The multi-input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries turns out to be the individual z-transform of a discrete-time impulse response sequence. However, when a MIMO system has a global feedback mechanism, which also involves multirelational signals, the aforementioned individual z-transforms of the overall transfer tensor are quite difficult to formulate. Therefore, a new mathematical framework to govern both feedforward and feedback MIMO systems is in crucial demand. In this work, we define the tensor z-transform to characterize a MIMO system involving multirelational signals as a whole rather than the individual entries separately, which is a novel concept for system analysis. To do so, we extend Cauchy’s integral formula and Cauchy’s residue theorem from scalars to arbitrary-dimensional tensors, and then, to apply these new mathematical tools, we establish to undertake the inverse tensor z-transform and approximate the corresponding discrete-time tensor sequences. Our proposed new tensor z-transform in this work can be applied to design digital tensor filters including infinite-impulse-response (IIR) tensor filters (involving global feedback mechanisms) and finite-impulse-response (FIR) tensor filters. Finally, numerical evaluations are presented to demonstrate certain interesting phenomena of the new tensor z-transform.
多输入多输出（MIMO）系统涉及由分布式信号源产生的多关系信号，在实践中已成为最通用的模型。现有的表征这种 MIMO 系统的工作是建立一个相应的变换张量，其每个条目都是离散时间脉冲响应序列的单独 Z 变换。然而，当 MIMO 系统具有全局反馈机制，同时涉及多关系信号时，上述整体传递张量的单个 z 变换就很难表述了。因此，亟需一种新的数学框架来管理前馈和反馈 MIMO 系统。在这项工作中，我们定义了张量 z 变换，以描述涉及多关系信号的 MIMO 系统的整体特征，而不是单独描述各个条目，这是系统分析的一个新概念。为此，我们将 Cauchy 积分公式和 Cauchy 残差定理从标量扩展到任意维张量，然后应用这些新的数学工具，建立反张量 z 变换并逼近相应的离散时间张量序列。我们在这项工作中提出的新张量 Z 变换可用于设计数字张量滤波器，包括无限脉冲响应（IIR）张量滤波器（涉及全局反馈机制）和有限脉冲响应（FIR）张量滤波器。最后，通过数值评估展示了新张量 Z 变换的某些有趣现象。
{"title":"Tensor <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"><mi>z</mi></math>-Transform","authors":"Shih Yu Chang, Hsiao-Chun Wu","doi":"10.1155/2024/6614609","DOIUrl":"https://doi.org/10.1155/2024/6614609","url":null,"abstract":"The multi-input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries turns out to be the individual z-transform of a discrete-time impulse response sequence. However, when a MIMO system has a global feedback mechanism, which also involves multirelational signals, the aforementioned individual z-transforms of the overall transfer tensor are quite difficult to formulate. Therefore, a new mathematical framework to govern both feedforward and feedback MIMO systems is in crucial demand. In this work, we define the tensor z-transform to characterize a MIMO system involving multirelational signals as a whole rather than the individual entries separately, which is a novel concept for system analysis. To do so, we extend Cauchy’s integral formula and Cauchy’s residue theorem from scalars to arbitrary-dimensional tensors, and then, to apply these new mathematical tools, we establish to undertake the inverse tensor z-transform and approximate the corresponding discrete-time tensor sequences. Our proposed new tensor z-transform in this work can be applied to design digital tensor filters including infinite-impulse-response (IIR) tensor filters (involving global feedback mechanisms) and finite-impulse-response (FIR) tensor filters. Finally, numerical evaluations are presented to demonstrate certain interesting phenomena of the new tensor z-transform.","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"38 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140728317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a study on the numerical solutions of the Burgers’ equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers’ equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers’ problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers’ equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering.
