate the predicted spatial non-uniformity that will appear after a preset control cycle. S5. When the predicted spatial non-uniformity does not fall within the preset uniformity allowable range, a pre-adjustment command is generated to adjust the foaming process parameters in advance. S6. The above steps are executed iteratively to update the spatial non-uniformity and ratio strain rate field with the newly generated equivalent foaming ratio distribution after adjustment, thereby updating the predicted spatial non-uniformity and the pre-adjustment command, forming a closed-loop control based on the ratio strain rate field calculation.

[0011] Specifically, S1 is: Along the substrate transport path, a reference measurement position before foaming and a real-time measurement position after foaming are set. A first synchronous multi-point thickness measurement component, composed of multiple point laser sensors, is configured at the reference measurement position. A second synchronous multi-point thickness measurement component, with the same spatial resolution and corresponding layout as the first synchronous multi-point thickness measurement component, is configured at the real-time measurement position. Simultaneously, an encoder that rotates synchronously with the substrate is installed along the substrate's travel direction to generate trigger pulses with equal spatial intervals. Based on the trigger pulses, a series of virtual trigger marks are generated along the substrate's width direction. Each virtual trigger mark corresponds to a fixed width position and moves synchronously with the substrate, thus forming a uniquely numbered virtual measurement grid across the entire width. When any of the virtual trigger marks reaches the reference measurement position, the virtual trigger mark triggers the first synchronous multi-point thickness measurement component to acquire a thickness data frame. The thickness data frame undergoes intra-frame interpolation to obtain a reference thickness value at a grid point, and the reference thickness value is assigned to the number of the virtual trigger mark. All marks of the virtual measurement grid are traversed to obtain the reference thickness distribution of the unfoamed substrate. When the same virtual trigger mark continues to travel to the real-time measurement position, the second synchronous multi-point thickness measurement component is triggered to acquire another thickness data frame. The same intra-frame interpolation is performed on the other thickness data frame to obtain a real-time thickness value at a grid point, and the real-time thickness value is assigned to the inherent number of the same virtual trigger mark. All marks of the virtual measurement grid are traversed to obtain the real-time thickness distribution of the foamed material. The reference thickness distribution and the real-time thickness distribution are spatially registered according to the number of the virtual trigger mark to form a thickness distribution data pair that is point-by-point aligned in the width direction and synchronously matched in the travel direction.

[0012] In practice, a reference measurement position is set before the foaming equipment inlet and a real-time measurement position is set after the foaming equipment outlet along the substrate conveying path. A first synchronous multi-point thickness measurement component is configured at the reference measurement position, and a second synchronous multi-point thickness measurement component is configured at the real-time measurement position. Both are composed of multiple point laser sensors arranged at equal intervals along the width direction, and have the same number of sensors, spacing, and spatial layout to ensure that the same width position is measured by the corresponding sensor before and after foaming. A rotary encoder is coaxially mounted on the substrate conveying roller shaft, rotating synchronously with the substrate. It outputs a trigger pulse every time it rotates through a preset equal spatial angle, and the interval of this trigger pulse corresponds to a fixed spatial step in the substrate's travel direction.

[0013] Each time the encoder emits a trigger pulse, the data processing unit generates a set of virtual trigger marks along the width of the substrate. The number of virtual trigger marks in each set is the same as the number of sensors in the first synchronous multi-point thickness measurement assembly. Each virtual trigger mark corresponds to a fixed sensor position along the width. Each set of virtual trigger marks moves synchronously with the substrate, so that during the entire measurement process, the width of the substrate is divided into a series of columns parallel to the direction of travel. Each column is formed by connecting virtual trigger marks at the same width position generated by consecutive trigger pulses end to end. The collection of all columns forms a virtual measurement grid covering the entire width of the substrate. Each virtual trigger mark in the virtual measurement grid is assigned a unique number, which contains both the column index of the width position and the row index of the direction of travel, thus uniquely determining the two-dimensional spatial position of the mark on the substrate plane.

[0014] The data acquisition and registration process is described in detail below. When a virtual trigger marker moves with the substrate to the reference measurement position, the arrival of the marker triggers the sensor corresponding to the width position of the marker in the first synchronous multi-point thickness measurement component to perform a thickness measurement. All markers in the same group as the virtual trigger marker synchronously trigger their corresponding sensors, thus all sensors corresponding to the markers in that group simultaneously acquire a frame of reference thickness data at the moving position. Since the sensors are point laser sensors, their measurement results are discrete points, and there is a physical distance between adjacent sensors, intra-frame interpolation is performed on the thickness data frame. The thickness value corresponding to the intermediate marker between adjacent markers in the virtual measurement grid is interpolated between the measurement values ​​of adjacent sensors, so that each virtual trigger marker in the frame is assigned a reference thickness value. The assignment method is as follows: the data processing unit maintains a two-dimensional data table in memory with the same structure as the virtual measurement grid. The rows and columns of this table correspond to the row index in the moving direction and the column index in the width direction, respectively. The reference thickness value corresponding to each virtual trigger marker after interpolation is stored in the data table at the row and column positions corresponding to the marker number. To illustrate with an example: Suppose that during the Nth trigger pulse of the encoder, virtual trigger markers numbered 1 to K are generated in the width direction. These markers travel to the reference measurement position, triggering the sensor to acquire a frame of raw data. After interpolation, K reference thickness values ​​are generated. The reference thickness value corresponding to the i-th marker is stored in the cell of the data table with the travel direction index N and the width direction index i. As the substrate continues to move, the encoder continuously generates trigger pulses, and the above process continues. After traversing all markers of the virtual measurement grid, the reference thickness values ​​in the data table are filled, thus obtaining the reference thickness distribution of the unfoamed substrate.

[0015] As the same set of virtual trigger markers continues to travel with the substrate to the real-time measurement position, the second synchronous multi-point thickness measurement component is triggered in the same manner to acquire real-time thickness data frames. After the same intra-frame interpolation processing, the real-time thickness value corresponding to each virtual trigger marker is stored in another real-time thickness data table with the same structure as the aforementioned data table. The real-time thickness value of the marker is stored in the corresponding row and column cells of this table. Since the number of the same virtual trigger marker remains unchanged during its movement, it occupies the same row and column positions in both the reference thickness data table and the real-time thickness data table. Therefore, based on the number of the virtual trigger marker, the spatial position of the reference thickness value before foaming and the real-time thickness value after foaming can be spatially registered one-to-one, forming a thickness distribution data pair that is point-by-point aligned in the width direction and synchronously matched in the direction of movement.

[0016] During the aforementioned data acquisition and registration process, the number and layout of point laser sensors in the first and second synchronous multi-point thickness measurement components remain consistent, ensuring that the corresponding sensor measures the same width position before and after foaming. The equally spaced trigger pulses generated by the rotary encoder ensure a constant spatial interval between two acquisitions in the traveling direction. Intra-frame interpolation processing improves the spatial resolution of the thickness data in the width direction, making the distribution density of the virtual trigger markers unrestricted by the physical spacing between sensors. Furthermore, the construction of the virtual measurement grid ensures that each set of associated thickness data carries clear spatial location information.

[0017] It should be noted that the point laser sensor, rotary encoder and data processing unit described in this embodiment are all commonly used hardware devices in the field. Their specific selection, installation method and communication interface can be adapted to the specifications and conditions of the actual production line, and will not be described in detail here.

[0018] Specifically, S2 is: The reference thickness value and real-time thickness value with the same virtual trigger mark number in the thickness distribution data pair are identified as associated point pairs; the numbers of all virtual trigger marks in the virtual measurement grid are traversed, and the real-time thickness value and reference thickness value in the associated point pair corresponding to each number are compared and calculated to generate an initial expansion ratio value corresponding to the number of the virtual trigger mark; the initial expansion ratio values ​​corresponding to the numbers of all virtual trigger marks are arranged according to the spatial position of each virtual trigger mark in the width direction and the travel direction to form an initial expansion ratio value distribution map; the initial expansion ratio value distribution map is subjected to background stripping processing, that is, the local bias component introduced by the difference in the thickness background of the incoming material in the initial expansion ratio value distribution map is identified, and the local bias component is removed from the initial expansion ratio distribution map to obtain the intrinsic expansion coefficient field stripped of the difference in the thickness background of the incoming material; the intrinsic expansion coefficient field is output as the equivalent foaming ratio distribution.

