Surface roughness is a key indicator of the quality of mechanical products, which can precisely portray the fatigue strength, wear resistance, surface hardness and other properties of the products. The convergence of current machine-learning-based surface roughness prediction methods to local minima may lead to poor model generalization or results that violate existing physical laws. Therefore, this paper combined physical knowledge with deep learning to propose a physics-informed deep learning method (PIDL) for milling surface roughness predictions under the constraints of physical laws. This method introduced physical knowledge in the input phase and training phase of deep learning. Data augmentation was performed on the limited experimental data by constructing surface roughness mechanism models with tolerable accuracy prior to training. In the training, a physically guided loss function was constructed to guide the training process of the model with physical knowledge. Considering the excellent feature extraction capability of convolutional neural networks (CNNs) and gated recurrent units (GRUs) in the spatial and temporal scales, a CNN–GRU model was adopted as the main model for milling surface roughness predictions. Meanwhile, a bi-directional gated recurrent unit and a multi-headed self-attentive mechanism were introduced to enhance data correlation. In this paper, surface roughness prediction experiments were conducted on the open-source datasets S45C and GAMHE 5.0. In comparison with the results of state-of-the-art methods, the proposed model has the highest prediction accuracy on both datasets, and the mean absolute percentage error on the test set was reduced by 3.029% on average compared to the best comparison method. Physical-model-guided machine learning prediction methods may be a future pathway for machine learning evolution. Advanced machining technology is highly integrated with the latest achievements in modern science and technology. The current demand for high efficiency and quality in the machining industry has made high precision and high speed the core goals. Milling is widely used as an advanced machining technology in machining processes due to its high efficiency and productivity ,2. To enThe quantification of surface roughness during milling can be broadly divided into direct methods based on measurement and indirect methods based on modeling ,13. DireTo address a series of problems associated with direct methods based on measurements, researchers proposed indirect methods based on modeling to predict surface roughness. Indirect methods based on modeling can be broadly classified as physical and data driven. Physical models typically focus on mining the explicit physical relationships between surface roughness and the relevant input parameters, including the process parameters, tool vibration and the geometrical characteristics of the tool. Zhuo et al.  proposedWith the rapid development of data science and computing capability, data-driven models have gradually entered the limelight. Data-driven methods employ statistical and machine learning models for end-to-end surface roughness predictions, enabling highly accurate predictions in scenarios without expert domain knowledge. Huang et al.  construc(1)The novel surface roughness prediction model integrates physical knowledge with deep learning to effectively solve the problem of data scarcity. In a limited data scenario, the proposed model has more accuracy and interpretability of prediction results than current methods.(2)The loss function guided by the physical model constructed in the deep learning model is able to use existing mechanistic knowledge to train the model and guide the model to converge the learning route to the region in the function space that conforms to the physical rules, which effectively improves the training efficiency of the model.(3)Physical knowledge was introduced as an input into the training process of the deep learning model to guide the learning process, resulting in a significant improvement in the accuracy of milling surface roughness predictions.In this paper, a novel prediction model for milling surface roughness was proposed by combining physical knowledge with deep learning. This study incorporates the physical mechanism model into the deep learning model in contrast to the existing work in this direction, which enables deep learning to perform surface roughness prediction under the constraints of physical laws. The embedding of physical knowledge enhances the generalization ability and prediction accuracy, providing a new method for surface roughness prediction. This model introduced the physical model in two phases of deep learning: before training and during training. Before training, a milling surface roughness mechanism model with tolerable accuracy was constructed to perform data augmentation on the limited experimental data. Data augmentation based on physical models not only expands the size of the input data, but also introduces the embedded mechanistic knowledge into the prediction model, subjecting it to potential mechanistic constraints. The proposed model guides the training process through the physically guided loss functions to take advantage of physical knowledge during training. This guidance mechanism enables the learning route of the model to converge to the region in the function space that conforms to the physical rules, improving the training efficiency and prediction rationality. Considering the excellent feature extraction capability of CNNs and GRUs in spatial and temporal dimensions, a CNN-GRU model was adopted as the main prediction model for milling surface roughness. In addition, a bidirectional gated recurrent unit model was introduced to combine the acquired forward and reverse output features into an output feature map for enhancing the correlation between the sequence data. Meanwhile, a multi-headed self-attention mechanism was introduced to aggregate the dimensional weights of the extracted features in the bidirectional gated recurrent unit layer to enhance the correlation between the data. The main contributions of this paper are as follows:The prediction of surface roughness during milling is critical to the design and manufacturing process of machine tools and has attracted many researchers to work on it. Surface roughness prediction methods can be broadly classified into physical models and data-driven models. Physical models typically focus on mining the explicit physical relationships between surface roughness and input parameters, including process parameters, tool geometry characteristics and tool vibration. Zhuo et al.  developeWith the rapid development of data science and computing capability, data-driven models have gradually entered the limelight. Data-driven methods employ statistical and machine learning models for end-to-end surface roughness predictions, enabling accurate predictions without domain experts. Kashyzadeh et al.  developeIn the current field of data science, the ability to collect and create observational data far exceeds the ability to understand data. Although purely data-driven models are suitable for data observation, observation bias will lead to poor model generalization and unreliable results. Therefore, it is urgent to integrate physical knowledge into relevant machine learning models to provide strong theoretical constraints. Physics-informed machine learning (PIML) enhances the capabilities of machine learning through building relevant models on the basis of empirical data and existing physical knowledge.Typical supervised learning tasks such as classification and prediction are widely deployed in numerous application scenarios such as computer vision, time series analyses and multi-intelligent-body systems. Although deep learning methods have been the main solution to domain problems in recent years, there has been little integration of physical knowledge into spatio-temporal data tasks. Spatio-temporal data are data with both temporal and spatial dimensions and can be generated from irregularly spaced sensor networks in areas such as traffic , weatherThe capability of data collection in the current data science field far exceeds the capability of data comprehension. The convergence of a purely data-driven model to local minima may lead to poor model generalization or results that violate existing physical laws. Therefore, it is urgent to integrate domain knowledge into relevant machine learning models to provide strong theoretical constraints and inductive biases on the basis of observations. This paper combined physical knowledge with deep learning to propose a surface roughness prediction model based on physics-informed deep learning (PIDL), the structure of which is shown in The structure of the proposed model can be divided into three main parts, which are data augmentation, physical deep learning and roughness prediction. Part I is mainly responsible for generating the data used throughout the model and performing data augmentation through physical mechanisms. Part II performs feature extraction and learning based on CNN–GRU, where physical constraints have been introduced through the physically guided constructed loss function. Part III feeds the milling features derived from Part II into the prediction model for surface roughness prediction. The interaction between the three parts can be summarized as a progression, where Part I provides the data containing the physical laws for Part II, and Part II provides the surface roughness prediction model under the constraints of the physics for Part III. The proposed model includes physical knowledge before and during training in deep learning. Before training, data enhancement was performed on limited experimental data by constructing a surface roughness mechanism model with a tolerable accuracy. During training, a physically guided loss function was constructed to guide the training process of the model with physical knowledge. Considering the excellent feature extraction capability of convolutional neural networks and gated recurrent units in the spatial and temporal dimensions, a CNN–GRU model was adopted as the main prediction model for surface roughness. Meanwhile, bidirectional gated recurrent units and a multi-headed self-attentive mechanism were introduced to enhance data correlation. The following section describes the sub-models and their components included in the