{"title":"A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term","authors":"Aysenur Busra Cakay, Selmahan Selim","doi":"10.1155/2024/5497604","DOIUrl":"https://doi.org/10.1155/2024/5497604","url":null,"abstract":"This paper presents a study on the numerical solutions of the Burgers’ equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers’ equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers’ problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers’ equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering.","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"21 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140754841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cryptography has recently become a critical area to research and advance in order to transmit information safely and securely among various entities, especially when the transmitted data is classified as crucial or important. This is due to the increase in the use of the Internet and other novel communication technology. Many businesses now outsource sensitive data to a third party because of the rise of cloud computing and storage. Currently, the key problem is to encrypt the data such that it may be stored on an unreliable server without sacrificing the ability to use it effectively. In this paper, we propose a graph encryption scheme by using cryptography and steganography. Data is encrypted using association schemes over finite abelian groups and then hiding the encrypted data behind randomly chosen cover image. We implemented and evaluated the efficiency of our constructions experimentally. We provide experimental results, statistical analysis, error analysis, and key analysis that demonstrates the appropriateness and efficiency of the proposed technique.
{"title":"Graph Crypto-Stego System for Securing Graph Data Using Association Schemes","authors":"Anuradha Sabharwal, Pooja Yadav, Kamal Kumar","doi":"10.1155/2024/2084342","DOIUrl":"https://doi.org/10.1155/2024/2084342","url":null,"abstract":"Cryptography has recently become a critical area to research and advance in order to transmit information safely and securely among various entities, especially when the transmitted data is classified as crucial or important. This is due to the increase in the use of the Internet and other novel communication technology. Many businesses now outsource sensitive data to a third party because of the rise of cloud computing and storage. Currently, the key problem is to encrypt the data such that it may be stored on an unreliable server without sacrificing the ability to use it effectively. In this paper, we propose a graph encryption scheme by using cryptography and steganography. Data is encrypted using association schemes over finite abelian groups and then hiding the encrypted data behind randomly chosen cover image. We implemented and evaluated the efficiency of our constructions experimentally. We provide experimental results, statistical analysis, error analysis, and key analysis that demonstrates the appropriateness and efficiency of the proposed technique.","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"37 14","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140081464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The complex behavior of shape memory alloys (SMAs), characterized by hysteresis and nonlinear dynamics, results in complex constitutive equations. To circumvent the complexity of solving these equations, a black box neural network (NN) has been employed in this research to model a rotary actuator actuated by an SMA wire. Considering the historical dependence of the pulley’s rotational angle on the applied voltage, a recurrent neural network (RNN) is suitable for capturing past information. Specifically, a long short-term memory (LSTM) neural network is selected due to its ability to address issues encountered in standard recurrent networks. There