[0019] It should be noted that in the thickness distribution data pairs output by step S1, each data set carries a unique virtual trigger marker number. This number contains a row index in the direction of travel and a column index in the width direction, ensuring a one-to-one correspondence between the pre-foaming reference thickness value and the post-foaming real-time thickness value in two-dimensional space. The data processing unit identifies reference thickness values ​​and real-time thickness values ​​with the same virtual trigger marker number as an associated point pair. After traversing all marker numbers in the virtual measurement grid, it obtains a set of associated point pairs equal to the number of grid points. Subsequently, a comparison operation is performed on each associated point pair, i.e., the real-time thickness value is divided by the reference thickness value, and the resulting ratio is used as the initial expansion ratio corresponding to that virtual trigger marker number. All the initial expansion ratios with different numbers are arranged in two dimensions according to the spatial coordinates of their corresponding virtual trigger markers in the width and travel directions, forming an initial expansion ratio distribution map of the same size as the virtual measurement grid.

[0020] The initial expansion ratio distribution map visually reflects the thickness change factor of the substrate before and after foaming at various locations. However, this change factor contains two pieces of information: first, the actual expansion capacity imparted to the material by the foaming process; and second, the background difference caused by the uneven thickness of the substrate itself. Specifically, in areas where the substrate is thinner, even if the foaming process conditions are completely identical, the thickness change factor is often higher than in areas where the substrate is thicker because the thinner areas are heated faster and the foaming agent decomposes more completely. This fluctuation in the ratio caused by the difference in substrate thickness is called the "background difference." If it is not separated, it will be directly mixed into the foaming ratio distribution, leading to misleading decisions regarding subsequent non-uniformity extraction and control.

[0021] To address the aforementioned issues, the specific implementation method of the background stripping process in this step is explained below. The data processing unit performs two-dimensional spatial smoothing filtering on the initial expansion ratio distribution map. The size of the filtering window is preset according to the typical spatial scale of the incoming material thickness fluctuation. After smoothing, a low-frequency component map reflecting the overall trend of the ratio distribution is obtained. This low-frequency component map is the local bias component introduced by the background difference in incoming material thickness. For example: assuming that there is a slow thickness gradient in the substrate material along the width direction, thicker in the middle and thinner on both sides, the initial expansion ratio distribution map will correspondingly show a macroscopic distribution trend of a lower ratio in the middle and a higher ratio on both sides. The low-frequency component map extracted by the smoothing filter can effectively capture this macroscopic trend. By subtracting this low-frequency component map point by point from the initial expansion ratio distribution map, the systematic influence of the macroscopic unevenness of the incoming material thickness on the expansion ratio is eliminated. The remaining part is the intrinsic expansion coefficient field stripped of the background difference in incoming material thickness. Among them, the intrinsic expansion coefficient field refers to the two-dimensional field quantity obtained after the above-mentioned background stripping treatment, which characterizes the spatial distribution of the expansion capacity contributed by the foaming process itself. Its physical meaning is fundamentally different from the initial expansion ratio: the initial expansion ratio reflects the total thickness change multiple including the inherent differences of the substrate, while the intrinsic expansion coefficient field eliminates the interference of the inherent differences of the substrate and only reflects the difference in expansion effect caused by the foaming process parameters at different spatial locations.

[0022] Specifically, S3 is: The equivalent foaming ratio values ​​corresponding to each grid point in the equivalent foaming ratio distribution are arranged according to the spatial coordinates of the virtual trigger markers corresponding to each grid point in the width and travel directions, constructing a two-dimensional spatial scalar field, where each grid point carries an equivalent foaming ratio value. Adjacent grid points with equal equivalent foaming ratio values ​​in the two-dimensional spatial scalar field are connected to form several closed or boundary-extending equipotential lines. All equipotential lines together constitute an equipotential line skeleton map describing the geometric shape of the spatial distribution of the foaming ratio. In the equipotential line skeleton map, the spatial distance between two adjacent equipotential lines in the normal direction is used as a scale, and the equivalent foaming ratio values ​​at each position of each equipotential line are determined along the normal direction. The rate of change of direction is calculated to obtain the directional derivative of expansion stress, which characterizes the steepness of the change in foaming ratio in space. The directional derivatives of expansion stress at all locations of all equipotential lines are summarized to form a directional derivative field of expansion stress. The numerical fluctuation amplitude of the directional derivatives of expansion stress at all locations in the directional derivative field of expansion stress is extracted as the spatial non-uniformity at the current moment. On a continuous time series, the rate of change of the area of ​​the closed region enclosed by the same equipotential line in the equipotential line skeleton map with time is continuously tracked to obtain the area change rate. The area change rate is used as the basis for constructing the ratio strain rate field to generate the ratio strain rate describing the evolution trend of the non-uniform state of foaming ratio in the time dimension.

[0023] In practice, the data processing unit arranges the equivalent foaming ratio value carried by each grid point in the equivalent foaming ratio distribution in a two-dimensional arrangement according to the column index in the width direction and the row index in the row direction of the virtual trigger mark corresponding to that grid point, forming a spatial two-dimensional scalar field. In the spatial two-dimensional scalar field, the coordinate position of each grid point is determined by both the column index and the row index, and the field value of each grid point is the equivalent foaming ratio value at that position. Based on the spatial two-dimensional scalar field, the data processing unit performs an equipotential line extraction operation. Specifically, by traversing the equivalent foaming ratio values ​​of all grid points in the spatial two-dimensional scalar field, adjacent grid points with equal field values ​​or falling within the same preset value range are grouped together, and lines are drawn along the adjacency relationship between the grid points within the group to form a continuous curve. This curve is an equipotential line. The shape of equipotential lines in a two-dimensional scalar field depends on the spatial distribution pattern of the equivalent foaming ratio: when the field value exhibits a closed peak or valley distribution in a certain region, the equipotential lines form closed loops; when the field value changes monotonically along a certain direction, the equipotential lines are open curves extending to the field boundary. Superimposing all equipotential lines in the same coordinate system constitutes an equipotential line skeleton map. The equipotential line skeleton map refers to a two-dimensional geometric map composed of a family of curves connected by grid points with equal equivalent foaming ratio values. Its topological structure reflects the spatial distribution pattern of high and low foaming ratio regions, the transition relationships between regions, and the geometric morphological characteristics of non-uniform states.

[0024] After obtaining the equipotential line skeleton map, the data processing unit further calculates the directional derivative of the expansion stress. Specifically, in the equipotential line skeleton map, for any point on an equipotential line, the normal direction at that point is determined. This normal direction is defined as perpendicular to the tangent direction of the equipotential line at that point and pointing in the direction of the fastest change in the equivalent foaming ratio. Along this normal direction, another equipotential line adjacent to the first equipotential line is searched, and the spatial distance between the two equipotential lines in the normal direction is obtained. The difference between the equivalent foaming ratio values ​​corresponding to the two equipotential lines is then obtained. Dividing this difference in equivalent foaming ratio values ​​by the spatial distance yields the directional derivative of the expansion stress at that point. The magnitude of the directional derivative of the expansion stress characterizes the steepness of the change in the foaming ratio in space. Its physical meaning is: during the foaming process, the rate at which the equivalent foaming ratio value decays or increases along the normal direction of the equipotential line is closely related to the non-uniformity of the expansion stress distribution within the foamed material. The expansion stress directional derivative field is formed by summing the expansion stress directional derivatives at all points along all equipotential lines in the equipotential skeleton map. This field is a two-dimensional field quantity of the same size as the two-dimensional scalar field in space, where the field value at each grid point is the expansion stress directional derivative at that location.

