Designing safety-critical controllers for the blast furnace is crucial yet challenging due to its complex inherent dynamics. Recent advances in reinforcement learning (RL) have shown promise in designing effective controllers for complex industrial systems. However, direct application of RL control to an on-site blast furnace is hindered by stringent safety requirements and model discrepancies. To address these challenges, this paper presents a duo-stage RL-based safety-critical control framework (DRLSC Framework) that integrates safety-critical offline pretraining and an online agent transfer mechanism for long-term safe control of the blast furnace. An offline pretraining algorithm is designed to provide an adequate initial policy for online control, which consists of a data-driven blast furnace environment and a Soft Actor-Critic (SAC) agent. The offline stage incorporates a neural control barrier function (NCBF) to guarantee system state safety, which is trained jointly with the RL agent. During the online stage, a transfer mechanism facilitates rapid agent adaptation to the actual system dynamics, mitigating performance degradation arising from model discrepancies. Finally, silicon content ([Si]) tracking control experiments are carried out using a real dataset obtained from the #2 blast furnace of Liuzhou Steel in China. Compared with model-predictive-control-based methods, our DRLSC Framework can decrease the long-term tracking error by 50.21% when facing simulation discrepancy and maintain state safety continuously.

Aviation electrical connectors are essential components in the aircraft electrical wiring interconnection system (EWIS), responsible for information and energy transmission. Even a minor fault in a connector pin can critically affect the reliability and stability of the EWIS. However, traditional faulty pin detection methods rely heavily on manual visual inspection, which is inefficient and susceptible to missed or false detections due to the inherent limitations of human observation. To address these challenges, this article introduces a novel system-level approach that transforms the defect detection process from the physical domain to a virtual one powered by digital twin (DT) technology. A general framework for DT-based defect detection is proposed and instantiated through the design and implementation of a DT-enabled automated faulty pin detection (DT-AFPD) system. The DT-AFPD system integrates 3-D machine vision into a complete detection pipeline encompassing equipment design, data acquisition, DT model construction, algorithm development, and system deployment. Specifically, a 4-degree-of-freedom (4-DOF) device equipped with a 3-D structured light camera is developed to acquire point cloud data of aviation connectors. Several preprocessing techniques are applied to reduce data volume and enhance point cloud quality. Based on this, a connector structure-aware faulty pin detection algorithm, named CSA-FPD, is designed to detect short and bent pins using limited data. The proposed DT-AFPD system is validated on 13 representative types of aviation electrical connectors, covering over 3600 pins. Experimental results demonstrate that the system achieves an average detection precision of 99.85%, effectively reducing the probability of EWIS reinstallation and enhancing the reliability of faulty pin detection.

Empowered by network function virtualization (NFV), mobile edge computing aims to provide low latency and ultra reliable network services to mobile end users, achieved as a service function chain (SFC) consisting of a series of ordered virtual network functions (VNFs). Due to user mobility, live SFC migration is imperative to avoid Quality of Service (QoS) degradation. Recent advances mainly make separate decisions on VNF node remapping and migration path routing in a heuristic manner, or implement both through reinforcement learning within a single agent of ill-defined policy and action space. In this paper, given next access node, we first formulate the live SFC migration problem as an integer linear programming (ILP) model to achieve optimal solutions. Then, we present HRL-QC, a hierarchical reinforcement learning framework that jointly optimizes VNF destination node remapping, migration path and post-migration service path selections for QoS-aware and cost-efficient live SFC migration. A GCN-Transformer block is introduced to capture long-range VNF-to-physical node dependencies, while a two-level actor-critic design couples the decision-makings through inter-level reward passing. Extensive evaluations show that HRL-QC outperforms the state-of-the-art in energy consumption, migration time, end-to-end service delay, and migration success rate, while remaining within a small margin of the optimal ILP solution.