are major drawbacks with modelling hysteresis with NNs that do not account for historical behavior. Traditional NNs, characterized by a one-to-one mapping, struggle to capture hysteresis loops wherein system behavior varies during loading and unloading cycles. Therefore, a single-tag data is used to determine the loading or unloading state, but tag signal causes discontinuity in network and omits various aspects of hysteresis in SMA, particularly within minor loops. In contrast, NNs incorporating past data to predict hysteresis behavior alleviate the need for tag data. However, such networks tend to have complex structures with a substantial number of neurons to effectively capture the inherent nonlinearity in SMAs. The long short-term memory (LSTM) neural network employed in this research, characterized by a simpler structure, achieves high accuracy in predicting hysteresis in SMAs without the need for tag data. In the proposed LSTM model, data related to the pulley’s rotational angle and the wire’s applied voltage from the current moment and the two previous moments serve as input. The data passes through a layer comprising three LSTM cells, and the output from the last LSTM cell is fed into a fully connected layer to predict the pulley’s rotational angle for the next moment. Training data are obtained by applying voltage at various frequencies and formats to the SMA wire while simultaneously recording the pulley’s angle with an encoder. Evaluation of the LSTM model is conducted in two configurations: online prediction (one-step ahead) and offline prediction (multistep ahead). In the online configuration where the model uses encoder data as angular inputs, the root mean square error (RMSE) of predictions for various input voltages is significantly low at about 0.1 degrees where the maximum rotational angle of pulley is 8 degrees. In the offline configuration when using the model’s predictions as angular inputs instead of encoder data, the RMSE rises to 0.3 degrees. To provide a clear demonstration of the LSTM model’s ability in this particular configuration, a comparison has been conducted between LSTM model and a rate-dependent Prandtl-Ishlinskii (RDPI) hysteresis model for predicting the pulley’s angle. The LSTM model outperforms the RDPI model by 70% in terms of accuracy. Overall, the LSTM model dem
形状记忆合金 (SMA) 具有滞后和非线性动力学特征，其复杂的行为导致了复杂的构成方程。为了避免求解这些方程的复杂性，本研究采用了黑盒神经网络（NN）来模拟由 SMA 线材驱动的旋转致动器。考虑到滑轮旋转角度对施加电压的历史依赖性，循环神经网络（RNN）适合捕捉过去的信息。具体来说，选择长短期记忆（LSTM）神经网络是因为它能够解决标准递归网络中遇到的问题。使用不考虑历史行为的神经网络来模拟滞后现象存在很大缺陷。传统 NN 的特点是一对一映射，难以捕捉系统行为在加载和卸载周期中发生变化的滞后回路。因此，单个标签数据被用来确定加载或卸载状态，但标签信号会导致网络的不连续性，并忽略 SMA 中滞后的各个方面，尤其是在次要环路中。与此相反，结合过去数据预测滞后行为的网络可减轻对标签数据的需求。不过，此类网络往往结构复杂，需要大量神经元才能有效捕捉 SMA 固有的非线性。本研究采用的长短期记忆（LSTM）神经网络的特点是结构较为简单，在预测 SMA 的滞后现象时具有较高的准确性，而无需标记数据。在所提出的 LSTM 模型中，与滑轮旋转角度以及当前时刻和前两个时刻的导线外加电压相关的数据作为输入。数据通过一个由三个 LSTM 单元组成的层，最后一个 LSTM 单元的输出被输入到一个全连接层，以预测下一时刻滑轮的旋转角度。训练数据通过向 SMA 线施加不同频率和格式的电压获得，同时用编码器记录滑轮的角度。LSTM 模型的评估在两种配置下进行：在线预测（提前一步）和离线预测（提前多步）。在在线配置中，模型使用编码器数据作为角度输入，在滑轮最大旋转角度为 8 度的情况下，各种输入电压的预测均方根误差 (RMSE) 明显较低，约为 0.1 度。在离线配置中，当使用模型的预测值作为角度输入而不是编码器数据时，均方根误差上升到 0.3 度。为了清楚地展示 LSTM 模型在这一特定配置中的能力，我们对 LSTM 模型和与速率相关的 Prandtl-Ishlinskii (RDPI) 迟滞模型进行了比较，以预测滑轮的角度。LSTM 模型的准确度比 RDPI 模型高出 70%。总体而言，LSTM 模型展示了在在线和离线配置中有效模拟 SMA 磁滞的能力。
{"title":"Modelling Hysteresis in Shape Memory Alloys Using LSTM Recurrent Neural Network","authors":"M. Zakerzadeh, Seyedkeivan Naseri, Payam Naseri","doi":"10.1155/2024/1174438","DOIUrl":"https://doi.org/10.1155/2024/1174438","url":null,"abstract":"The complex behavior of shape memory alloys (SMAs), characterized by hysteresis and nonlinear dynamics, results in complex constitutive equations. To circumvent the complexity of solving these equations, a black box neural network (NN) has been employed in this research to model a rotary actuator actuated by an SMA wire. Considering the historical dependence of the pulley’s rotational angle on the applied voltage, a