[0025] After obtaining the expansion stress directional derivative field, the data processing unit extracts the spatial non-uniformity at the current moment. The extraction method involves statistically analyzing the numerical distribution of the expansion stress directional derivative at all grid points in the expansion stress directional derivative field, calculating its standard deviation or interquartile range as a measure of numerical fluctuation amplitude. This numerical fluctuation amplitude is the spatial non-uniformity at the current moment. A larger spatial non-uniformity indicates a more drastic spatial variation in the foaming ratio and poorer product uniformity; conversely, a smaller spatial non-uniformity indicates a more uniform distribution of the foaming ratio.

[0026] The above steps complete the extraction of spatial inhomogeneity at the current moment. Based on this, step S3 further constructs a scaling strain rate field to capture the evolution trend of the inhomogeneity state over time. The specific implementation is as follows: The data processing unit repeatedly performs the above operations of constructing a two-dimensional spatial scalar field and generating equipotential line skeleton maps at each sampling moment in a continuous time series, thereby obtaining a series of equipotential line skeleton maps arranged in chronological order. For the same equipotential line in this series of equipotential line skeleton maps, the evolution of the area of ​​the closed region enclosed by the equipotential line over time is tracked. The area tracking method is as follows: In each frame of the equipotential line skeleton map, the closed equipotential line corresponding to the same equivalent foaming scaling value is identified, and the area of ​​the closed region enclosed by it is calculated; with time as the horizontal axis and area as the vertical axis, the area values ​​of the closed equipotential line in multiple frames are fitted, and the rate of change of the area of ​​the closed region enclosed by the equipotential line is obtained from the slope of the fitted curve. The area tracking and rate calculation operations described above are performed on all equipotential lines. The area change rate corresponding to each equipotential line is then arranged according to the equivalent foaming ratio value of that equipotential line, thus obtaining the ratio strain rate field. Here, the "ratio strain rate field" refers to a two-dimensional field quantity with the area change rate of the enclosed region by the equipotential lines as the field quantity and the equivalent foaming ratio value corresponding to the equipotential lines as the index. Its physical meaning is the rate distribution of area expansion or contraction of different ratio regions in a state of non-uniform foaming ratio. Summarizing the area change rates corresponding to all equipotential lines constitutes the ratio strain rate field. The ratio strain rate field describes the evolution trend of the non-uniform foaming ratio state in the time dimension, that is, whether different ratio regions are expanding or contracting and the speed of evolution.

[0027] It should be noted that the specific numerical calculations involved in this step, such as equipotential line extraction, normal direction determination, area tracking, and rate fitting, can all be implemented using image processing and numerical analysis methods known in the art. The specific algorithm implementation details are not limited to the scope of this invention and will not be elaborated here.

[0028] Specifically, S4 is: Traversing the closed region enclosed by each equipotential line in the strain rate field, the area change rate of each equipotential line is converted into topological tension acting on the boundary of the region enclosed by that equipotential line. The direction of the topological tension is outward along the normal to each point of the equipotential line, and the magnitude of the topological tension is proportional to the area change rate. The topological tension of all equipotential lines constitutes a topological tension field acting on the equipotential line skeleton pattern. Using the current equipotential line skeleton pattern as the initial configuration, the topological tension field is used as the driving force to drive the position migration of each equipotential line in the equipotential line skeleton pattern. During the migration, elastic constraints are applied between equipotential lines to maintain the topological relationship between adjacent equipotential lines. This is achieved through iterative deformation simulation. The deformation accumulation time is carried out until a preset control period is reached, and a predicted equipotential line skeleton map is obtained. In the predicted equipotential line skeleton map, the spatial distance between two adjacent equipotential lines in the normal direction is remeasured to obtain the predicted distance distribution. Using the predicted distance distribution as a scale and combined with the equivalent foaming ratio value, the rate of change of the equivalent foaming ratio value along the normal direction at each position of each equipotential line is recalculated to obtain the updated expansion stress directional derivative. All updated expansion stress directional derivatives are summarized into a predicted expansion stress directional derivative field. The numerical fluctuation amplitude of the expansion stress directional derivative at all positions in the predicted expansion stress directional derivative field is extracted as the predicted spatial non-uniformity that will appear after the preset control period.

[0029] It should be noted that in the strain rate field obtained in step S3, each equipotential line is associated with an area change rate. The sign and magnitude of the area change rate characterize the direction and speed of expansion or contraction of the region enclosed by the equipotential line, respectively. Traversing the closed region enclosed by each equipotential line in the strain rate field, the area change rate of each equipotential line is converted into topological tension acting on the boundary of the region enclosed by that equipotential line. Topological tension is a vector field quantity, its direction being outward along the normal at each point on the equipotential line, and its magnitude being proportional to the area change rate of the region enclosed by the equipotential line. Here, the physical meaning of topological tension needs further explanation: during the foaming process, due to the uneven distribution of expansion stress within the material, there is an objective mechanical action of mutual compression or stretching between regions of different strain rates. Topological tension is precisely an abstract representation of this mechanical action at the geometric level of the equipotential line framework; its unit is the area change rate unit, representing the equivalent driving force per unit arc length of the equipotential line that causes a change in the area of ​​the region enclosed by that equipotential line. By summing up the topological tensions corresponding to all equipotential lines in the scaling strain rate field, while preserving their respective spatial positions and directional distributions, a topological tension field is constructed that acts on the entire equipotential line skeleton pattern. This topological tension field describes the direction and intensity distribution of the expansion or contraction driving forces experienced at the current moment by the boundaries of different scaling regions in a state of non-uniform foaming scaling.

[0030] After obtaining the topological tension field, using the current equipotential line skeleton map as the initial configuration, a step-by-step iterative deformation simulation is performed to obtain the predicted equipotential line skeleton map. The specific implementation of the deformation simulation is as follows: Each equipotential line in the equipotential line skeleton map is discretized into a series of control points. Each control point carries its spatial coordinates and the equivalent foaming ratio of the equipotential line to which it belongs. In each iteration step, a driving force from the topological tension at that location in the topological tension field is applied to each control point, causing the control point to produce a small displacement along the normal direction. The displacement is proportional to the magnitude of the topological tension and the iteration step size. At the same time, elastic constraints are applied between adjacent equipotential lines. That is, in each iteration step, the distance between any two adjacent equipotential lines in the normal direction is detected. When the distance is less than the preset minimum allowable value, a repulsive force is applied between the two equipotential lines to correct it. When the distance is greater than the preset maximum allowable value, an attractive force is applied to pull it back, thereby ensuring that adjacent equipotential lines do not cross or separate during the deformation process, maintaining the original topological relationship unchanged. It should be noted that the introduction of elastic constraints between equipotential lines is to simulate the objectively existing mechanical continuity between adjacent regions within the foamed material, making the deformation simulation of the equipotential line skeleton more closely resemble the physical reality of material continuum deformation. After each iteration step, the spatial coordinates of all control points are updated to obtain the deformed equipotential line skeleton corresponding to that iteration step, which is then used as the input configuration for the next iteration step. The above iterative process is repeated, and the product of the iteration step size and the number of iterations is the deformation accumulation time. When the deformation accumulation time reaches a preset control period, the iteration stops, and the equipotential line skeleton map at this point is the predicted equipotential line skeleton map.