physically guided deep prediction model in detail.During mechanical milling, the machined surface of the workpiece tends to not be completely flat. As an indicator to monitor the quality of workpiece machining, surface roughness is directly related to micro-geometry errors such as small spaces and micro-scale peak and valley inhomogeneities in the machining plane ,37. The a to b is visualized in The physical modeling of surface roughness was carried out by combining geometric and mechanical models for the residual areas of the machined surfaces mentioned above. The specific flow of this physical modeling process is referred to in , and thiIn the actual machining process, the surface roughness of the workpiece is related to geometric factors and the mechanical properties of the material. The extrusion phenomenon of the tool on the surface of the workpiece during the cutting process results in elastic–plastic deformations of the machined surface. In view of the above influence of the mechanical properties on surface roughness, a corresponding physical model is constructed for the residual height, z-axis cutting force during cutting, The plastic deformation height, Deep learning has been demonstrated to be robust in terms of feature extraction, time series prediction, etc. This paper combines physical knowledge with deep learning to propose a physically guided surface roughness prediction model. The proposed prediction model consists of physical knowledge and neural networks. Considering the convincing feature extraction capability of CNNs for time-series data, this paper adopted a CNN for milling feature extraction to track the spatial pattern variations caused by process parameter differences. The CNN can feed pre-processed data directly into the model and extract the deep non-linear features hidden in it by combining multiple convolution and pooling layers. Since GRUs are capable of learning sequential and time-varying patterns from the original dataset, this paper employed a GRU for the dynamic tracking of surface changes during the tool life cycle. Compared with the LSTM architecture, GRUs have the advantages of faster convergence and a fewer number of parameters. This paper adopted a hybrid deep learning model, CNN–GRU, as the neural network part of the prediction model. Physical knowledge was mainly integrated in the training phase and the loss function part in order to use the physical knowledge to guide the training process of the model.CNNs are one of the most successful deep learning methods, and their network structure can be divided into 1D-CNNs, 2D-CNNs and 3D-CNNs. A 1D-CNN is very suitable for time series analyses of sensor data or analyses of periodic signal data, and its specific structure is shown in M, then the dimension of the extracted convolutional features is The CNN architecture typically consists of a convolutional layer and a pooling layer to filter and extract useful features from the input data. The leftmost part of the figure shows the input multi-dimensional time series data. Each convolutional layer has a corresponding convolutional kernel, and each colored box on the input data in the figure represents a filter. The filter slides top-down over the entire input matrix, producing convolutional features of the input data through a contained coefficient matrix. The dimension of the convolutional features extracted by the filter is Recurrent neural networks (RNNs) are capable of memorizing historical data features and are applicable to a variety of sequential data problems. Gated recurrent unit (GRU) neural networks selectively filter and remember historical information through update gates and reset gates, solving the problems of gradient disappearance, gradient explosion and poor long-term memory of traditional RNNs. Due to the temporal characteristics of vibrational data, this paper employs a GRU to extract vibration sequence features. The detailed internal structure of the GRU model is shown in A multi-headed self-attention mechanism can enhance the correlation between dimensions of sequence data, and is widely used in the fields of natural language processing and image processing. This paper set the head number of the multi-headed self-attentive mechanism to eight, and its specific structure is shown in The output feature mapping of the bidirectional GRU served as its input tensor, and the three feature matrices Query, Key and Value were calculated with The multiple Z matrices calculated above were executed and transformed into the target output tensor X*. The input tensor and the output tensor have the same data structure, and all dimensions of the output tensor take the relevance of the remaining dimensions into account. This mechanism can fully extract and enhance the correlation between the vibrational data dimensions.The bidirectional GRU mainly consists of three GRU layers and a multi-headed self-attentive mechanism. In order to ensure a reasonable training time at the current model parameter scale, the first GRU layer is a bidirectional GRU layer that extracts vibrational data location features in both the forward and reverse dimensions. Then, a multi-headed self-attentive mechanism was placed after the bidirectional GRU layer for weight setting. This mechanism enables aggregating the dimensional weights of the extracted output features in the bidirectional GRU layer to enhance data correlation. Finally, the remaining GRU layers were set as normal GRU layers for extracting the features of the output matrix from the multi-headed self-attentive mechanism.Physical knowledge was introduced in the training phase and loss function part of the above deep learning model to guide the training process of the model with physical knowledge. Compared to the conventional loss function, the proposed physically guided loss function was capable of capturing dynamic patterns that can be generalized in the framework of defined physical laws, and its mathematical model is shown in Equation (19).The specific expression of physical knowledge normally depends on factors such as milling material, milling environment, static parameter settings, etc. Take S45C steel as an example. In the practical milling process, the surface roughness is positively correlated with the vibration frequency, i.e., the surface roughness increases with the increase in vibration frequency. The vibration frequency decreases as the spindle speed increases. When the feed speed falls within the range of In this section, the proposed method was compared with existing methods on open datasets. Moreover, ablation experiments were conducted to demonstrate the effectiveness of each component of the physically guided surface roughness prediction method.https://drive.google.com/drive/folders/1M6dgDNYhHbp8iFmNXeV9HrL6x6AvSshx?usp=share_link, accessed on 3 January 2019. The tool specifications used in the experiments are shown in The experiments verify the effectiveness of the proposed method with two open-source milling datasets. The first dataset was the open-source S45C steel milling dataset from Chung Hsing University, Taiwan, China, which performed cutting operations on S45C steel with tungsten carbide milling cutters . T. Wu aIn order to verify the proposed approach and evaluate the accuracy of predicting the workpiece surface roughness, a tungsten carbide milling cutter was utilized to cut the steel S45C in the experiment. A total of nine steel blocks with three different clamping torque straight cutting trajectories were used in the experiments. The cutting parameters include various spindle speeds, feed speeds, feed volumes, etc. The different combinations of cutting parameters and the expected material removal rate (MRR) of the cutter are shown in x-axis and 8/80/800 μm in the y-axis with the corresponding resolutions of 0.0001/0.001/0.01 μm.A priori knowledge shows that the vibration signal is sensitive to tool wear and is therefore a good indicator of tool wear and surface roughness. The first and second rows in x, y and z axes vibrations during micro milling operation were collected. The monitoring system platform is equipped with a Deltatron 4519-003 Brüel and Kjaer monoaxial accelerometer on the x, y, z-axes, with a sensitivity of 10.58 mV/g and a bandwidth of 20 kHz. The sensor was connected to a PCB PZT 482A22 amplifier to supply power in order to obtain an acceptable voltage measurement which can be easily processed by most data acquisition systems. The vibration signals were fed into an NI 6251 National Instruments data acquisition card, with a sampling frequency of 50 kHz, and processed in a National Instruments high performance PXI 8187 embedded controller. Both of the above open-source datasets are allowed to be replicated, manipulated and used in other works, and since the same techniques and processes are reproducible in practical applications in manufacturing engineering, the above datasets were used for experiments in this study.The second dataset was the micromechanical milling dataset from the GAMHE 5.0 pilot line. The Workstation 4 (WS4) of the GAMHE 5.0 pilot line is a Kern Evo high precision machining center with a maximum spindle speed of 50,000 rpm, equipped with a Blum laser system for running micro-milling and micro-drilling operations. This dataset contains results from micro-milling operations on a sintered tungsten–copper alloy (W78Cu22) with 0.3 mm, 0.5 mm, 0.8 mm and 1 mm diameter milling cutters. Five cutting parameters were considered during the experimental machining process: spindle speed, feed rate, axial depth of cut, radial depth of cut and milling tool radius. The different combinations of cutting parameters are shown in i-th sample and Three performance evaluation metrics, namely the coefficient of determination (R2), the root mean square error (RMSE) and the mean absolute percentage error (MAPE), were adopted to assess the performance of the proposed model. The root mean square error is a typical metric for regression models to indicate the extent of the error that the model will produce in its predictions, representing the sample standard deviation of the difference between the predicted and true values. The mean absolute percentage error is a relative error measure that avoids positive and negative errors canceling each other out through absolute values, and is commonly adopted to measure the accuracy of various time series prediction models. The coefficient of determination reflects how accurately the model fits the data and usually takes a value in the range [0–1]. The coefficient of determination is positively correlated with the explanation ability of the current equation variables of the target quantity. The computation of R2, RMSE and MAPE is shown in Equations (23)–(25), where In this paper, the following models were used as for comparison to verify the validity of the proposed model:(1) ANNs : This s(2) LSTM : This s(3) GRU : This s(4) SAE–LSTM : This s(5) CNN–LSTM : This s(6) CNN–BGRU : This sThe proposed surface roughness prediction model was validated on the open-source S45C steel milling dataset and the micromechanical milling dataset in the GAMHE 5.0 test line and compared with classical and state-of-the-art methods proposed in recent years. Considering the generalization and persuasiveness of the two dataset divisions, we selected 80% of the dataset as the training set and 20% as the test set based on 10-fold cross-validation to fully demonstrate the performance of the proposed model. The comparison results of the evaluation metrics for each prediction method are shown in As can be seen from In the following, the experimental results of each prediction model on the training and testing sets are combined to analyze the reasons for the improvement presented by the proposed model.