The training scale of large language models (LLMs) has reached tens of thousands of GPUs and is still continuously expanding, enabling faster learning of larger models. Accompanying the expansion of the resource scale is the prevalence of failures (CUDA error, NaN values, job hang, etc.), which poses significant challenges to training stability. Any large-scale LLM training infrastructure should strive for minimal training interruption, efficient fault diagnosis, and effective failure tolerance to enable highly efficient continuous training. This paper presents ByteRobust, a large-scale GPU infrastructure management system tailored for robust and stable training of LLMs. It exploits the uniqueness of LLM training process and gives top priorities to detecting and recovering failures in a routine manner. Leveraging parallelisms and characteristics of LLM training, ByteRobust enables high-capacity fault tolerance, prompt fault demarcation, and localization with an effective data-driven approach, comprehensively ensuring continuous and efficient training of LLM tasks. ByteRobust is deployed on a production GPU platform with over 200,000 GPUs and advances the state of the art in training robustness by achieving 97% ETTR for a three-month training job on 9,600 GPUs.

State monitoring plays an important role in industrial automation, where smart sensors are deployed in production sites to estimate the state of physical processes. Critical monitoring information is expected to be delivered timely to ensure estimation performance. However, ensuring the necessary data rate for such data transmission under bandwidth limitations requires larger transmission power, increasing the energy consumption of devices. To address this contradiction between estimation performance and energy consumption, a distributed estimation and communication co-design scheme using a cross-layer optimization technique is proposed in this paper. An event-triggered quantized distributed estimation algorithm is proposed to reduce energy consumption. Then, the impact of system dynamics, event-triggered threshold, and number of quantization bits on the convergence of estimation errors is investigated. Based on this relationship, the quantization in the application layer, event-triggered communications in the transport layer, and transmission power in the physical layer are jointly optimized. This constrained minimization problem is formulated as a mixed-integer nonlinear programming problem and solved with a cross-layer optimization method based on the alternating direction method of multipliers. The global convergence of the optimization method is analyzed. Finally, a numerical case study in the hot rolling process shows the superiority of the co-design scheme in balancing estimation accuracy and energy consumption.

Flight control system testing (FCST) is one of the most important process to check whether flight control surfaces can operate properly according to commands during aircraft assembly. Traditional testing method relies heavily on manual labor, leading to low efficiency and inconsistent quality. In this paper, we apply digital twin (DT) technology to the FCST process for the first time. We firstly design an architecture of DT-enabled FCST including four layers to support further development. Then, we present a triangular mesh alignment-based angle measurement (TMA-AM) algorithm to efficiently collect deflection angle data for DT-enabled FCST. Extensive experiments conducted on a aircraft wing subassembly platform show that the TMA-AM algorithm achieves an average angular measurement error of less than 0.1°, outperforming existing methods. Moreover, we develop a virtual experimental platform named DT-FCST aligned with a real aircraft wing subassembly platform. In addition, TMA-AM algorithm is integrated with the DT-FCST platform. By integrating real-time data from the cockpit, real-time physical-virtual interaction of aircraft control sticks and flight control surfaces are achieved, ensuring consistency between physical and virtual movements. The integration of DT technology with the TMA-AM algorithm enables real-time synchronization, monitoring, and unified data management, significantly enhancing the efficiency and accuracy of the FCST. Note to Practitioners—To address the inefficiencies and low monitoring quality associated with traditional manual testing methods in flight control system testing (FCST), we firstly introduce digital twin (DT) technology to this process. To support effective and accurate measurement during the FCST, we propose a vision-based method tailored to accurately measure deflection angles of flight control surfaces. This method replaces manual measurements with a non-contact approach, significantly improving measurement accuracy and efficiency. We provide a detailed description of the construction process of the DT-FCST platform including requirement analysis, DT model construction, and on-site experiments. This DT-based approach achieves real-time synchronization between virtual and physical testing processes, enhancing monitoring quality and overall testing effectiveness. Specifically, it can achieve a 90% reduction in the number of operators and shorten the single testing time to 16.7% of the traditional testing method.

Joint estimation is crucial in the industrial Internet of Things (IIoT) by integrating data from diverse devices to improve monitoring accuracy. Network slicing can meet the heterogeneous needs of devices through logical isolation. However, existing methods often overlook the interaction of multiple slices on estimation performance, leading to potential estimation bias and ineffective resource costs. To address this, we propose an Age of Task (AoT)-driven associated network slicing method tailored for joint estimation scenarios. Specifically, we design an association-oriented slicing architecture for joint estimation that considers both the heterogeneous requirements of individual slices and the interactive effects of multiple slices. We define slice association based on the AoT to quantify the coupling relationship between slicing strategies and estimated performances. Moreover, we develop a dynamic-fitness multivariable particle swarm optimization algorithm to achieve associated slicing. Simulation results show that the associated slicing scheme achieves a flexible balance between timeliness and accuracy.