[0031] The deformation simulation process is illustrated by the following example: Assume that at the current moment, there exists a closed equipotential line in the equipotential line skeleton map with an equivalent foaming ratio of five times. Its area change rate is positive, indicating expansion, and the resulting topological tension direction is outward along the normal. Simultaneously, there exists a closed equipotential line outside this equipotential line with an equivalent foaming ratio of four times. Its area change rate is negative, indicating contraction, and the resulting topological tension direction is inward along the normal. During the deformation simulation, the control points of the five-times-equipotential line migrate outward under the outward topological tension, while the control points of the four-times-equipotential line migrate inward under the inward topological tension, gradually reducing the distance between the two equipotential lines. When the distance approaches the preset minimum allowable value, elastic constraints intervene to prevent them from approaching further, thus reflecting the dynamic mechanical equilibrium between the five-times-equipotential and four-times-equipotential regions. After deformation accumulation over a preset control period, each equipotential line migrates to a new spatial position, forming the predicted equipotential line skeleton map.

[0032] After obtaining the predicted equipotential line skeleton map, the predicted expansion stress directional derivative field is reconstructed based on it. Specifically, the operation is as follows: In the predicted equipotential line skeleton map, the spatial distance between any two adjacent equipotential lines along the normal direction is remeasured to obtain the predicted spacing distribution; using the predicted spacing distribution as a benchmark, and combining the equivalent foaming ratio value corresponding to each equipotential line, the rate of change of the equivalent foaming ratio value along the normal direction at each position of each equipotential line is recalculated using the same method as in step S3. That is, the difference in the equivalent foaming ratio values ​​of adjacent equipotential lines is divided by the predicted spacing between them to obtain the updated expansion stress directional derivative; the updated expansion stress directional derivatives at all positions of all equipotential lines are summarized to form the predicted expansion stress directional derivative field. The predicted expansion stress directional derivative field describes the steepness of the expected expansion stress change at each position in the foaming ratio spatial distribution after a preset control period.

[0033] Finally, the predicted spatial non-uniformity is extracted from the predicted expansion stress directional derivative field. The extraction method is the same as that in step S3, which extracts the spatial non-uniformity from the expansion stress directional derivative field at the current moment. That is, the numerical fluctuation range of the expansion stress directional derivative at all positions in the predicted expansion stress directional derivative field is statistically analyzed and used as the predicted spatial non-uniformity that will appear after the preset control period.

[0034] Specifically, S5 is: The predicted spatial non-uniformity is compared with a preset allowable range of uniformity. When the predicted spatial non-uniformity exceeds the allowable range of uniformity, the excess deviation is calculated, and a pre-regulation command generation process is triggered. From the predicted expansion stress directional derivative field, regions where the directional derivative exceeds a preset threshold are searched row by row along the width direction. Regions that continuously exceed the preset threshold are marked as excess deformation zones. The center position coordinates of each excess deformation zone in the width direction and the position of the equipotential line crossed by the excess deformation zone in the predicted equipotential line skeleton map are extracted. Based on the position of the equipotential line crossed by each excess deformation zone in the predicted equipotential line skeleton map, the topological stress borne by the corresponding equipotential line is extracted from the topological tension field. The peak tension is calculated, and the topological tension peaks corresponding to all over-limit deformation zones are weighted and synthesized to obtain the desired correction tension field for driving deformation correction. Using this desired correction tension field as input, the expected tension change is converted into the process parameter compensation amount to be applied to each independent adjustment segment in the foaming process execution unit by inverse deduction along the normal direction of each equipotential line in the predicted equipotential line skeleton map. The process parameter compensation amount corresponds one-to-one with the width position covered by the independent adjustment segment. The process parameter compensation amount is superimposed with the current process parameter setting value of the corresponding independent adjustment segment to generate a pre-control command for each independent adjustment segment, which is then output to the foaming process execution unit for advance adjustment.

[0035] It should be noted that the preset uniformity allowable range is a numerical range pre-set and stored in the system according to the product process requirements. This range represents the upper limit of the allowable spatial non-uniformity for a qualified product. In this embodiment, an example value for the uniformity allowable range is a spatial non-uniformity of no more than 0.15. When the predicted spatial non-uniformity output in step S4 exceeds 0.15, the excess deviation is calculated, which is the difference between the predicted spatial non-uniformity and 0.15, and the pre-control instruction generation process is triggered. If the predicted spatial non-uniformity does not exceed this range, it indicates that no early intervention is needed within the preset control cycle. The subsequent operations of this step are skipped, and the process directly enters the cyclic update process of step S6.

[0036] After the pre-regulation instruction generation process is initiated, the spatial region requiring intervention is located in the predicted expansion stress directional derivative field. Specifically, the predicted expansion stress directional derivative field is scanned line by line along the width direction, and the continuous grid point region in each line where the directional derivative exceeds a preset threshold is marked as an over-limit deformation zone. In this embodiment, the preset threshold is 0.2 / cm, which means that when the expansion stress directional derivative exceeds 0.2 / cm, it indicates that the spatial variation of the foaming ratio at that location has exceeded an acceptable steepness, and it is a region requiring close attention and correction. Each over-limit deformation zone consists of grid points that continuously exceed this threshold. The data processing unit extracts the center coordinates of each over-limit deformation zone in the width direction. Simultaneously, based on the grid point positions covered by the over-limit deformation zone, the equipotential lines crossed by the over-limit deformation zone are determined in the predicted equipotential line skeleton map, i.e., the equipotential lines contained within the spatial range of the over-limit deformation zone.

[0037] Subsequently, for each exceeding deformation zone, based on the position of the equipotential line it crosses in the predicted equipotential line skeleton map, the peak value of the topological tension borne by the corresponding equipotential line is extracted from the topological tension field constructed in step S4. The "peak value of topological tension" refers to the maximum value of the topological tension at all points along an equipotential line. It is extracted by iterating through the topological tension values ​​at each control point of the equipotential line and taking the maximum value. The reason for choosing the peak value instead of the average value is that the peak value reflects the local location on the equipotential line where the deformation driving force is strongest. This local location is the core driving source that exacerbates the uneven deformation. Correcting based on the peak value can specifically eliminate the main driving force of the uneven state. After extracting the peak values ​​of the topological tension corresponding to all exceeding deformation zones, the center position coordinates of each exceeding deformation zone in the width direction are used as the weight calculation basis. The topological tension peak values ​​are weighted and synthesized by multiplying the amplitude of each topological tension peak value by the normalized weight coefficient at the center position of the exceeding deformation zone and then summing the results to obtain a desired corrected tension field that simultaneously carries information on the tension direction and amplitude. The expected correction tension field describes the spatial distribution of the deformation correction driving force required to eliminate the predicted non-uniformity exceeding the limit.

[0038] After obtaining the desired tension field, a reverse deduction from tension to process parameters is performed. The deduction path is as follows: along the normal direction of each equipotential line in the predicted equipotential line skeleton map, the desired tension change at each location in the desired tension field is converted into the process parameter compensation amount to be applied to each independent adjustment section in the foaming process execution unit through a preset tension-parameter mapping relationship. The foaming process execution unit is divided into several independent adjustment sections along the width direction, and each section corresponds to a set of independently adjustable foaming process parameters. The tension-parameter mapping relationship is established based on the fact that in the EVA foaming process, the foaming ratio has a spatial correspondence to the response of process parameters. The tension change in the normal direction of the equipotential line at a certain width position is mainly determined by process parameters such as foaming temperature or hot air flow rate corresponding to that width position. The mapping relationship between the two can be obtained through offline process test calibration and stored in the data processing unit in the form of a mapping table. The reverse deduction process is a table lookup process: using the desired tension change as an index, the corresponding process parameter compensation amount is found in the mapping table. The obtained process parameter compensation amounts correspond one-to-one with the width and position covered by the independent adjustment section.

[0039] To illustrate the reverse deduction process above, consider the following example: Suppose that an over-limit deformation band crosses an equipotential line located in the center region of the width, corresponding to an equivalent foaming ratio of 5.0. It is expected that the tension field indicating the tension in the outward direction of the normal to this equipotential line needs to be reduced by 0.1 cm² / sampling period. By looking up the table, it is found that this reduction in tension corresponds to a reduction in the foaming temperature of 3.5℃ in the center region of the width. This center region of the width happens to fall within the coverage of the third independent adjustment section in the foaming process execution unit, so a temperature compensation of -3.5℃ is generated for this section.