(1) In this paper, a model of the physical mechanisms of milling surface roughness was constructed by combining geometric and mechanical models of the residual area of the machined surface. The data augmentation of the milling experimental dataset through this mechanism model introduced the physical knowledge on the milling process while enhancing the data size to make the learning of the prediction model more reasonable and adequate.(2) The spatial and temporal features were extracted from the input time series data via convolutional neural networks and gated recurrent units. To enhance the correlation between the sequence data, a bidirectional GRU was introduced to combine the acquired forward and reverse output features into an output feature map. In addition, a multi-headed self-attention mechanism was introduced to aggregate the dimensional weights of the extracted output features in the bidirectional GRU layer to enhance the correlation between the data.(3) Physical knowledge was introduced into the training phase and loss function part of deep learning to guide the training process of the model. The physically guided loss function can help the model to converge the learning route to the region in the function space that conforms to physical laws, which effectively improves the training efficiency and physical rationality of the model.Time series segments of the same size as S45C were cropped from the GAMHE 5.0 dataset to verify the correlation between model effects and dataset size, as well as to check the stability of the proposed model with different data sizes. Surface roughness prediction experiments were conducted on the reduced-scale GAMHE 5.0 dataset, and a comparison of R2 metric values for each model on the test set is shown in To investigate the effect of different components in the proposed model, the following ablation experiments were performed, where “-” indicates the removal of the current component. “Origin” denotes the complete model proposed in this paper. “-Augmentation” denotes the removal of the data augmentation based on the physical mechanism model. “-CNN” denotes the removal of the feature extraction module based on a CNN. “-Bidirectional” denotes the removal of the bidirectional mechanism of the gated recurrent units. “-Multi-headed” denotes the removal of the multi-headed self-attentive mechanism. “-Loss” denotes the removal of the physical guided loss function. “-Physical” denotes the removal of all physical knowledge introduced in the prediction model, specifically data augmentation and physically guided learning. The proposed model was used on two open-source datasets after removing different components, and a comparison of the model effects on the test set is shown in Through the above ablation experiments, it can be seen that each component of the proposed model contributes to prediction, and the removal of any component will lead to a decrease in the prediction accuracy. Specifically, the prediction accuracy of the model decreased the most for both datasets when the CNN or all physical knowledge introduced into the model was removed. The proposed method performed milling feature extraction by the excellent feature extraction capability of the CNN on time-series data to track the spatial pattern variation caused by the process parameter differences. In terms of the introduction of physical knowledge, it was specifically manifested in data augmentation based on physical mechanism models and physically guided loss functions. Data augmentation enabled the introduction of physical knowledge embedded in the milling process while enhancing the data size, allowing for more adequate learning of the predictive model and improving the generalization of the model. The physically guided loss function can help the model to converge the learning route to the region in the function space that conforms to physical laws, which effectively improves the training efficiency and physical rationality of the model. Then, the performance of the model degrades to some extent both when the data augmentation based on the physical mechanism model and the physically guided loss function were dropped separately. Finally, when the microscopic component of the gating recurrent units was dropped, the model performance slightly degraded. Concretely, the multi-headed component can sufficiently extract correlations between vibrational data dimensions to enhance the connections between them. The bidirectional component can extract the position features of the input data along the forward and reverse dimensions, effectively improving the prediction accuracy. In summary, the removal of the above six components individually in turn all resulted in different degrees of degradation in the model performance. In particular, the absence of the introduced physical knowledge resulted in a clear decrease in accuracy. The reason for the reduction in the training efficiency and physical reasonability is that the model has lost the ability to converge the learning route to a physically lawful region of the function space. Therefore, each component of the model contributes to the model performance.Through validation experiments on two open-source datasets, the proposed surface roughness prediction model based on deep learning of physical information has better performance compared to comparison models. The experiments demonstrate that convolutional neural networks possess stronger feature extraction capabilities than stacked autoencoders. At the same time, the bidirectional gated recurrent unit can improve the efficiency and learn patterns and connections embedded in time series data for both forward and reverse sequences more comprehensively. The values of both the RMSE and the MAPE increase from the training phase to the testing phase, while the opposite occurs for R2, indicating that the model may be underfitted or overfitted. This is primarily due to the weak generalization ability of the model, resulting from the small training sample. This paper mitigated the overfitting problem of the model by a 10-fold cross-validation. The data augmentation with the constructed physical mechanism model introduced physical knowledge of the milling process while augmenting the amount of data to make the learning of the prediction model more reasonable and adequate. Then, spatial and temporal features were extracted from the input time series data by a convolutional neural network and a gated recurrent unit. To enhance the correlation between sequence data, a bidirectional GRU and a multi-headed self-attention mechanism were applied. Physical knowledge was introduced into the training phase and the loss function part of deep learning to guide the training process of the model, leading the model to converge the learning route to the region in the function space that conforms to physical laws, effectively improving the training efficiency and physical rationality. In the future, this work will further explore surface roughness prediction in online scenarios to ensure the applicability of the proposed model to most manufacturing scenarios such as industrial, medical, and military scenarios.Surface roughness has a serious impact on surface hardness, wear resistance and fatigue strength of the product and is a key evaluation index for evaluating the quality of mechanical products, making it significant to study the prediction and optimization of surface roughness. The convergence of machine learning methods to local minima may lead to poor model generalization or results that violate existing physical laws. Therefore, this paper combined physical knowledge with deep learning to propose a physics-guided surface roughness prediction model. This model introduced physical knowledge before and during the training of deep learning. Before training, data augmentation was performed on the limited experimental data by constructing a surface roughness mechanism model with tolerable accuracy. During training, a physically guided loss function was constructed to guide the training process of the model with physical knowledge. Considering the excellent feature extraction capability of convolutional neural networks and gated recurrent units in spatial and temporal dimensions, a CNN–GRU model was adopted as the main prediction model for surface roughness. In addition, the bidirectional GRU and multi-headed self-attention mechanism were introduced to enhance the data intercorrelation. Surface roughness prediction experiments were conducted on the open source S45C steel milling dataset and the micromechanical milling dataset in the GAMHE 5.0 test line. Through a comparison with the prediction results of the latest methods, the proposed model has the highest prediction accuracy on both open source datasets, with the mean absolute percentage error on the test set reduced by 3.029%. In addition, ablation experiments were also conducted on different components to verify the validity of the proposed model and its components.In the future, our research interests lie in investigating how to achieve precise and efficient optimization of the machining parameters in order to greatly improve the product quality and production efficiency while reducing costs. Moreover, although the proposed prediction model has the highest prediction accuracy on two open-source datasets, it did not directly consider tool wear as a model parameter, but indirectly characterized it through the material removal rate. Therefore, an exploration of the direct relationship between tool wear and surface roughness predictions will be carried out in a follow-up work. The prediction model proposed in this paper is temporarily applicable to surface roughness prediction in offline scenarios where online real-time inspection tasks are not possible. Therefore, the direction of future research lies in how to implement online inspection techniques in this framework. This study performed experimental validation using two existing open-source datasets. Subsequent milling experiments will be conducted and data will be collected personally, and whether there is a correlation between the selection of cutting parameters and surface roughness predictions will be explored within the course of the study. The application of physical knowledge to problems related to overfitting and underfitting will also be of interest.