recurrent neural network (RNN) is suitable for capturing past information. Specifically, a long short-term memory (LSTM) neural network is selected due to its ability to address issues encountered in standard recurrent networks. There are major drawbacks with modelling hysteresis with NNs that do not account for historical behavior. Traditional NNs, characterized by a one-to-one mapping, struggle to capture hysteresis loops wherein system behavior varies during loading and unloading cycles. Therefore, a single-tag data is used to determine the loading or unloading state, but tag signal causes discontinuity in network and omits various aspects of hysteresis in SMA, particularly within minor loops. In contrast, NNs incorporating past data to predict hysteresis behavior alleviate the need for tag data. However, such networks tend to have complex structures with a substantial number of neurons to effectively capture the inherent nonlinearity in SMAs. The long short-term memory (LSTM) neural network employed in this research, characterized by a simpler structure, achieves high accuracy in predicting hysteresis in SMAs without the need for tag data. In the proposed LSTM model, data related to the pulley’s rotational angle and the wire’s applied voltage from the current moment and the two previous moments serve as input. The data passes through a layer comprising three LSTM cells, and the output from the last LSTM cell is fed into a fully connected layer to predict the pulley’s rotational angle for the next moment. Training data are obtained by applying voltage at various frequencies and formats to the SMA wire while simultaneously recording the pulley’s angle with an encoder. Evaluation of the LSTM model is conducted in two configurations: online prediction (one-step ahead) and offline prediction (multistep ahead). In the online configuration where the model uses encoder data as angular inputs, the root mean square error (RMSE) of predictions for various input voltages is significantly low at about 0.1 degrees where the maximum rotational angle of pulley is 8 degrees. In the offline configuration when using the model’s predictions as angular inputs instead of encoder data, the RMSE rises to 0.3 degrees. To provide a clear demonstration of the LSTM model’s ability in this particular configuration, a comparison has been conducted between LSTM model and a rate-dependent Prandtl-Ishlinskii (RDPI) hysteresis model for predicting the pulley’s angle. The LSTM model outperforms the RDPI model by 70% in terms of accuracy. Overall, the LSTM model dem","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139837944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The complex behavior of shape memory alloys (SMAs), characterized by hysteresis and nonlinear dynamics, results in complex constitutive equations. To circumvent the complexity of solving these equations, a black box neural network (NN) has been employed in this research to model a rotary actuator actuated by an SMA wire. Considering the historical dependence of the pulley’s rotational angle on the applied voltage, a recurrent neural network (RNN) is suitable for capturing past information. Specifically, a long short-term memory (LSTM) neural network is selected due to its ability to address issues encountered in standard recurrent networks. There are major drawbacks with modelling hysteresis with NNs that do not account for historical behavior. Traditional NNs, characterized by a one-to-one mapping, struggle to capture hysteresis loops wherein system behavior varies during loading and unloading cycles. Therefore, a single-tag data is used to determine the loading or unloading state, but tag signal causes discontinuity in network and omits various aspects of hysteresis in SMA, particularly within minor loops. In contrast, NNs incorporating past data to predict hysteresis behavior alleviate the need for tag data. However, such