[0040] Finally, the aforementioned process parameter compensation amounts are superimposed on the current process parameter settings of the corresponding independent adjustment sections segment by segment to generate pre-control instructions for each independent adjustment section. These pre-control instructions contain the identification information of each independent adjustment section and the corresponding process parameter adjustment target value. These are packaged into an instruction message and output to the foaming process execution unit for advance adjustment. Upon receiving the pre-control instructions, the foaming process execution unit adjusts the process parameters according to the instruction target value within each independent adjustment section, thereby completing preventative intervention before the uneven foaming ratio actually deteriorates.

[0041] Specifically, S6 is: After the foaming process execution unit completes the adjustment according to the pre-control instructions, S1 to S2 are executed again to obtain the newly generated equivalent foaming ratio distribution, i.e., the adjusted intrinsic expansion coefficient field. Using the adjusted intrinsic expansion coefficient field as input, the steps in S3 are executed to construct the adjusted equipotential line skeleton map, and the adjusted spatial non-uniformity is extracted from the adjusted equipotential line skeleton map. The adjusted equipotential line skeleton map is compared line by line with the predicted equipotential line skeleton map in S4, and the difference between the predicted value and the actual value of the area of ​​the closed region enclosed by the same equipotential line is calculated to obtain the skeleton deformation residual. The distribution of the skeleton deformation residuals is fed back to the topological tension field. The topological tension corresponding to each equipotential line in the topological tension field is corrected to obtain the corrected topological tension field after historical residual correction. The corrected topological tension field is used to drive deformation simulation in the next cycle S4. The adjusted spatial non-uniformity is used as the current spatial non-uniformity in the next cycle. The topological tension field is updated with the corrected topological tension field. Then, the steps of constructing the scaling strain rate field in S3 and the prediction calculation steps in S4 are re-executed to generate the updated prediction spatial non-uniformity and pre-control command, forming a closed-loop control based on skeleton deformation residual feedback.

[0042] It should be noted that after the foaming process execution unit completes the adjustment of the process parameters of each independent adjustment section according to the pre-control instructions generated in step S5, the foaming equipment continues to operate under the new process parameter conditions, and the substrate continues to pass through the foaming area and completes foaming. At this time, steps S1 and S2 are re-executed. Specifically, in step S1, the first synchronous multi-point thickness measurement component and the second synchronous multi-point thickness measurement component, under the synchronous triggering of the virtual trigger mark, re-collect the baseline thickness distribution before foaming and the real-time thickness distribution after foaming under the new process parameter conditions, and form a new thickness distribution data pair through spatial position registration; step S2 takes this new thickness distribution data pair as input, and after determining the associated point pair, calculating the initial expansion ratio, constructing the initial expansion ratio distribution map, and performing background stripping processing, outputs the newly generated equivalent foaming ratio distribution after adjustment. This equivalent foaming ratio distribution is the adjusted intrinsic expansion coefficient field. The adjusted intrinsic expansion coefficient field reflects the actual state of the spatial distribution of the foaming ratio after the process parameter adjustment and is the basis for evaluating the effect of this round of control.

[0043] After obtaining the adjusted intrinsic expansion coefficient field, it is used as input to perform the relevant operations for constructing the equipotential line skeleton map in step S3. Specifically, the equivalent foaming ratio values ​​carried by each grid point in the adjusted intrinsic expansion coefficient field are arranged according to spatial coordinates into a two-dimensional spatial scalar field. Equipotential lines are extracted, and the adjusted equipotential line skeleton map is constructed. From this adjusted equipotential line skeleton map, the adjusted spatial non-uniformity is calculated using the same extraction method as in step S3. This adjusted spatial non-uniformity characterizes the actual level of foaming ratio uniformity after the process parameters are adjusted.

[0044] Subsequently, the distribution of skeleton deformation residuals is calculated. The calculation method is as follows: the adjusted equipotential skeleton map is compared line by line with the predicted equipotential skeleton map generated in step S4. The comparison benchmark is the area of ​​the closed region enclosed by the same equipotential line. In the predicted equipotential skeleton map, each closed equipotential line has a predicted area; in the adjusted equipotential skeleton map, the closed equipotential line corresponding to the same equivalent foaming ratio value has an adjusted actual area. The data processing unit traverses all closed equipotential lines, calculating the difference between the predicted area and the adjusted actual area for each line, and uses this difference as the skeleton deformation residual corresponding to that equipotential line. The set of skeleton deformation residuals for all equipotential lines is the skeleton deformation residual distribution.

[0045] The skeleton deformation residual distribution is essentially a geometrical measure of the deviation between the predicted results of the prediction model and the actual adjustment effect of process parameters. When the predicted area of ​​an equipotential line is greater than the actual area after adjustment, it indicates that the deformation simulation in step S4 overestimated the expansion degree of the equipotential line, that is, the topological tension corresponding to the equipotential line in the topological tension field was overestimated; conversely, when the predicted area is smaller than the actual area after adjustment, it indicates that the topological tension was underestimated. The skeleton deformation residual distribution provides a quantitative correction basis for each equipotential line for the correction of the topological tension field.

[0046] Based on the skeleton deformation residual distribution, the data processing unit corrects the topological tension field. The correction method is as follows: for each equipotential line in the topological tension field, its amplitude is correlated with the skeleton deformation residual corresponding to that equipotential line in the skeleton deformation residual distribution. Specifically, the correction direction is determined by the sign of the skeleton deformation residual, and the magnitude of the correction is determined by the absolute value of the skeleton deformation residual, which is proportional to the absolute value of the residual. When the predicted area is larger than the actual area, the topological tension amplitude of the equipotential line is reduced; when the predicted area is smaller than the actual area, the topological tension amplitude of the equipotential line is increased. After completing the above correction for the topological tension of all equipotential lines, the corrected topological tension field after historical residual correction is obtained.

[0047] The above correction process is illustrated by an example: Suppose that step S4 predicts the area enclosed by an equipotential line with an equivalent foaming ratio of 5.0 will be 12.1 cm² after one preset control cycle. However, the actual measured area enclosed by the equipotential line after adjustment is 11.5 cm². The predicted area is 0.6 cm² larger than the actual area, meaning the residual deformation of the equipotential line's skeleton is +0.6 cm². This indicates that the topological tension applied to the equipotential line in step S4 is too large. The data processing unit then corrects the amplitude of the topological tension corresponding to the equipotential line downwards, with the correction amount proportional to 0.6 cm². This makes the deformation driving force of the equipotential line in the next cycle closer to the physical response of the actual foaming process.

[0048] The corrected topological tension field will be used for deformation simulation in step S4 of the next cycle, replacing the original topological tension field in the previous cycle. Simultaneously, the data processing unit uses the adjusted spatial non-uniformity as the current spatial non-uniformity for the next cycle, updating the starting point for constructing the scaling strain rate field in step S3. It then re-executes the steps for constructing the scaling strain rate field in S3 and the prediction calculation steps in S4, generating updated prediction spatial non-uniformity and pre-control commands. This forms a closed-loop control based on skeleton deformation residual feedback. In each cycle, this closed-loop control not only detects, predicts, and adjusts the non-uniformity but also self-calibrates the topological tension field through skeleton deformation residuals. This allows the deformation driving force of the prediction model to gradually approach the physical laws of the actual foaming process as the number of cycles increases, continuously improving prediction accuracy.