Efficient routing in urban vehicular networks is essential for timely and reliable safety message transmission, and the selection of paths and relays greatly affects the quality of routing. However, existing routing methods usually face difficulty in finding the globally optimal transmission path due to their greedy search strategies or the lack of effective ways to accurately evaluate relay performance in intricate traffic scenarios. Therefore, we present a vehicular safety message routing method based on heuristic path search and multi-attribute decision-making (HMDR). Initially, HMDR utilizes a heuristic path search, focusing on road section connectivity, to pinpoint the most favorable routing path. Subsequently, it employs a multi-attribute decision-making (MADM) technique to evaluate candidate relay performance. The subjective and objective weights of the candidate relays are determined using ordinal relationship analysis and the Criteria Importance Through Intercriteria Correlation (CRITIC) weighting methods, respectively. Finally, the comprehensive utility values of the candidate relays are calculated in combination with the link time and the optimal relay is selected. In summary, the proposed HMDR method is capable of selecting the globally optimal transmission path, and it comprehensively considers multiple metrics and their relationships when evaluating relays, which is conducive to finding the optimal relay. The experimental results show that even if the path length is long, the proposed HMDR method gives preference to the path with better connectivity, resulting in a shorter total transmission delay for safety messages; in addition, HMDR demonstrates faster propagation speed than the other evaluated methods while ensuring better one-hop distance and one-hop delay. Therefore, it helps to improve the performance of vehicular safety message transmission in intricate traffic scenarios, thus providing timely data support for secure driving. Over the past few decades, the surge in urban vehicle numbers has intensified challenges in city transportation systems, notably in traffic safety and efficiency. The Vehicular Ad Hoc Network (VANET) has emerged as a cornerstone for Intelligent Transportation Systems (ITSs), offering a promising solution to these challenges . VANET hOwing to the dynamic nature of the VANET topology, vehicles often grapple with inconsistent communication links amongst themselves. Such instability can quickly escalate to network congestion, packet loss, and significant end-to-end delays. In crisis situations like traffic accidents, it is imperative that safety messages are dispatched promptly and reliably to the intended destination or vehicle ,12. The As shown in For path selection, many current routing approaches tend to concentrate solely on identifying the next best intersection from a given point. This narrow focus can result in localized optimization issues. Regarding relay selection, these methods often prioritize one-hop distance or link quality, neglecting the interplay and significance of multiple evaluation metrics. This oversight complicates the accurate evaluation of relay performance in intricate traffic scenarios. Existing routing methods, therefore, face difficulty in determining a globally optimal transmission path or are unable to accurately assess relay performance, leading to poor performance in routing in intricate traffic scenarios, which affects timely data support for secure driving. It is, therefore, necessary to design an efficient routing method that takes into account the selection of globally optimal transmission paths and relays for timely and reliable transmission.For the identification of a global optimal transmission path, HMDR initiates by constructing a path tree model spanning from the destination intersections back to the source intersection. It then pinpoints the most efficient route through a heuristic path search that emphasizes road section connectivity;To accurately evaluate the performance of candidate relays, the HDMR method employs a multi-attribute decision-making-based approach. This calculates the subjective and objective weights of candidate relays based on the ordinal relationship analysis method and the Criteria Importance Through Intercriteria Correlation (CRITIC) weighting method, respectively;The innovative HMDR strategy synergistically blends path search with relay selection, equipping it to manage safety message transmission in multifaceted traffic scenarios. Our experimental results underscore the effectiveness and proficiency of HMDR in this domain.In this study, we design a heuristic path search and multi-attribute decision-making-based routing (HMDR) method for vehicular safety messages. Our main contributions are as follows:The remainder of this paper is organized as follows: Existing VANET routing methods are mainly categorized into network-topology-based, cluster-based, and geographic-location-based types. Network-topology-based routing methods store the routing information of the network nodes in a routing table and periodically update the routing table to maintain its availability. Perkins et al. proposed a hop-by-hop Destination Sequenced Distance Vector (DSDV) routing method , whereinCluster-based routing approaches use hierarchical network organization by cluster head nodes in order to reduce network communication and, thus, network resource overhead. Knowing how to select cluster head nodes and optimize the data transmission path to balance the network load and reduce the network communication overhead are urgent optimization problems. Zhang et al. proposed a new algorithm—AODV-MEC—for AODV clustering based on an edge computing strategy , which cLocation-based routing methods use the location information between vehicles to decide the relay of route transmission without consideration of route discovery, route maintenance, and network topology, which is more suitable for high-mobility vehicular networks. Karp et al. proposed a greedy peripheral stateless routing protocol named Greedy Perimeter Stateless Routing (GPSR) , which cThe key to the safety message routing problem in vehicular network environments lies in knowing how to select appropriate routing paths and relays and is reflected in the transmission delay, speed, and coverage of safety messages. However, combined with the above description and summary, existing routing methods often fail to solve the key problem described above, which affects timely data support for secure driving. In this study, we intend to study and propose the method of combining path and relay selection, which integrates the effects of transmission paths and relay nodes on the quality of communication links and message propagation speed to adapt to intricate traffic scenarios.The system model and related assumptions are given in this section, and the meaning of the main variables frequently used in this paper are listed in Initially, the vehicles with Poisson distribution travel in the urban traffic scenario with multiple lanes in both directions;Each vehicle is equipped with an On-board Unit (OBU) that utilizes a Dedicated Short-range Communication (DSRC) interface for Vehicle-to-vehicle (V2V) communication and obtains location information based on GPS and electronic map;Each vehicle receives road section information from the equipped electronic map, as well as periodical Beacon messages  from its neighbors, and maintains a road section information table and a neighbor information table;Both buildings and trees attenuate the RSS of safety messages, and each vehicle records the RSS values of monitored neighboring vehicles in the neighbor information table.In our study, an urban vehicular network environment within a certain geographical range is analyzed as a whole. A system model of our study is shown in The main process of HMDR is shown in In the optimal path search phase, the source and destination intersections are firstly determined based on the geographical location of the source node and the target node; then, the path model, based on the connectivity of road sections, is established according to road-section-related information, and the least costly path is finally selected as the optimal routing path.In the optimal relay selection phase, the current forwarding vehicle first calculates the utility values of the four evaluation metrics of candidate delays by using corresponding utility functions and constructs the decision matrix. Then, it calculates the subjective and objective weights of the metrics of candidate relays by using ordinal relationship analysis and CRITIC weighting methods, respectively, and obtains the comprehensive utility values reflecting the performance of candidate relays. Finally, it calculates the node with the biggest comprehensive utility value combined with the link time as the next forwarding node of the safety message.When a candidate relay is selected as the optimal relay and receives the safety message, it becomes a new forwarding node and continues to select a new relay based on HDMR. The above steps are repeated until the safety message reaches the destination after multi-hop forwarding.Intersections in urban environments are usually regarded as key nodes for