Network function virtualization (NFV) empowered by virtualization technology can achieve flexible virtual network function (VNF) placement. To improve resource utilization and energy efficiency, different VNFs tend to be co-located on common servers, which inevitably intrigues VNF performance degradation induced by hardware resource competition. The problem of energy-efficient and interference-aware service function chain (SFC) provisioning is considered in this paper and envisioned to yield minimum activated servers and maximum average throughput. It is formulated as a mixed integer linear programming (MILP) model to achieve optimal solutions. Then, a gale-shapley based offline approximation algorithm is designed through bipartite matching, to yield an SFC allocation decision in one go with proved competitive ratio. In online scenario, Transformer and its efficient model Fastformer, combined with Graph Attention Network (GAT) respectively, are introduced into deep reinforcement learning (DRL) structure for the first time to quickly and accurately abstract features of substrate network and SFC. A DRL-based Fastformer-assisted energy efficient and interference aware SFC provisioning (DRL-EI) algorithm is proposed with an elaborately designed reward function to balance energy consumption and VNF interference. Simulations indicate the gap between DRL-EI and MILP is marginal. DRL-EI outperforms state-of-art work in terms of energy consumption, VNF normalized throughput and acceptance rate.

Flight control system testing (FCST) is one of the most important testing during aircraft final assembly, however, traditional testing method highly rely on manual labor, resulting in low testing quality and efficiency. Targeting at improving the testing quality and efficiency, in this paper, we apply digtial twin (DT) technology in the FCST process for the first time. A virtual experimental platform named DT-FCST is constructed which is identical to a real wing experimental platform, supporting testing elements management and physical-virtual mapping experiments. The development process of the DT-FCST platform is detailed. Several experiments are conducted to demonstrate the effectiveness of DT technology. Based on motion scripts integrating real-time data from the cockpit, the cockpit control sticks and flight control surfaces are driven to achieve consistency between physical and virtual motions. With the help of DT technology, the entire movement process of control sticks and flight control surfaces are mapped with high fidelity in the DT-FCST platform, greatly enhancing the testing efficiency and monitoring quality.

During the flight control system testing (FCST), it is crucial to accurately measure the deflection angles of flight control surfaces to determine whether they respond precisely to commands. However, the traditional measurement method, which relies on the manual use of angle measuring rulers for inspections, is inefficient, prone to wear, and lacks precision. To address this issue, we introduce a vision-based angle measurement method for FCST that replaces manual measurements, thereby significantly enhancing testing efficiency and accuracy. Our proposed triangular mesh alignment based angle measurement algorithm (TMA-AM) is a non-contact measurement method that involves two key procedures, capturing 3D coordinates from images and calculating deflection angles. The TMA-AM algorithm converts deflected angles into angular differences between two coordinate systems, while accounting for the curved characteristics of control surfaces. We evaluate TMA-AM algorithm on a 3D-printed wing test platform and an aircraft wing test platform. Experimental results demonstrate that our method achieves high accuracy, with an average angular measurement error below 0.05○.

During the flight control system testing (FCST) process, multiple testing tasks should be completed. Battery-powered wireless sensors are used to measure the motion state of each flight control surface. In this paper, we investigate a multi-task scheduling problem to enhance overall monitoring accuracy during the FCST process. However, the decline in sensor battery levels, along with limited time slot resources, impacts the transmission quality of measurement data, leading to reduction in monitoring accuracy. Thus, we analyze the relationship among battery levels, transmission power, and monitoring accuracy to transform the original problem into an expectation probability maximization problem. Three important factors of monitoring accuracy are identified, based on which, we present the accuracy-oriented testing task scheduling (AOTS) algorithm. To validate the effectiveness of AOTS algorithm, we compare its performance among three different scheduling orders. Simulation results demonstrate that AOTS algorithm can not only improve the testing accuracy, but also reduce the fluctuation in accuracy among all testing tasks. Additionally, there are various elements in the FCST process that need to be uniformly managed to enhance the level of digitization. To address this issue, we design a digital twin enabled FCST (DT-FCST) system to manage data, models and algorithms in the FCST process. Finally, we implement the AOTS algorithm into developed DT-FCST system.