[0049] In another preferred embodiment of the present invention, the method further includes: obtaining the current spatial position of each equipotential line in the predicted equipotential line skeleton map, and defining a target uniform configuration in which each equipotential line is arranged in parallel at equal intervals along the width direction; performing shape feature point registration between the predicted equipotential line skeleton map and the target uniform configuration, extracting the deformation offset of each equipotential line in the width direction, and constructing a time-varying deformation mapping describing the deformation of the current configuration to the target uniform configuration, wherein the time-varying deformation mapping satisfies the large deformation differential homeomorphism constraint to ensure that the equipotential lines are not torn or folded during the deformation process; solving for the minimum energy functional of the time-varying deformation mapping over the entire deformation time interval, extracting the velocity vector field at each moment, and integrating the velocity vector field along the deformation time axis to generate a line connecting the current configuration and the target uniform configuration. The optimal deformation path for a uniform configuration is defined as the optimal geodesic flow. The velocity direction and magnitude of each equipotential line at each deformation moment are extracted along the optimal geodesic flow, and the velocity vectors are orthogonally projected according to the normal directions of each point on the equipotential line to obtain the transient deformation driving force vectors acting on each point on each equipotential line. The transient deformation driving force vectors of each equipotential line at all deformation moments along the optimal geodesic flow are time-domain integrally averaged to obtain a static conformal driving force field that does not change with time. The static conformal driving force field is fed forward into the topological tension field derived from the scaling strain rate field in a point-by-point superposition manner to generate a composite driving force field after conformal compensation. The composite driving force field replaces the original topological tension field, driving the equipotential line skeleton map to perform the step-by-step iterative deformation simulation described in S4.

[0050] It should be noted that the target uniform configuration is a spatial arrangement template of equipotential lines pre-defined based on the ideal uniform foaming ratio of the product. In this configuration, the equipotential lines are arranged in parallel at equal intervals along the width direction, and the normal spacing between adjacent equipotential lines is equal everywhere. The method for registering the shape feature points between the predicted equipotential line skeleton map and the target uniform configuration is as follows: extract a set of feature points on each equipotential line in the predicted equipotential line skeleton map, and extract the same number of feature points on the corresponding equipotential lines in the target uniform configuration, establishing a one-to-one correspondence between the two sets of feature points. Based on the feature point correspondence, the deformation offset of each equipotential line in the width direction is extracted, that is, the displacement vector required for each feature point to migrate from its current position to the corresponding position in the target uniform configuration. Based on this deformation offset, a time-varying deformation mapping describing the deformation required for the current configuration to transform into the target uniform configuration is constructed. It should be noted here that time-varying deformation mapping refers to parameterizing the deformation process as a function of time variables. At the start of deformation, it maps to the current configuration; at the end of deformation, it maps to the target uniform configuration; and each intermediate moment corresponds to an intermediate configuration that gradually transitions from the current configuration to the target uniform configuration. This time-varying deformation mapping satisfies the large deformation homeomorphism constraint, meaning that the deformation mapping is continuously differentiable at every moment and has a continuously differentiable inverse mapping. This ensures that equipotential lines do not tear, fold, or self-intersect during deformation, maintaining their topological integrity.

[0051] After constructing the deformation mapping, the energy functional minimum of the mapping is solved over the entire deformation time interval. The energy functional is defined as the kinetic energy integral of the velocity vector field at each moment during deformation. The physical meaning of solving for the energy functional minimum is to find the deformation path with the least resistance and the shortest path among all deformation paths satisfying the differential homeomorphism constraint. In the solution process, the variational principle is used to transform the energy functional minimization problem into an optimization problem of the velocity vector field, extracting the velocity vector field that minimizes the energy functional at each moment. The velocity vector field is integrated moment-by-moment along the deformation time axis to generate an optimal deformation path connecting the current configuration and the target homogeneous configuration. This optimal deformation path is the optimal geodesic flow. The optimal geodesic flow describes the optimal evolution trajectory of the equipotential skeleton converging from the current non-uniform configuration to the target homogeneous configuration.

[0052] After obtaining the optimal geodesic flow, the velocity direction and magnitude of each equipotential line at each deformation moment are extracted along the optimal geodesic flow, resulting in the velocity vector at each point on each equipotential line at each deformation moment. The velocity vector at each point at each deformation moment is orthogonally projected along the normal direction of the equipotential line to which that point belongs, i.e., the component of the velocity vector in the normal direction is taken, yielding the transient deformation driving force vector acting on each point on each equipotential line. The reason for taking the normal direction component is that the area change of the equipotential line is only contributed by the displacement in its normal direction; the displacement in the tangential direction does not change the area of ​​the region enclosed by the equipotential line. After obtaining the transient deformation driving force vectors for all deformation moments, they are integrated and averaged in the time domain. Specifically, for each point on each equipotential line, the transient deformation driving force vectors at that point across all deformation moments covered by the optimal geodesic flow are summed, and then divided by the total number of deformation moments to obtain a static vector that does not change with time. The static vector is the static conformal driving force at that point. The sum of the static conformal driving forces at all points on all equipotential lines constitutes the static conformal driving force field. The physical meaning of the static conformal driving force field is: without considering the historical evolution trend of the foaming process, starting only from the current non-uniform configuration, the distribution of conformal driving forces that need to be continuously applied at each point on the equipotential lines to make it converge toward the target uniform configuration.

[0053] Finally, the static conformal driving force field can be fed forward into the topological tension field derived from the strain rate field in step S4 by point-by-point superposition. That is, for each point in the equipotential skeleton pattern, the static conformal driving force is vector-summed with the original topological tension at that point to generate a composite driving force field after conformal compensation. This composite driving force field replaces the original topological tension field, driving the equipotential skeleton pattern to perform the step-by-step iterative deformation simulation described in S4. In the composite driving force field, the topological tension field component provides the evolutionary driving force based on historical trends, while the static conformal driving force field component provides the conformal guiding force pointing towards the uniform convergence target. The combined effect of the two makes the deformation simulation both continue the physical evolution inertia of the foaming process and have an inherent guidance towards the uniform target, thereby further improving the physical rationality and prediction accuracy of the predicted equipotential skeleton pattern.

[0054] In another preferred embodiment of the invention, the method further includes: extracting the directional derivative values ​​of the expansion stress at each sampling point along each equipotential line from the expansion stress directional derivative field, defining each sampling point as a node, and assigning each node an equipotential line number and a position index along the arc length of that equipotential line; using the sequence of the expansion stress directional derivative of each node changing with time as the stress fluctuation time sequence of that node; for any two nodes, calculating the stress fluctuation conditional mutual information transmitted between the two nodes under the condition of given stress fluctuation time sequences of all other nodes, and using the conditional mutual information as the directed connection weight to construct a directed weighted directional derivative conditional mutual information network; traversing all nodes in the directional derivative conditional mutual information network to find all node subsets that satisfy the following conditions: the bidirectional conditional mutual information between any two nodes in the subset exceeds a preset causal association threshold, and the subset is not included by any other subset that satisfies the condition. Each such subset is defined as a causal cluster; the sum of all directed connection weights within each causal cluster is calculated, and the causal cluster with the largest sum is determined as the maximal independent causal cluster. The maximal independent causal cluster represents the core information interaction loop that triggers the concentrated propagation of non-uniform stress in the medium; for each node in the maximal independent causal cluster, its criterion in all connection paths within the cluster is calculated, and the node with the highest criterion is determined as the medium bottleneck node. The medium bottleneck node corresponds to the blocking point on the stress concentration propagation path; taking the main peak frequency of the stress fluctuation time series of the medium bottleneck node as the center, the random perturbation frequency range is limited to a preset proportional bandwidth of the main peak frequency to generate a bandwidth-limited pseudo-random perturbation sequence, and the pseudo-random perturbation sequence is used as a micro-perturbation signal and superimposed in an aperiodic manner onto the process parameter setting value of the independent adjustment section at the spatial location corresponding to the medium bottleneck node to induce random resonance dissipation of local stress accumulation.