finding routing paths. In the first optimal path search phase, the HMDR method determines the source and destination intersections based on the geographical locations of the source and target vehicles and constructs a path tree model to obtain all possible transmission paths; then, the connectivity value of each road section is calculated. Finally, the optimal path with the least cost is determined using a heuristic search method.V is the number of vertices  and E is the number of edges . As the number of intersections increases, the path tree still traverses all of them quickly. In addition, the path tree avoids path loops so that there are no duplicate segments in the routing path.In this subsection, we use a tree structure to search for routing paths, called the path tree model. The path tree stores the relationship between intersections in the form of an adjacency table and adopts the depth-first search method for traversing the routing paths. The traversal time complexity of a path tree is The selection of source and destination intersections is related to the geographic location of the source and target vehicles. If the source or target vehicle is located at an intersection, the intersection is considered the source or destination intersection; if the source vehicle is located at a road section, the intersection closer to the destination intersection is considered the source intersection; and if the target vehicle is located at a road section, the intersections at both ends of the road section are considered the destination intersections. As shown in After determining the optimal path search area, the path model can be created by extending from the source node to the destination node. The source node is recorded as layer 0 nodes, its adjacent intersections as layer 1 nodes, the subsequent adjacent intersections as layer 2 nodes, etc., until the destination intersection is covered, and a completed path model is shown in Each path tree starts at a destination intersection and ends at the source intersection;When an intersection is added to a path tree, it is removed from the set of intersections to which it belongs;i;The children of an intersection in layer The construction of all path trees is completed when the intersection set is After completing the path model, path trees are constructed using the following rules:L and H. A depth-first traversal algorithm is used to find all available paths starting from the root and ending at the leaf nodes, the reverse paths of which are feasible paths from the destination to the source node, including According to the above rules, path trees with the destination intersection as the root are generated. As shown in In addition to the length of the routing path, the connectivity of the routing path, also known as the connectivity of the communication link, is an important factor that affects the message transmission delay, and since the routing path is composed of multiple road sections, it is necessary to analyze the connectivity of road sections in order to select a more reliable routing path.R is the communication radius of vehicles and In a V2V communication-based vehicular network environment, each vehicle obtains road and vehicle-related information based on an installed electronic map and receives periodic Beacon messages from its neighbors. Referring to the literature , the spek vehicles occurring in the counting interval on According to the Poisson distribution, the probability of To calculate the connectivity of Equation : (3)Pijconis and d are the start and end of the selected segment, respectively. Combining the path tree model and connectivity analysis of road sections, the optimal routing path from the source node to the target node is selected using the following criteria. The specific selection criterion for the optimal path is that the path has connectivity above a certain level and is of the shortest length. The selection of the optimal path can be formalized as follows: During the multi-hop transmission of safety messages on the optimal routing path, the optimal relay node is selected at each hop. The HMDR method selects four important metrics for relay selection and analyzes the subjective and objective weights of each metric to obtain a table for the comprehensive utility values of all candidate relays. After excluding the “edge nodes” that are about to leave the communication range, the optimal relay node is selected as the node that forwards the safety message, i.e., the node with the first rank in the comprehensive utility value transmits the message.m vehicles are recorded in the neighbor information table of the current forwarding vehicle and constitute the set of candidate relays n metrics for evaluating its performance and constitutes the set of evaluation metrics H for relay selection after data preprocessing based on related utility functions.To facilitate the description and solution of the problem, it is assumed that The proposed HDMR method selects four important metrics, i.e., the relative distance and relative speed between the current forwarding vehicle and the candidate relay, the RSS of the candidate relay, and area density, to evaluate the performance of the candidate relay. First, the data need to be preprocessed using the corresponding utility functions for the four detected metrics.(1) Relative distancejth candidate relay is The relative distance between the current forwarding vehicle and the candidate relay is the next hop distance for safety message forwarding. If the relative distance is shorter, the safety message requires more hops to cover the target area or reach the destination. Assuming that the location of the current forwarding vehicle is R is the transmission radius.The utility value of the relative distance (2) Relative speedjth candidate relay is i and the candidate relay j can be expressed as follows: The relative speed between the current forwarding vehicle and the candidate relay is another important factor that affects the link stability. The smaller the relative speed, the less likely the candidate relay will easily leave the communication range of the current forwarding vehicle, which also means the higher the link stability. Assuming that the speed of the current forwarding vehicle is The utility value of the relative speed (3) RSSjth candidate relay recorded in the neighbor information table of the current forwarding vehicle is The RSS of the candidate relay is a key factor in ensuring whether the safety message can be successfully received or not, and a larger RSS value indicates that the safety message is more likely to be successfully received. The value of RSS depends on the distance between vehicles; the further the distance, the smaller the RSS value. Assuming that the RSS value of the (4) Area densityj and its utility value are as follows: The area density of a candidate relay is the number of neighboring vehicles per unit distance length, which reflects the degree of traffic congestion on the lanes; too small an area density may lead to link interruptions, while too large an area density is more likely to generate message conflicts. Only if the density is appropriate can it be guaranteed that the selected relay will achieve a higher quality of routing. The area density of candidate relay (5) Decision matrixH is obtained as follows: j-th evaluation metric of the i-th candidate relay.The initial evaluation metrics of candidate relays are normalized using the above utility functions, and the initial standardized decision matrix In this section, we select four evaluation metrics and use a multi-attribute decision-making-based method to evaluate the performance of candidate relays. In order to obtain comprehensive utility values that can more accurately reflect the performance of candidate relays, the subjective and objective weights of the candidate relays are comprehensively considered and calculated based on ordinal relationship analysis and CRITIC weighting methods, respectively.(1) Subjective weightsOrdinal relationship analysis is an improved Analytic Hierarchy Process (AHP) method without a consistency test, which is adopted to calculate the subjective weights of evaluation metrics of candidate relays in this section.n evaluation metrics, which indicates that the adjacent evaluation metric Assuming that there is an ordinal relationship The subjective weight of metric According to the proportional relationship between adjacent evaluation metrics, the subjective weight of metric Equation : (16)wsu(2) Objective weightsThe CRITIC weighting method is an objective empowerment method based on the volatility and the correlation of data, which is utilized to calculate the objective weights of evaluation metrics of candidate relays in this section.j-th metric; then, calculate j-th metric of candidate relays.The volatility of data is expressed in the form of standard deviation. The larger the standard deviation of a metric, the more information it reflects and the more weight it should be assigned. Let i-th and j-th metrics; then, the correlation coefficient for metric The correlation of data is expressed by the correlation coefficient. The larger the correlation coefficient of a metric, the more it reflects the same information and the less weight it should be assigned. Let Considering the above two aspects together, the objective weight of metric (3) Comprehensive weights and utility valuesWe evaluate the performance of candidate relays from both subjective and objective aspects. After obtaining the subjective and objective weights of the metrics, the adjustment coefficient Similarly, the comprehensive weights of other metrics can be calculated. Finally, the comprehensive utility values of candidate relays are calculated by combing the comprehensive weights and the decision matrix through the simple weighting method, and the comprehensive utility value Candidate relays with higher comprehensive utility values exhibit better performance, as evidenced by better link quality while maintaining longer one-hop distances.Theoretically, selecting the candidate relay with the higher comprehensive utility value can result in greater gains; however, due to the high-speed mobility of vehicles, candidate relays at the edge of the current forwarding vehicle’s communication range may leave that range in a short period of time, resulting in the interruption of the communication link between them and routing failures. Therefore, these “edge” vehicles need to be removed before selecting a relay. Here, the method of the literature ,26 is utA, B, C, and D are calculated as follows: Suppose that we now need to estimate the link time between vehicles The current forwarding vehicle In this section, we implemented and evaluated our proposed HDMR method and comparative methods in a simulated vehicular environment built via MATLAB, an experimental urban traffic scenario with multiple lanes in both directions, as shown in In this study, the subjective and objective weights of four metrics are comprehensively considered in evaluating candidate relays and weighted via the adjustment factor It can be seen that the subjective weight calculation method based on the ordinal relationship method combined with expert experience can better reflect the performance of candidate relays and effectively improve the performance of HDMR as its proportion increases. The objective weight calculation method based on CRITIC cannot effectively reflect the importance of the evaluation metrics of candidate relays and has difficulty in accurately selecting relays with better overall performance when their proportion is higher; however, it can effectively differentiate between the metrics that have a certain degree of correlation, and when the objective weight obtained based on it is combined with the subjective weight, it can, to a certain extent, improve the accuracy of the evaluation of the candidate relay, which is conducive to the selection of the best relay.From In the second case, the length of the road section ranges from 2500 to 6500 m and the transmission radius is 100, 200, and 300 m. From The selection of paths and the selection of relays are the two main factors that affect the performance of routing methods; therefore, several typical methods that can improve the selection strategies of both are selected for comparison and, thus, used to verify the effectiveness of the proposed HMDR. Specifically, four methods, namely, BRAVE ; MISR 2; IGCR 2; OPBR 3; and simThe transmission of safety messages is a multi-hop forwarding process; first of all, the total hops, the total transmission distance, and the total transmission delay of the safety message from the source node to the destination node are compared under different traffic density conditions. The comparison results are shown in BRAVE always selects the shortest routing path as it uses Dijkstra’s algorithm, and OPBR is also able to find the shortest path in our experimental scenario by continuously selecting relays. Both of them try to select a candidate relay with a longer distance within the one-hop range; thus, they have smaller total hops and transmission distances. By considering the connectivity of road segments, the other four methods select relatively long routing paths. In addition, compared with MISR and IGCR, our proposed HMDR and simplified HMDR methods further enhance the transmission reliability at the expense of reducing the one-hop distance, which is beneficial to reducing transmission delay, despite having the maximum number of hops. Specifically, the total number of hops for the proposed HMDR method is similar to that of the simplified HMDR method, while it is approximately 10.00% to 21.55% more than BRAVE and OPBR and approximately 5.22% to 13.71% more than MISR and IGCR. Moreover, the total transmission distance of HMDR is also similar to that of the simplified HMDR, MISR, and IGCR, while it is approximately 5.40% to 6.34% more than BRAVE and OPBR. Moreover, as can be seen in The average one-hop distance, the average one-hop delay, and the average speed of the safety message from the source node to the destination node are then compared under different traffic density conditions. The comparison results are shown in As traffic density increases, the average one-hop distance increases for all methods. Different from other methods, our proposed HMDR and simplified HMDR simultaneously consider multiple evaluation metrics when selecting relays and use a combination of subjective and objective methods to evaluate the comprehensive weights of candidate relays; thus, the link stability is higher, although the one-hop distance of the selected relay is shorter. The other methods also simultaneously consider multiple factors to select a path but lack a predictive method for link stability and, therefore, have a higher probability of link interruption. Specifically, the average one-hop distance and one-hop delay of HMDR are similar to simplified HMDR; the average one-hop distance is approximately 3.43% to 10.03% shorter than the other methods, and the average one-hop delay of HMDR is approximately 8.55% to 26.98% lower than the other methods. Furthermore, as traffic density increases, the average transmission speed of each method decreases, which is mainly due to the increase in transmission delay. As can be seen in Currently, urban traffic safety is an important research topic, and safety message routing for vehicular networks is of great research interest and application value. The V2V-based routing method is more suitable for application in urban scenarios and does not have to rely on the construction of infrastructure, such as RSUs, reflecting its advantages of low latency and low cost. In this study, we proposed a vehicular safety message routing method HMDR that combines path search and relay selection for timely and reliable transmission in intricate traffic scenarios. It uses a heuristic path search method based on road connectivity to select the optimal global routing path to avoid the local optimal problem of path searching. It selects the optimal relay node based on multi-attribute decision-making to accurately evaluate the relay performance in intricate traffic scenarios. The experimental results show that the proposed HMDR tends to choose the path with better connectivity and shorter length, with the characteristics of a lower delay and a higher speed, which is superior to other methods. Therefore, it can be said that HMDR helps to improve the performance of vehicular safety message transmission in intricate traffic scenarios and provides timely data support for secure driving.However, V2V communication-based routing methods face the problem of poor road connectivity when in sparse traffic and message collisions when in dense traffic, which is less suitable for long-distance transmission in urban scenarios; while V2I-based communication effectively improves communication quality and transmission distance, V2I-assisted safety message routing methods will be studied in future work.


This study investigates the mechanical behavior of the Physiomesh/Securestrap system, a hernia repair system used for IPOM procedures associated with high failure rates. The study involved conducting mechanical experiments and numerical simulations to investigate the mechanical behavior of the Physiomesh/Securestrap system under pressure load. Uniaxial tension tests were conducted to determine the elasticity modulus of the Physiomesh in various directions and the strength of the mesh-tissue-staple junction. Ex-vivo experiments on porcine abdominal wall models were performed to observe the system's behavior under simulated intra-abdominal pressure load. Numerical simulations using finite element analysis were employed to support the experimental findings. The results reveal nonlinearity, anisotropy, and non-homogeneity in the mechanical properties of the Physiomesh, with stress concentration observed in the polydioxanone (PDO) stripe. The mesh-tissue junction exhibited inadequate fixation strength, leading to staple pull-out or breakage. The ex-vivo models demonstrated failure under higher pressure loads. Numerical simulations supported these findings, revealing the reaction forces exceeding the experimentally determined strength of the mesh-tissue-staple junction. The implications of this study extend beyond the specific case of the Physiomesh/Securestrap system, providing insights into the mechanics of implant-tissue systems. By considering biomechanical factors, researchers and clinicians can make informed decisions to develop improved implants that mimic the mechanics of a healthy abdominal wall. This knowledge can contribute to better surgical outcomes and reduce complications in abdominal hernia repair and to avoid similar failures in future. Each variant of implant location will differ in terms of both the forces acting on the implant, the healing and recovery process, and the risk of recurrence.Surgeons operating on anterior abdominal wall defects have at their disposal several potential spaces to insert the implant. The mesh can be placed between the fascia and the subcutaneous tissue (onlay), under the rectus abdominis muscles (retrorectus), between the fascia and the peritoneum (underlay), and directly in the peritoneal cavity 4. IPOM requires placing the implant directly into the peritoneal cavity, to cover the abdominal wall defect with an appropriate margin, and fixing it with special joints . That is connected with exposing the abdominal organs to direct contact with the mesh. Main concern of this way is a possible organ-mesh adhesion, which in turn may disturb the intestinal passage, and in critical situations even lead to fully symptomatic obstruction5. Initial optimism related to the introduction of IPOM was constantly reduced due to long-term post-operative complications. Currently, IPOM has its place in surgery of the anterior abdominal wall, but the indications have been severely limited and confined.The original description of laparoscopic treatment of abdominal hernias concerned mainly the IPOM method6. The literature describes further innovative solutions, including eMILOS  and eTEP , which aim to avoid the need to implant the mesh directly into the peritoneal cavity9. This does not detract from the fact that IPOM is still the most frequently performed minimally invasive technique in terms of abdominal hernias.In the time of development of minimally invasive techniques and robotic techniques, the possibilities of implant placement have expanded11. With this in mind, medical companies produce next generation of implants and fixation materials to find the golden standard. There has been a lot of expectations for covering the implant with an absorbable, non-adhesive barrier resorbed within 30–240 days after treatment. Examples of such meshes include Parietex Composite , Sepramesh IP Composite , Ventrio ST , and Ventralight ST . Fixation mechanisms also include a whole range of non-absorbable and absorbable materials.Implant manufacturers focus on continuous development and improvement of their products by creating next generations of composite meshes to reduce the risk of adhesions. As yet, no ideal implant for use inside the peritoneal cavity has been found. The assumptions for an ideal mesh include withstanding forces induced by intraperitoneal pressure and body movement until the mesh is overgrown with the tissue, adapting to the shape of the abdominal cavity taking into account its specific structure, stimulating the body to overgrow the mesh with its own tissues, not generating adhesions with organs located inside the peritoneal cavity, and not leading to intra-abdominal injuries. In fact, the whole biomechanical system composed of abdominal tissues, mesh and staples must create a mechanically reliable system, which would mimic mechanics of healthy abdominal wall12. The history of IPOM also involves spectacular failures that required stopping and re-evaluating the direction of development.New generations of meshes and modifications of the procedures, including closing the cavities before inserting the implant , robotic procedures, changing the staple arrangement scheme (single crown vs double crown technique) are intended to improve surgical outcome. However, not every novelty translates into improved clinical outcome14. In a randomized clinical trial (RCT) conducted in 2015, Pawlak et al. compared Physiomesh/Securestrap with Ventralight ST/Sorbafix systems in terms of pain, recurrence, and perioperative complications in laparoscopic IPOM procedures15. The trial was stopped for safety reasons, resulting from the high recurrence rate (20%) in the group of patients operated on with the Physiomesh/Securestrap system. In addition, patients in this group complained of much greater pain compared to patients operated with Ventralight ST/Sorbafix. Data from the German Herniamed Registry and Danish Hernia Database coincided with the results presented in the above RCT16.One example is implementation of the Physiomesh (PH) implant with a dedicated Securestrap fixation device by Ethicon  in 2014. This supposedly improved generation of mesh, pushed the company out of the market and resulted in the need to pay out high, often multi-million, compensation17. A decisive advantage of Ventralight ST/Sorbafix was the lower percentage of inflammation, fibrosis, bleeding and angiogenesis at the implantation site, and a much greater overgrowth with host tissues in the 14-day period after implantation. However, no difference was observed in terms of adhesions, collagen deposition, necrosis generation, and mesh shrinkage percentage17. Alone, biological factors did not fully justify such a spectacular failure of Physiomesh/Securestrap system. This suggests that, apart from biological components, failure of Physiomesh/Securestrap is also due to biomechanical factors. Given this hypothesis, we investigated the mechanical behavior of the implant itself and the mesh-staple-tissue system ex-vivo. The objective was to establish the potential causes of such a high recurrence rate and pain. Our philosophy is that drawing conclusions from the failure of the Physiomesh/Securestrap system will not only help to avoid similar situations in the future, but it will also provide a better understanding of the mechanics of implant fixed to the human anterior abdominal wall.The biological reasons for failure of the Physiomesh/Securestrap system were partly explained in animal models where, similarly to the aforementioned RTC, it was compared with a Ventralight ST implant and a dedicated Sorbafix fixation systemTo understand the cause of the very high rate of failures related to the use of the Physiomesh/Securestrap system in patients undergoing anterior abdominal wall reconstruction, we conducted mechanical experiments as well as numerical simulations of the system behaviour under physiological pressure load.Three kinds of experiments were performed of Physiomesh. The elasticity of the implant was determined based on the uniaxial tension tests. The strength of the mesh-tissue junction made by the staple was identified in the uniaxial tests. Finally, ex-vivo experiments on the Physiomesh/Securestrap implanted to porcine abdominal wall were performed. Within the experiments the behaviour of that system under simulated ‘intra-abdominal’ pressure load was observed. All that allowed us to gain knowledge on the mechanical performance of the Physiomesh/Securestrap system and delivered data for numerical simulations of the system behaviour, complementing the biomechanical analysis.17. Physiomesh was available in a kit with Securestrap staples, which allow fixation of the implant to the anterior abdominal wall in the IPOM Procedure. Securestrap staple’s substance is a mixture of PDO and lactide copolymer, which should be completely resorbed in vivo 12 months after implantation. Its shape resembles very narrow ‘U’ letter with two “teeth” of 6.7 mm length each, which provide two sites of fixation18. Both materials are shown in Fig. The study involved the Physiomesh implant made from monofilament polypropylene surrounded by polydioxanone (PDO) on both sides in order to maintain the bond with two poliglecaprone-25 absorbable barrier layers on both sides of the implant. Their purpose is to minimize adhesions between the mesh and the abdominal organs. In principle, the barrier should dissolve in vivo a few weeks after implantation. A violet stripe made of PDO is inserted through the center of the implant on one side, as an orientation marker. According to the manufacturer recommendation the mesh should be oriented in the abdominal wall so that the stripe aligns the cranio-caudal direction of the body21. The device allows simulating a static or dynamic action of an ‘intra-abdominal’ pressure on a sample placed inside. Three similar models were built of a side part of the porcine abdominal wall, in which hernia orifice (diameter of 5 cm) was cut. The hernia was ‘operated’ with the use of Physiomesh attached by Securestrap staples to the fascia. Twelve staples were used in each model, evenly spaced in a single crown layout. The crown had a diameter of 12 cm, so there was 3.5 cm of tissue and mesh overlap around the hernia orifice , 18 kPa (jumping or coughing) and 25 kPa In each case the deflection of the mesh center was measured, and then used for numerical models validation. The main question, however, concerned the possible failure of the mesh-tissue junction under the given load.23. In the present study, the implant is modelledd by a polygonal membrane structure supported in 12 points representing the joints of implant and the abdominal wall (staples). The elasticity of the abdominal wall is represented by elastic springs at joints—in the membrane plane and by elastic foundation around the hernia orifice24. To reduce the simulation time, only half of the implant was implemented using symmetry along the center of the PDO stripe. The material of the implant was modelled using a dense net model24. This type of modelling has been developed for technical fabrics and described in Ambroziak et al.25 Stiffness functions, determined for two directions in uni-axial tensile tests, were used this way to define the implant’s stiffness in two orthogonal directions like in a net of two types of threads. The finite element model was built using 4-node membrane elements with 3 translational degrees of freedom in each node.The study was supplemented by numerical simulations of the Physiomesh implanted in the abdominal wall and subjected to intraabdominal pressure. The numerical model was defined by means of Finite Element Method (FEM) using Marc (Hexagon) commercial software. The idea of modelling of implanted surgical mesh was proposed by LubowieckaThe model was loaded respectively to the experimental conditions. This way one simulation referred to the load of 14 kPa and the second one to 25 kPa according to the experiment. Reaction forces calculated in the mesh supporting points, representing the forces in each joint, were calculated in both cases and compared to the experimentally assessed limit load of the tissue and implant connection.26. On the other hand, there was a problem in the present study with samples fixation in the machine jaws. Grooved inserts to machine jaws (flat gripping system) have been used and there was a problem with the samples slipping of the fixation, due to slippery mesh coating. Curved gripping system, dedicated to membranes, could not be used in this study because of too small samples. Taking all this into account, standard static tests were performed.Knowing that the largest stretch of the abdominal wall is in the cranio-caudal direction, it was reasonable to recognize the mesh mechanical properties in the direction of the violet stripe, which was recommended by the producer to be aligned with this direction. Axis of the stripe is named here as direction ‘1’. Moreover, the mesh reveals anisotropic properties which can be sensed manually—it stretches differently in different directions. Thus, the direction perpendicular to the violet stripe (direction ‘2’) was selected for the investigation as well. With this background, three pieces of two kinds of rectangular samples were cut from one piece of the implant (only one piece was available for the tests): with the longer side aligned to the violet stripe and perpendicularly to the stripe  obtained in uni-axial tests vs engineering strains  are presented in Fig. The initial stiffness of the mesh is the most important, because that property is valid at the beginning of the mesh load . One can notice the highest initial stiffness of the PDO stripe, intermediate value for the mesh in ‘2’ direction, while initial stiffness is the smallest in the ‘1’ direction Fig. b. HoweveThe reaction forces of the model that represent the forces acting on a staples were calculated in two load scenarios, with pressure value of 14 kPa and of 25 kPa, referring to the experiment. In those two cases the maximum forces obtained were 3.79 N and 5.29 N respectively. That means that if applying the higher pressure, the forces in joints exceed their strength identified experimentally. The maximum displacements (while the mesh bulges under pressure) in both simulations were 17 mm and 23 mm respectively to the loading conditions . The