networks tend to have complex structures with a substantial number of neurons to effectively capture the inherent nonlinearity in SMAs. The long short-term memory (LSTM) neural network employed in this research, characterized by a simpler structure, achieves high accuracy in predicting hysteresis in SMAs without the need for tag data. In the proposed LSTM model, data related to the pulley’s rotational angle and the wire’s applied voltage from the current moment and the two previous moments serve as input. The data passes through a layer comprising three LSTM cells, and the output from the last LSTM cell is fed into a fully connected layer to predict the pulley’s rotational angle for the next moment. Training data are obtained by applying voltage at various frequencies and formats to the SMA wire while simultaneously recording the pulley’s angle with an encoder. Evaluation of the LSTM model is conducted in two configurations: online prediction (one-step ahead) and offline prediction (multistep ahead). In the online configuration where the model uses encoder data as angular inputs, the root mean square error (RMSE) of predictions for various input voltages is significantly low at about 0.1 degrees where the maximum rotational angle of pulley is 8 degrees. In the offline configuration when using the model’s predictions as angular inputs instead of encoder data, the RMSE rises to 0.3 degrees. To provide a clear demonstration of the LSTM model’s ability in this particular configuration, a comparison has been conducted between LSTM model and a rate-dependent Prandtl-Ishlinskii (RDPI) hysteresis model for predicting the pulley’s angle. The LSTM model outperforms the RDPI model by 70% in terms of accuracy. Overall, the LSTM model dem
形状记忆合金 (SMA) 具有滞后和非线性动力学特征，其复杂的行为导致了复杂的构成方程。为了避免求解这些方程的复杂性，本研究采用了黑盒神经网络（NN）来模拟由 SMA 线材驱动的旋转致动器。考虑到滑轮旋转角度对施加电压的历史依赖性，循环神经网络（RNN）适合捕捉过去的信息。具体来说，选择长短期记忆（LSTM）神经网络是因为它能够解决标准递归网络中遇到的问题。使用不考虑历史行为的神经网络来模拟滞后现象存在很大缺陷。传统 NN 的特点是一对一映射，难以捕捉系统行为在加载和卸载周期中发生变化的滞后回路。因此，单个标签数据被用来确定加载或卸载状态，但标签信号会导致网络的不连续性，并忽略 SMA 中滞后的各个方面，尤其是在次要环路中。与此相反，结合过去数据预测滞后行为的网络可减轻对标签数据的需求。不过，此类网络往往结构复杂，需要大量神经元才能有效捕捉 SMA 固有的非线性。本研究采用的长短期记忆（LSTM）神经网络的特点是结构较为简单，在预测 SMA 的滞后现象时具有较高的准确性，而无需标记数据。在所提出的 LSTM 模型中，与滑轮旋转角度以及当前时刻和前两个时刻的导线外加电压相关的数据作为输入。数据通过一个由三个 LSTM 单元组成的层，最后一个 LSTM 单元的输出被输入到一个全连接层，以预测下一时刻滑轮的旋转角度。训练数据通过向 SMA 线施加不同频率和格式的电压获得，同时用编码器记录滑轮的角度。LSTM 模型的评估在两种配置下进行：在线预测（提前一步）和离线预测（提前多步）。在在线配置中，模型使用编码器数据作为角度输入，在滑轮最大旋转角度为 8 度的情况下，各种输入电压的预测均方根误差 (RMSE) 明显较低，约为 0.1 度。在离线配置中，当使用模型的预测值作为角度输入而不是编码器数据时，均方根误差上升到 0.3 度。为了清楚地展示 LSTM 模型在这一特定配置中的能力，我们对 LSTM 模型和与速率相关的 Prandtl-Ishlinskii (RDPI) 迟滞模型进行了比较，以预测滑轮的角度。LSTM 模型的准确度比 RDPI 模型高出 70%。总体而言，LSTM 模型展示了在在线和离线配置中有效模拟 SMA 磁滞的能力。
{"title":"Modelling Hysteresis in Shape Memory Alloys Using LSTM Recurrent Neural Network","authors":"M. Zakerzadeh, Seyedkeivan Naseri, Payam Naseri","doi":"10.1155/2024/1174438","DOIUrl":"https://doi.org/10.1155/2024/1174438","url":null,"abstract":"The complex behavior of shape memory alloys (SMAs), characterized by hysteresis and nonlinear dynamics, results in complex constitutive equations. To circumvent the complexity of solving these equations, a black box neural network (NN) has been employed in this research to model a rotary actuator actuated by an SMA wire. Considering the historical dependence of the pulley’s rotational angle on the applied voltage, a recurrent neural network (RNN) is suitable for capturing past information. Specifically, a long short-term memory (LSTM) neural network is selected due to its ability to address issues encountered in standard recurrent networks. There are major drawbacks with modelling hysteresis with NNs that do not account for historical behavior. Traditional NNs, characterized by a one-to-one mapping, struggle to capture hysteresis loops wherein system behavior varies during loading and unloading cycles. Therefore, a single-tag data is used to determine the loading or unloading state, but