[0055] In actual continuous foaming processes, although the directional derivative of expansion stress in some areas may not continuously exceed the preset threshold, it may still show a gradual accumulation trend over a long time scale. This stress accumulation often does not occur independently at each location, but rather propagates spatially through a specific path, eventually releasing in a localized area, leading to unpredictable local density anomalies in the product. Simply relying on threshold-triggered control is insufficient to capture this potential stress propagation path and its key blockage points. Therefore, this embodiment first extracts the directional derivative values ​​of expansion stress at each sampling point along each equipotential line at fixed arc length intervals from the expansion stress directional derivative field. Each sampling point is defined as a node, and each node is assigned its corresponding equipotential line number and a position index along the arc length of that equipotential line. The equipotential line number uses the equivalent foaming ratio value corresponding to the equipotential line in step S3 as an identifier, and the arc length position index increases incrementally along the direction of the equipotential line, with the starting point of the equipotential line as zero. Thus, each node has a unique two-dimensional spatial location and stress characteristic value in the equipotential line skeleton map.

[0056] Then, a directional derivative conditional mutual information network is constructed: the directional derivative values ​​of the expansion stress at multiple consecutive sampling times of each node are arranged in chronological order to form the stress fluctuation time series of that node. For any two nodes, given the stress fluctuation time series of all other nodes, the stress fluctuation conditional mutual information transmitted between these two nodes is calculated. The conditional mutual information measures the degree to which the uncertainty of the stress fluctuation of one node on the stress fluctuation of another node is reduced after excluding the influence of all other nodes; a larger value indicates a stronger direct causal relationship between the two nodes. Using this conditional mutual information as the directed connection weight from the first node to the second node, the above calculation is performed pairwise on all nodes to construct a directed weighted directional derivative conditional mutual information network.

[0057] After obtaining the directional derivative conditional mutual information network, causal clusters are searched within it. The search method is as follows: traverse all nodes and find all subsets of nodes that satisfy the following conditions: the bidirectional conditional mutual information between any two nodes in the subset exceeds a preset causal association threshold, and the subset is not contained in any other subset that also satisfies the condition. An example value for the preset causal association threshold is 0.3. Each such subset of nodes found is defined as a causal cluster. The physical meaning of a causal cluster is: during the propagation of expansion stress, the nodes within the cluster form a highly coupled, mutually causal information interaction dense region, where stress fluctuations propagate and amplify rapidly. Among all identified causal clusters, the sum of the weights of all directed connections within each causal cluster is calculated, representing the total intensity of information interaction within the causal cluster. The causal cluster with the largest sum is determined as the maximally independent causal cluster, which represents the core information interaction loop that triggers the concentrated propagation of non-uniform stress in the medium.

[0058] For the largest independent causal cluster, the bottleneck node is further located. The calculation method is as follows: For each node in the largest independent causal cluster, traverse the shortest connection paths between all node pairs within the cluster, count the number of times the node appears on these shortest paths, and divide this count by the total number of node pairs in the cluster to obtain the node's traversability. The higher the traversability, the more indispensable the node is in the cluster's information interaction. The node with the highest traversability is identified as the bottleneck node. The bottleneck node corresponds to the blockage point on the stress concentration propagation path, i.e., the local location where stress accumulation is most likely to occur and outward release is most constrained. The calculation of traversability is illustrated with an example: Assume there are four nodes A, B, C, and D in the largest independent causal cluster, with a total of six node pairs. In the shortest paths of each of the six pairs of nodes, node B appears on five of the pairs' shortest paths; therefore, node B's traversability is 5 / 6. If this value is the highest among the four nodes, then node B is identified as the bottleneck node.

[0059] After locating the bottleneck node in the medium, the main peak frequency of the stress fluctuation time series of that node is extracted, which is the frequency component with the highest energy after spectral analysis of the historical stress fluctuation time series. Centered on this main peak frequency, the random disturbance frequency range is limited to a preset proportional bandwidth of this main peak frequency; in this embodiment, the proportional bandwidth is taken as ±15%. Within this limited frequency range, a bandwidth-limited pseudo-random disturbance sequence is generated by applying random phases to each frequency component within the limited frequency range and then superimposing them. The amplitude of this pseudo-random disturbance sequence is controlled within a preset proportion of the normal fluctuation range of the process parameters, for example, not exceeding 2% of the steady-state setpoint of the corresponding process parameters, to ensure that it does not have a macroscopic impact on the overall product quality. This pseudo-random disturbance sequence is used as a micro-disturbance signal and superimposed aperiodically onto the process parameter setpoint of the independent adjustment segment at the spatial location corresponding to the bottleneck node in the medium.

[0060] The mechanism of the micro-perturbation signal is based on the principle of stochastic resonance: in a nonlinear system, noise perturbation of appropriate intensity can enhance the system's response to weak periodic signals, allowing accumulated stress that was originally below the dissipation threshold to be released by the noise energy overcoming the dissipation barrier. The bottleneck node of the medium, as a blockage point for stress accumulation, is in a metastable local stress state. A micro-perturbation with a frequency matching the main peak frequency can induce stress dissipation at this point with minimal energy input, thereby eliminating local stress concentration in a non-destructive manner before the stress reaches the threshold, and improving the long-term stability of the foaming ratio uniformity.

[0061] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A closed-loop control method for the uniformity of EVA foam expansion ratio based on thickness feedback, characterized in that, Includes the following steps: S1. During the continuous foaming process, the baseline thickness distribution of the unfoamed substrate before foaming and the real-time thickness distribution of the foam after foaming are collected simultaneously to form a thickness distribution data pair. S2. Perform a point-by-point correlation calculation between the real-time thickness distribution and the reference thickness distribution to generate an equivalent foaming ratio distribution; S3. Perform spatiotemporal gradient analysis on the equivalent foaming ratio distribution, extract the spatial non-uniformity of the equivalent foaming ratio distribution, and construct the ratio strain rate field in the foaming process based on the change of the spatial non-uniformity in the time series. S4. Calculate the predicted spatial non-uniformity that will occur after a preset control cycle using the aforementioned strain rate field. S5. When the predicted spatial non-uniformity does not fall within the preset uniformity allowable range, a pre-adjustment instruction to adjust the foaming process parameters in advance is generated. S6. Repeat the above steps to update the spatial non-uniformity and the expansion ratio field with the newly generated equivalent foaming ratio distribution after adjustment, and then update the predicted spatial non-uniformity and the pre-control command to form a closed-loop control based on the expansion ratio field calculation.

2. The method according to claim 1, characterized in that, Specifically, S1 is: On the substrate conveying path, a reference measurement position before foaming and a real-time measurement position after foaming are set. A first synchronous multi-point thickness measurement component composed of multiple point laser sensors is configured at the reference measurement position. A second synchronous multi-point thickness measurement component with the same spatial resolution and corresponding layout as the first synchronous multi-point thickness measurement component is configured at the real-time measurement position. At the same time, an encoder that rotates synchronously with the substrate is installed along the substrate traveling direction to generate trigger pulses with equal spatial intervals. Based on the trigger pulse, a series of virtual trigger marks are generated in the width direction of the substrate. Each virtual trigger mark corresponds to a fixed width position and moves synchronously with the substrate, thereby forming a uniquely numbered virtual measurement grid across the entire width. When any of the virtual trigger markers travels to the reference measurement position, the virtual trigger marker triggers the first synchronous multi-point thickness measurement component to acquire a thickness data frame, performs intra-frame interpolation processing on the thickness data frame to obtain a reference thickness value at a grid point, and assigns the reference thickness value to the number of the virtual trigger marker; By traversing all markers of the virtual measurement grid, the baseline thickness distribution of the unfoamed substrate is obtained; When the same virtual trigger mark continues to travel to the real-time measurement position, the second synchronous multi-point thickness measurement component is triggered to acquire another thickness data frame. The same intra-frame interpolation processing is performed on the other thickness data frame to obtain the real-time thickness value at a grid point, and the real-time thickness value is assigned the inherent number of the same virtual trigger mark. All marks of the virtual measurement grid are traversed to obtain the real-time thickness distribution of the foam after foaming. The reference thickness distribution and the real-time thickness distribution are spatially registered according to the number of the virtual trigger marker to form a thickness distribution data pair that is point-by-point aligned in the width direction and synchronously matched in the travel direction.