stripe changes a basic demanded property of the mesh, which states the mesh stiffness should be smaller in the cranio-caudal direction (after implantation) than in the lateral direction. That property would be preserved for Physiomesh without the stripe. As Fig. The uni-axial tests of the mesh-staples-tissue system provided valuable information about the strength of the mesh-tissue junction. The results indicated that the limit load of the junction was identified as 4 N, with failure primarily occurring due to the staple pull-out from the tissue. This finding suggests that the fixation strength of the Physiomesh/Securestrap system may not be sufficient, especially under higher loads. As a comparison, we identified the junction capacity of Ventralight/Sorbafix system experimentally in uni-axial tensile test according to the testing protocol described above. The limit load of the Ventralight/SorbaFix system is 9.7N  meaning the capacity of this system is greater than Physiomesh/Securestrap. This result can be one pieces of evidence standing for the Ventralight/SorbaFix system as a solution better than Physiomesh/SecureStraps.The ex-vivo experiments on the porcine abdominal wall models implanted with the Physiomesh/Securestrap system under simulated intra-abdominal pressure load provided insights into the mechanical behavior of the system. The models exhibited failure, such as staple breakage or pull-out, particularly under higher pressure loads. The failure modes consistently occurred in an oblique direction relative to the PDO stripe of the mesh. However, the highest calculated junction force value is observed in the joint on the PDO stripe. One should be aware of the fact that the material model of implant identified in uni-axial test does not fully describe the multi-axial behaviour of knitted surgical mesh. In particular, since the specimens were cut correspondingly to the indications regarding the direction of mesh implantation , which does not necessarily correspond to the structure of the implant.The numerical simulations further supported the experimental findings, demonstrating reaction forces on the staples and maximum displacements under different pressure scenarios. The results indicated that the forces in the fixation points exceeded their experimentally determined strength, particularly under higher pressure loads.The numerical simulations showed how the PDO stripe itself seems to influenced negatively the implanted mesh behaviour, causing a stress concentration along this stiffer part of the material under pressure Fig. b. It mayThe mesh exhibited high flexibility in transverse projection – probably intended to compensate for rotation movements of the torso. In turn, in the longitudinal projection the flexibility was insufficient. The reason for that is the strengthening of the mesh with the extra stripe which influenced negatively overall behaviour of the implanted mesh.22. This is also confirmed by experimental research conducted by our team. There are not enough strong threads within the knitting pattern, which directly contributed to breakage of the implant from the fixation system. To be strict, the mesh is knitted out of threads of two diameters  in 2010. In its application, the manufacturer (Ethicon) declared that it is essentially similar to three other implants of this company, which have been approved for use before. Very quickly, the FDA began receiving information about a disturbingly high complication rate after Physiomesh/Securestrap system use. Physiomesh was permanently withdrawn from the market in May 2016 and as of today, the number of court claims has reached nearly 300031.The implications of this study extend beyond the specific case of the Physiomesh/Securestrap system. By understanding the mechanical behavior of implant-tissue systems, researchers and clinicians can make informed decisions regarding implant selection, design modifications, and fixation techniques to improve surgical outcomes. This knowledge may guide the development of future generations of implants, with a focus on creating mechanically reliable systems compatible with the mechanics of a healthy abdominal wallThe analysis described above was partially inspired by the discussions with patients who experienced hernia recurrence after surgery with a Physiomesh/Securestrap system. The patients were often able to accurately describe the moment of recurrence. The recurrence most often appeared in a moment of and abrupt sneeze, cough or sudden twisting of the torso accompanied by severe pain and/or feeling of “tearing tissue”. In our model we took into account one of the situations, generating a high intraabdominal pressure, and explain the biophysical reasons for this state of affairs.33. This is a period when there is no incorporation of the mesh, and the implant itself is held in its position solely by the force of the mesh-fascia-staple junction. In the case of the Physiomesh/Securestrap system, we believe that most of such a high percentage of recurrences (compared to other implants) occurred within the first few days after the procedure. Patients described this as a sudden pain after performing activities that increased intra-abdominal pressure . However, the moment when they reported to the doctor was delayed by several months—this was due to the fact that the patient began to realize the recurrence only when he noticed a protrusion on the previously torn implant15. In the past, our team has conducted repetitive stress tests on individual implants to replicate as closely as possible the conditions prevailing, for example, during a coughing fit, persistent vomiting, etc.21. However, in the case of Physiomesh/Securestrap, this was not necessary. The mesh fixation broke under physiological pressure (apart from 14 kPa) in single load tests that we did not study mechanics under repetitive load.”Based on experimental studies on an animal model , we know that the first two weeks are crucial for maintaining position in the case of defects repaired using the IPOM technique34.Without maintaining mechanical stability, the initiation of the incorporation process is impossible. We derive our knowledge on this from, among others, the works of Kallinowski et al. focusing on the GRIP CONCEPT, that is, critical resistance to impacts36. The presented analysis proves that the animal model cannot be directly translated into humans. The reasons include both phylogenesis with the adoption of a vertical body posture by humans, as well as the anatomy of the anterior abdominal wall, and thus a different distribution of forces and directions of their action in the event of a sudden increase of intraabdominal pressure than in the case of four-legged animals.In preclinical studies on the porcine and rabbit model, Ethicon presented great results in terms of both fixation, maintenance in the operating field, and tissue incorporation17. Additionally, it shows the complexity and importance of this mechanism in terms of the mesh-fascia system on the level of physics.Comparing with a number of studies referring to biological aspects of mesh implantation based on an animal model, the above results also allow to understand the biomechanical cause of such rapid and spectacular hernia recurrences in patients operated on with the Physiomesh implant. We believe that this will prevent similar errors in the future, while also serving as a valuable reference for the development of new implantsIt is important to note that this study has certain limitations. Uni-axial tests of the mesh and fixation system have been made for limited number of samples. However, the results considered as average confirm the mesh anisotropy and significant mesh stiffening in the area covered by PDO stripe. That features could be sensed manually for this mesh. On the other hand, nowadays bi-axial tests are considered more suitable than uni-axial to identify parameters of constitutive law of material working in a complex stress state. Thus, in the future study such test should be performed in a case of abdominal mesh. Also, dense net model for implant material does not considers either yarns interactions or the textile coating. Moreover, the cut of specimens of the mesh prepared for the material identification corresponded to the indications regarding the direction of mesh implantation , which did not necessarily correspond to the structure of the implant.The experiments and simulations were conducted under controlled laboratory conditions, which may not fully replicate the complex in vivo environment. There was porcine tissue used in an ex vivo experiment that may lead to slightly different results comparing to in vivo human tissue behaviour. Additionally, the study focused on mechanical aspects and did not consider other factors that may contribute to the clinical failures observed with the Physiomesh/Securestrap system, such as biological responses and host reactions.16.What is interesting, when Physiomesh/Securestrap entered the market in 2013, surgical meshes were classified as category II according to the regulations of the European Union and the USA. This meant that the introduction of a new product for general use required no clinical trials. From October 2021, upon entry into force of the MDR , hernia meshes have been included in group III, which means that randomized control tests must be carried out before placing the implant on the marketWe firmly believe that the presented findings will prevent similar mistakes in the future, but at the same time will allow us to supplement our current knowledge in the field of the anterior abdominal wall mechanics.In conclusion, this study provides a critical insights into the mechanical behavior of the Physiomesh/Securestrap system, highlighting the need for better understanding of the biomechanical factors involved in abdominal hernia repair. By considering these factors, future advancements in implant design and fixation techniques may be made, ultimately leading to better surgical outcomes and reduced complications for patients undergoing abdominal hernia repair surgeries. Hopefully, results allow avoiding similar errors in the future. The findings not only expose the cause for the failure of Physiomesh/Securestrap system, but also allow better understanding of the forces acting within the abdominal cavity, as well as to broaden our knowledge in this field. Optimally, this will prevent similar situations from arising in the future.