The booming development of 5G and Internet of Things (IoT) technologies significantly promotes the revolution of industrial IoT systems. Age of Information (AoI) is expected to play a critical role in industrial IoT systems, especially for time-sensitive monitoring and control applications. In addition, edge computing (EC) will be leveraged to effectively support industrial tasks in the limited communication and computing resources environment, bringing threefold benefits of shorter end-to-end delay, improving information timeliness, and reduced communication burden. Thus, we propose an edge-assisted co-design architecture of sampling-scheduling-control to improve the overall system performance. Under this architecture, a new definition, Age of Task (AoT), is proposed first to measure the timeliness of multielement and compute-intensive monitoring tasks in the industry. By analyzing the coupling relationship between AoT and control performance, an analytical expression of estimation error based on AoT is derived. Furthermore, we prove that the optimal control law could be expressed in a certain equivalent form, making it possible to decompose the design of control and network resource allocation (sampling, scheduling). According to the relation between AoT and estimation error, a co-design method, event-triggered sampling and max-age-reduce-first scheduling (ETMA), is proposed to minimize the system cost, including control cost and network energy consumption. The simulation results show that our co-design scheme has the optimal system cost among the four state-of-the-art schemes.

In the industrial Internet of Things, state estimation of large-scale physical systems is performed by multiple sensors in a distributed manner. However, frequent communications during an estimation interval can increase the energy consumption of sensors, causing thermal stress and reliability issues. Although system reliability can be improved by data compression, the compression-induced distortion may lead to the divergence of the estimation error. To address these challenges, a distributed estimation and communication co-design scheme is proposed in this article, which balances estimation performance and energy efficiency under the system lifetime reliability constraint. First, a consensus-based distributed estimation algorithm is proposed to adapt the data compression configuration. Then, the impact of system dynamics, network connectivity, and data compression configuration on estimation performance is investigated. Based on the relationship, the distributed estimation algorithm and the channel allocation with power control are jointly optimized to minimize the estimation error and energy cost under the system lifetime reliability constraint. This constrained minimization problem is formulated as a mixed-integer nonlinear programming problem and solved with the designed decomposition method. Finally, simulation results demonstrate that the proposed co-design scheme shows superiority in improving both the estimation accuracy and energy efficiency under the system lifetime reliability constraint.

In recent times, it has been witnessed that an increasing number of Internet connected devices impose an huge challenge on Internet of Thing (IoT) networks, which provides diverse and complex network services through edge computing empowered IoT terminals. A network service can be formally represented by Virtualized Network Function Forwarding Graph (VNF-FG) with the advent of Network Function Virtualization (NFV) technology. Previous researches mainly focus on VNF-FG embedding (VNF-FGE) and take VNF-FG as the input. In this paper, we investigate the design of VNF-FG, which is required to be a Directed Acyclic Graph (DAG) and achieved by the IoT terminal, in two scenarios. In static scenario, for a set of traffic flows arriving at the IoT terminal and requesting different network services, an ILP model Ps including loop prevention constraints is well formulated. An approximation algorithm AFGC with competitive ratio O(1+(|R|-1)α), α (0, 1) is then designed, which thoroughly search all key instances to run comprehensive loop break. In dynamic scenario for a flow request on the fly reaching an IoT terminal, an ILP model Pd and another approximation algorithm DFGU with a competitive ratio O(1 + |SCr||Fr| ) are developed, giving priority to reuse of existing topology to generate an augmented VNF-FG of minimum size. Simulation results indicate AFGC and DFGU outperform state-of-art work, and the gap between each of the two algorithms and their respective ILP models is marginal.

Joint monitoring, by integrating observations from multiple types of equipment, is essential for a thorough understanding of physical processes in the Industrial Internet of Things (IIoT). However, it does demand sufficient resources to ensure reliable and timely delivery of such observations. Although network slicing is widely used to meet such heterogeneous requirements, it falls short in this system, because it causes interconnected impacts on system performance across multiple slices. In this paper, we introduce an innovative associated network slicing framework for joint monitoring, which focuses on system cost minimization while accounting for slice associations. Particularly, to better understand the characteristics, we introduce a new concept, Age of Inexact Task (AoIT), to capture inter-slice associations. We then decompose the optimization variables to facilitate efficient Associated Network Slicing (ANS) algorithmic design, leading to a closed-form solution for intra-slice small-timescale resource allocation and an iterative block coordinate gradient descent algorithm for inter-slice large-timescale resource allocation. Simulation results demonstrate that our proposed ANS balances heterogeneous requirements and associations, showing significant reductions in system costs compared to existing solutions.