tag signal causes discontinuity in network and omits various aspects of hysteresis in SMA, particularly within minor loops. In contrast, NNs incorporating past data to predict hysteresis behavior alleviate the need for tag data. However, such networks tend to have complex structures with a substantial number of neurons to effectively capture the inherent nonlinearity in SMAs. The long short-term memory (LSTM) neural network employed in this research, characterized by a simpler structure, achieves high accuracy in predicting hysteresis in SMAs without the need for tag data. In the proposed LSTM model, data related to the pulley’s rotational angle and the wire’s applied voltage from the current moment and the two previous moments serve as input. The data passes through a layer comprising three LSTM cells, and the output from the last LSTM cell is fed into a fully connected layer to predict the pulley’s rotational angle for the next moment. Training data are obtained by applying voltage at various frequencies and formats to the SMA wire while simultaneously recording the pulley’s angle with an encoder. Evaluation of the LSTM model is conducted in two configurations: online prediction (one-step ahead) and offline prediction (multistep ahead). In the online configuration where the model uses encoder data as angular inputs, the root mean square error (RMSE) of predictions for various input voltages is significantly low at about 0.1 degrees where the maximum rotational angle of pulley is 8 degrees. In the offline configuration when using the model’s predictions as angular inputs instead of encoder data, the RMSE rises to 0.3 degrees. To provide a clear demonstration of the LSTM model’s ability in this particular configuration, a comparison has been conducted between LSTM model and a rate-dependent Prandtl-Ishlinskii (RDPI) hysteresis model for predicting the pulley’s angle. The LSTM model outperforms the RDPI model by 70% in terms of accuracy. Overall, the LSTM model dem","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"48 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139778004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.
{"title":"Inexact Exponential Penalty Function with the Augmented Lagrangian for Multiobjective Optimization Algorithms","authors":"Appolinaire Tougma, K. Some","doi":"10.1155/2024/9615743","DOIUrl":"https://doi.org/10.1155/2024/9615743","url":null,"abstract":"This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"3 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139439345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In recent decades, the study of non-Newtonian fluids has attracted the interest of numerous researchers. Their study is encouraged by the significance of these fluids in fields including industrial implementations. Furthermore, the importance of heat and mass transfer is greatly increased by a variety of scientific and engineering processes, including air conditioning, crop damage, refrigeration, equipment power collectors, and heat exchangers. The key objective of this work is to use the mathematical representation of a chemically reactive Casson-Maxwell fluid over a stretched sheet circumstance. Arrhenius activation energy and aspects of the magnetic field also have a role. In addition, the consequences of both viscous dissipation, Joule heating, and nonlinear thermal radiation are considered. The method transforms partial differential equations originating in fluidic systems into nonlinear differential equation systems with the proper degree of similarity which is subsequently resolved utilizing the Lobatto IIIA technique’s powerful computing capabilities. It is important to recall that the velocity profile drops as the Maxwell fluid parameter increases. Additionally, the increase in the temperature ratio parameter raises both the fluid’s temperature and the corresponding thickness of the boundary layer.