3. The method according to claim 1, characterized in that, Specifically, S2 is: The reference thickness value and the real-time thickness value with the same virtual trigger mark in the thickness distribution data pair are identified as associated point pairs; Traverse the numbers of all virtual trigger markers in the virtual measurement grid, compare the real-time thickness value in the associated point pair corresponding to each number with the reference thickness value, and generate an initial expansion ratio value corresponding to the number of the virtual trigger marker; Arrange the initial expansion ratio values ​​corresponding to the numbers of all virtual trigger markers according to the spatial positions of each virtual trigger marker in the width direction and the travel direction to form an initial expansion ratio distribution map; The initial expansion ratio distribution map is subjected to background stripping processing, that is, the local bias component introduced by the difference in the thickness of the incoming material in the initial expansion ratio distribution map is identified, and the local bias component is removed from the initial expansion ratio distribution map to obtain the intrinsic expansion coefficient field stripped of the difference in the thickness of the incoming material. The intrinsic expansion coefficient field is output as an equivalent foaming ratio distribution.

4. The method according to claim 1, characterized in that, Specifically, S3 is: The equivalent foaming ratio value corresponding to each grid point position in the equivalent foaming ratio distribution is arranged according to the spatial coordinates of the virtual trigger mark corresponding to each grid point in the width direction and the travel direction to construct a spatial two-dimensional scalar field, wherein each grid point carries an equivalent foaming ratio value. Connect adjacent grid points with equal equivalent foaming ratio values ​​in the two-dimensional scalar field of space to form several equipotential lines that are closed or extend to the boundary. All equipotential lines together constitute an equipotential line skeleton map describing the geometric shape of the spatial distribution of foaming ratio. In the equipotential line skeleton map, the spatial distance between two adjacent equipotential lines in the normal direction is used as the scale. The rate of change of the equivalent foaming ratio value at each position of each equipotential line along the normal direction is calculated to obtain the expansion stress directional derivative, which characterizes the steepness of the change of the foaming ratio in space. The expansion stress directional derivatives at all positions of all equipotential lines are summarized to form the expansion stress directional derivative field. The numerical fluctuation amplitude of the directional derivative of the expansion stress at all locations in the expansion stress directional derivative field is extracted as the spatial non-uniformity at the current moment. In a continuous time series, the rate of change of the area of ​​the closed region enclosed by the same equipotential line in the equipotential line skeleton map with time is continuously tracked to obtain the area change rate. Using the area change rate as the basis for constructing the scaling strain rate field, the scaling strain rate is generated to describe the evolution trend of the non-uniform foaming scaling state in the time dimension.

5. The method according to claim 1, characterized in that, Specifically, S4 is: Traversing the closed region enclosed by each equipotential line in the strain rate field, the area change rate of each equipotential line is converted into topological tension acting on the boundary of the region enclosed by the equipotential line. The direction of the topological tension is outward along the normal of each point of the equipotential line, and the magnitude of the topological tension is proportional to the area change rate. The topological tension of all equipotential lines constitutes a topological tension field acting on the equipotential line skeleton pattern. Using the current equipotential line skeleton map as the initial configuration, the topological tension field is used as the driving force to drive the position migration of each equipotential line in the equipotential line skeleton map. During the migration, elastic constraints are applied between equipotential lines to keep the topological relationship between adjacent equipotential lines unchanged. Through iterative deformation simulation, until the deformation accumulation time reaches a preset control period, the predicted equipotential line skeleton map is obtained. In the predicted equipotential line skeleton map, the spatial distance between two adjacent equipotential lines in the normal direction is remeasured to obtain the predicted spacing distribution. Using the predicted spacing distribution as a benchmark and combining it with the equivalent foaming ratio value, the rate of change of the equivalent foaming ratio value along the normal direction at each position of each equipotential line is recalculated to obtain the updated expansion stress directional derivative, and all the updated expansion stress directional derivatives are summarized into the predicted expansion stress directional derivative field. The numerical fluctuation amplitude of the expansion stress directional derivative at all locations in the predicted expansion stress directional derivative field is extracted as the predicted spatial non-uniformity that will appear after a preset control period.

6. The method according to claim 1, characterized in that, Specifically, S5 is: The predicted spatial non-uniformity is compared with the preset uniformity allowable range. When the predicted spatial non-uniformity exceeds the uniformity allowable range, the excess deviation is calculated, and the pre-control instruction generation process is triggered. From the predicted expansion stress directional derivative field, search for regions where the directional derivative exceeds a preset threshold row by row along the width direction. Mark the regions that continuously exceed the preset threshold as over-limit deformation zones, and extract the center position coordinates of each over-limit deformation zone in the width direction, as well as the position of the equipotential line crossed by the over-limit deformation zone in the predicted equipotential line skeleton map. Based on the position of the equipotential lines crossed by each over-limit deformation zone in the predicted equipotential line skeleton map, the peak value of the topological tension borne by the corresponding equipotential line is extracted from the topological tension field, and the peak values ​​of the topological tension corresponding to all over-limit deformation zones are weighted and synthesized to obtain the desired correction tension field used to drive deformation correction. Using the desired tension field as input, the process is reversed along the normal direction of each equipotential line in the predicted equipotential line skeleton map, and the desired tension change is converted into the process parameter compensation amount that needs to be applied to each independent adjustment section in the foaming process execution unit. The process parameter compensation amount corresponds one-to-one with the width position covered by the independent adjustment section. The compensation amount of the process parameters is superimposed with the current process parameter setting value of the corresponding independent adjustment section to generate a pre-adjustment command for each independent adjustment section, and output to the foaming process execution unit for advance adjustment.

7. The method according to claim 1, characterized in that, Specifically, S6 is: After the foaming process execution unit completes the adjustment according to the pre-control instructions, S1 to S2 are executed again to obtain the newly generated equivalent foaming ratio distribution after adjustment, that is, the intrinsic expansion coefficient field after adjustment. Using the adjusted intrinsic expansion coefficient field as input, the steps in S3 are executed to construct the adjusted equipotential line skeleton map, and the adjusted spatial inhomogeneity is extracted from the adjusted equipotential line skeleton map. The adjusted equipotential line skeleton map is compared line by line with the predicted equipotential line skeleton map in S4. The difference between the predicted value and the actual value of the area of ​​the closed region enclosed by the same equipotential line is calculated to obtain the skeleton deformation residual distribution. The skeleton deformation residual distribution is fed back to the topological tension field, and the topological tension corresponding to each equipotential line in the topological tension field is corrected to obtain the corrected topological tension field after historical residual correction. The corrected topological tension field is used to drive deformation simulation in the next cycle S4. The adjusted spatial nonuniformity is used as the current spatial nonuniformity of the next cycle. The corrected topological tension field is used to update the topological tension field. Then, the steps of constructing the scaling strain rate field in S3 and the prediction calculation in S4 are re-executed to generate the updated prediction spatial nonuniformity and pre-control command, forming a closed-loop control based on skeleton deformation residual feedback.

8. The method according to claim 5, characterized in that, The topological tension is a vector field quantity with the rate of change of the area of ​​the closed region enclosed by the equipotential line as its amplitude and the direction of the outward direction of the normal to each point of the equipotential line as its direction. Its physical meaning is the equivalent driving force that causes the area of ​​the region enclosed by the equipotential line to change per unit arc length of the equipotential line.

9. The method according to claim 7, characterized in that, The skeleton deformation residual distribution is a set of area deviations obtained by performing line-by-line difference calculations on the predicted area of ​​the closed region enclosed by each equipotential line in the predicted equipotential line skeleton map and the measured area of ​​the closed region enclosed by the same equipotential line in the adjusted equipotential line skeleton map, using the area of ​​the closed region enclosed by each equipotential line as the comparison quantity.

Images