Aircraft final assembly line (AFAL) is a typical complex manufacturing system with multiple installation and test processes operating simultaneously at each workstation. Lots of robots and sensors are connected and operated for heterogeneous processes by sharing limited communication and computing resources. How to manage devices and resources in a coordinated and efficient way is thus very challenging. Digital twin (DT) is a powerful technology for multiple objects management in the complex assembly system. It enables us to coordinate various devices and allocate communication and computing resources at workstations. In this article, two main processes, i.e., vision-assisted installation and flight control system test, are considered in the AFAL. We introduce a DT-enabled AFAL system and propose a DT-assisted heterogeneous processes coordinated (DT-HPC) framework to coordinate various devices and resources at each workstation. The wirelessly connected robots and sensors are applied for perception and information fusion. In order to minimize the total energy consumption and computing resources of all the wireless devices, joint design of the wireless channel allocation, transmission power, and computing resource allocation are proposed to satisfy the diverse Quality-of-Service (QoS) requirements. First, we propose a priority-aware channel assignment (PACA) algorithm to allocate channels for sensors and robots. Then, the optimal computing resource allocation strategy for two processes is derived while guaranteeing the processing latency requirements. Next, we derive the minimum transmission power of wireless sensors to guarantee the monitoring accuracy and calculate the transmission power of robots to obtain the satisfied transmission rate. Finally, we apply the DT-HPC framework in the DT-enabled AFAL system. The simulation results prove that our proposed algorithms can save energy while guaranteeing different QoS requirements.

To comply with security and performance policies, multicast communication requires inline service that chains an ordered sequence of virtualized network functions (VNFs) with an emerging paradigm of network function virtualization (NFV). As many-to-many multicast pattern is widely used especially in current MEC, multimedia and big data industries, in this paper we focus on multi-source NFV-enabled multicasting in software-defined networks (SDN). By jointly investigating VNF placement and multicast tree routing strategy, it is envisioned to construct service function tree with minimum network resource expenditure while yielding lower delay. We explore this problem in static and online scenarios individually. In static situation with a bunch of multicast requests to be served, a column generation based computing model is first formulated, based on which an approximation algorithm MDNM that leverages logic of Viterbi and puts emphasis on resource optimization and segmental delay satisfaction is developed. In online situation with a single incoming request, another ILP model to enhance VNF instance reuse is established. Then, SVM algorithm giving importance to source determination and VNF instance consolidation is designed to efficiently construct a service function tree with minimum resource usage and delay. Simulation results show that the gap between each of the two proposed algorithms and their respective ILP models is marginal, and both MDNM and SVM outperform the state-of-art work.

To minimize operating expenses (OPEX) and enhance flexibility for network service provisioning, network function virtualization (NFV) is used to chain an ordered sequence of virtualized network functions (VNFs), also known as service function chain (SFC), which can be placed on commodity servers. As a highly complementary to NFV, software-defined networking (SDN) can offer an agile way of VNF orchestration with the capability of fine-granularity network control over flows. However, one-step migration to SDN is impossible in ISP networks. Thus, in this paper we resort to hybrid SDN to implement SFC provisioning. By jointly optimizing SDN deployment that largely determines capital expenditures (CAPEX) and VNF placement that decides OPEX, we manage to make a tradeoff between CAPEX and OPEX, and to find out an appropriate SDN deployment rate for network operator. We first formulate the problem as an integer linear programming (ILP) model$\boldsymbol {P}$, then, reformulate it with column generation technique and decomposition theory to develop a distributed approximation algorithm CGPD which obtains a tight upper bound of optimal solution to$\boldsymbol {P}$. In order to better decrease CAPEX, a dynamic programming-based heuristic EOES giving importance to effective resource sharing is further designed based on a hidden Markov model. The simulation results indicate that, the difference between$\boldsymbol {P}$and CGPD is marginal. CGPD outperforms EOES in terms of total cost, average delay and bandwidth utilization, and EOES on the other hand demands 20% SDN deployment rate which is half of that of CGPD.