{"title":"Importance of Activation Energy on Magnetized Dissipative Casson-Maxwell Fluid through Porous Medium Incorporating Chemical Reaction, Joule Heating, and Soret Effects: Numerical Study","authors":"Nesreen Althobaiti","doi":"10.1155/2024/5730530","DOIUrl":"https://doi.org/10.1155/2024/5730530","url":null,"abstract":"In recent decades, the study of non-Newtonian fluids has attracted the interest of numerous researchers. Their study is encouraged by the significance of these fluids in fields including industrial implementations. Furthermore, the importance of heat and mass transfer is greatly increased by a variety of scientific and engineering processes, including air conditioning, crop damage, refrigeration, equipment power collectors, and heat exchangers. The key objective of this work is to use the mathematical representation of a chemically reactive Casson-Maxwell fluid over a stretched sheet circumstance. Arrhenius activation energy and aspects of the magnetic field also have a role. In addition, the consequences of both viscous dissipation, Joule heating, and nonlinear thermal radiation are considered. The method transforms partial differential equations originating in fluidic systems into nonlinear differential equation systems with the proper degree of similarity which is subsequently resolved utilizing the Lobatto IIIA technique’s powerful computing capabilities. It is important to recall that the velocity profile drops as the Maxwell fluid parameter increases. Additionally, the increase in the temperature ratio parameter raises both the fluid’s temperature and the corresponding thickness of the boundary layer.","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"11 20","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139382892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Pareto dominance-based evolutionary algorithms can effectively address multiobjective optimization problems (MOPs). However, when dealing with many-objective optimization problems with more than three objectives (MaOPs), the Pareto dominance relationships cannot effectively distinguish the nondominated solutions in high-dimensional spaces. With the increase of the number of objectives, the proportion of dominance-resistant solutions (DRSs) in the population rapidly increases, which leads to insufficient selection pressure. In this paper, to address the challenges on MaOPs, a knee point-driven many-objective evolutionary algorithm with adaptive switching mechanism (KPEA) is proposed. In KPEA, the knee points determined by an adaptive strategy are introduced for not only mating selection but also environmental selection, which increases the probability of generating excellent offspring. In addition, to remove dominance-resistant solutions (DRSs) in the population, an interquartile range method is adopted, which enhances the selection pressure. Moreover, a novel adaptive switching mechanism between angle-based selection and penalty for selecting solutions is proposed, which is aimed at achieving a balance between convergence and diversity. To validate the performance of KPEA, it is compared with five state-of-the-art many-objective evolutionary algorithms. All algorithms are evaluated on 20 benchmark problems, i.e., WFG1-9, MaF1, and MaF4-13 with 3, 5, 8, and 10 objectives. The experimental results demonstrate that KPEA outperforms the compared algorithms in terms of HV and IGD in most of the test instances.
{"title":"A Knee Point-Driven Many-Objective Evolutionary Algorithm with Adaptive Switching Mechanism","authors":"Maowei He, Xu Wang, Hanning Chen, Xuguang Li","doi":"10.1155/2024/4737604","DOIUrl":"https://doi.org/10.1155/2024/4737604","url":null,"abstract":"The Pareto dominance-based evolutionary algorithms can effectively address multiobjective optimization problems (MOPs). However, when dealing with many-objective optimization problems with more than three objectives (MaOPs), the Pareto dominance relationships cannot effectively distinguish the nondominated solutions in high-dimensional spaces. With the increase of the number of objectives, the proportion of dominance-resistant solutions (DRSs) in the population rapidly increases, which leads to insufficient selection pressure. In this paper, to address the challenges on MaOPs, a knee point-driven many-objective evolutionary algorithm with adaptive switching mechanism (KPEA) is proposed. In KPEA, the knee points determined by an adaptive strategy are introduced for not only mating selection but also environmental selection, which increases the probability of generating excellent offspring. In addition, to remove dominance-resistant solutions (DRSs) in the population, an interquartile range method is adopted, which enhances the selection pressure. Moreover, a novel adaptive switching mechanism between angle-based selection and penalty for selecting solutions is proposed, which is aimed at achieving a balance between convergence and diversity. To validate the performance of KPEA, it is compared with five state-of-the-art many-objective evolutionary algorithms. All algorithms are evaluated on 20 benchmark problems, i.e., WFG1-9, MaF1, and MaF4-13 with 3, 5, 8, and 10 objectives. The experimental results demonstrate that KPEA outperforms the compared algorithms in terms of HV and IGD in most of the test instances.","PeriodicalId":509379,"journal":{"name":"Journal of Applied Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139389300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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