Various wireless manufacturing sensors facilitate the monitoring and control process in industrial internet of things (IIoT) systems. Control and image data generated by control and vision-based monitoring systems are transmitted over shared wireless network to edge computing devices for further decisions. Due to limited spectrum resources and sensor transmit power, it is challenging to design efficient quality of service (QoS) aware transmission mechanism for heterogeneous data with diverse real-time and system performance requirements. In this paper, in order to guarantee control system performance and image clarity, the time slot allocation and transmission power for heterogeneous sensors are jointly optimized by formulating a mixed integer nonlinear programming (MINLP) problem. The MINLP problem is decomposed into multiple subproblems which are proved to have strong duality and can be solved in dual domain. Then, according to acknowledge (ACK) signals, we propose a priority-aware heuristic algorithm to solve each subproblem. Finally, the simulation results prove that our proposed mechanism can reduce the number of slots and energy consumption for transmission while guaranteeing performance requirements of heterogeneous data.

Assessment and reporting of skills is a central feature of many digital learning platforms. With students often using multiple platforms, cross-platform assessment has emerged as a new challenge. While technologies such as Learning Tools Interoperability (LTI) have enabled communication between platforms, reconciling the different skill taxonomies they employ has not been solved at scale. In this paper, we introduce and evaluate a methodology for finding and linking equivalent skills between platforms by utilizing problem content as well as the platform’s clickstream data. We propose six models to represent skills as continuous real-valued vectors, and leverage machine translation to map between skill spaces. The methods are tested on three digital learning platforms: ASSISTments, Khan Academy, and Cognitive Tutor. Our results demonstrate reasonable accuracy in skill equivalency prediction from a fine-grained taxonomy to a coarse-grained one, achieving an average [email protected] of 0.8 between the three platforms. Our skill translation approach has implications for aiding in the tedious, manual process of taxonomy to taxonomy mapping work, also called crosswalks, within the tutoring as well as standardized testing worlds.

Hybrid Software-Defined Networking (HSDN) is a transitional networking form of SDN where SDN elements are partially deployed in traditional networks. Previous researches show that redirecting every flow of source-destination pair through at least one SDN switch can obtain flow manageability, e.g., access control and traffic measurement. Intuitively, the selection of SDN switch as the waypoint for every flow has a significant effect on the Traffic Engineering (TE) performance, such as maximum link utilization and routing efficiency. And it is worth noting that SDN switch can split traffic to the outgoing links to exactly profit the TE performance. In this paper, from the perspective of TE performance, we propose a flow routing and splitting (FRS) algorithm whereby we jointly determine an appropriate SDN switch for every flow as the waypoint, as well as optimizing the traffic splitting fractions for every SDN switch among its outgoing links to minimize the maximum link utilization. We conduct simulations with different SDN deployment rate. The results indicate that, when 20% of the SDN switches are deployed, the proposed FRS algorithm can obtain a lower maximum link utilization compared with other state-of-art works. Not only that, FRS algorithm can also generate a little longer paths for every flow on the average, which has a limited influence on the routing efficiency.

Abstract Continuous constrained optimization is a powerful tool for synthesizing novel human motion segments that are short. Graph‐based motion synthesis methods such as motion graphs and move trees are popular ways to synthesize long motions by playing back a sequence of existing motion segments. However, motion graphs only support transitions between similar frames, and move trees only support transitions between the end of one motion segment and the start of another. In this paper, we introduce an optimization‐based graph that combines continuous constrained optimization with graph‐based motion synthesis. The constrained optimization is used to create a vast number of complex realistic‐looking transitions in the graph. The graph can then be used to synthesize long motions with non‐trivial transitions that for example allow the character to switch its behavior abruptly while retaining motion naturalness. We also propose to build this graph semi‐autonomously by requiring a user to classify generated transitions as acceptable or not and explicitly minimizing the amount of required classifications. This process guarantees the quality consistency of the optimization‐based graph at the cost of limited user involvement.

The rate equations for the intracavity-frequency-doubled quasi-three-level lasers are developed. By normalizing the related parameters, it is shown that the general solution to the rate equations is dependent upon four dimensionless parameters: the normalized reabsorption loss, the pump to laser-mode size ratio, the normalized pump level, and a parameter written as η shg , which is related to the ability of the nonlinear crystal to convert the fundamental to the second harmonic. By numerically solving these rate equations, a group of general curves are obtained to express the relations between the solution and the four dimensionless parameters.