A Novel Mathematical Formulation for Density-based Topology Optimization Method Considering Multi-Axis Machining Constraint

This paper proposes a novel density-based method for structural design considering restrictions of multiaxis machining processes. A new mathematical formulation based on Heaviside function is presented to transform the design field into a geometry which can be manufactured by multi-axis machining process. The formulation is developed for 5-axis machining, which can be also applied to 2.5D milling restriction. The filter techniques are incorporated to effectively control the minimum size of void region. The proposed method is demonstrated by solving the compliance minimization problem for different machinable freeform designs. Several two and three-dimensional numerical examples are presented and discussed in detail.

Modeling of thermal stresses during hardening the product surface by thermal pulse

A particular solution of a linear variant of the dynamic thermal elasticity problem is considered in application to modeling the conditions of surface hardening of metal products by an energy pulse. The authors determined the equation of medium motion with the model of temperature pulse tested earlier for compatibility with special cases of the equations of parabolic and hyperbolic thermal conductivity. The problem of loading a flat plane of a short circular cylinder (disk) with a temperature pulse is presented. Pulse is a consequence of adopted structure of the volumetric power density of the heat flux, the time multiplier of which has the form of a single wave of the Heaviside function. Classical thermoelastic displacement potential and the method of its division into the product of independent variables functions were used to construct the thermal stress tensor. Differential equations for multiplier functions and their general solutions were found. Natural boundary conditions were set for the components of thermal stress tensor, and their tasks were solved. The obtained solutions are in the form of segments of functional series (the Bessel function in radial coordinate and the exponential function in axial coordinate). The article considers a numerical example of loading a disk made of 40KhN steel which has the mechanical properties sensitive to temperature treatment. Maple computer mathematics package was used in the calculations. Approximate solutions take into account the first 24 terms of the functional series. Estimation of the example makes it possible to explain the presence of stress peaks and stress intensity as a consequence of mutually inverse processes of temperature stress growth and reduction of elasticity coefficients with temperature rise. The numerical example warns against relying only on estimates of solutions to thermoelasticity problems without taking into account the plastic and viscous properties of the material.

Propagation of Electrical Pulse in the Finite Interval of Perfect Electrical Conductivity and its Full Absorption at the End of Interval

The propagation of an electric pulse in a finite interval is investigated in the case when the pulse is generated at the input of the interval and absorbed at the end of the interval. The pulse propagation is described by a hyperbolic equation with regard for dissipation. The pulse generation at the input is specified as a Heaviside function, and the absorption at the output is set by a permanent magnet. The model describes the propagation of disturbances with a finite speed. A formulation of the corresponding initial boundary value problem is given, for the solution of which the Laplace transform in time is applied in the case of arbitrary coefficients. An exact analytical solution in the Laplace image space was obtained, and other applications with the complete absorption are presented. A general solution is constructed, and the case of low dissipation is considered for some values of the coefficients characterizing real situations.

Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application

Integral transformations are essential for solving complex problems in business, engineering, natural sciences, computers, optical science, and modern mathematics. In this paper, we apply a general integral transform, called the Jafari transform, for solving a system of ordinary differential equations. After applying the Jafari transform, ordinary differential equations are converted to a simple system of algebraic equations that can be solved easily. Then, by using the inverse operator of the Jafari transform, we can solve the main system of ordinary differential equations. Jafari transform belongs to the class of Laplace transform and is considered a generalization to integral transforms such as Laplace, Elzaki, Sumudu, G\_transforms, Aboodh, Pourreza, etc. Jafari transform does not need a large computational work as the previous integral transforms. For the Jafari transform, we have studied some valuable properties and theories that have not been studied before. Such as the linearity property, scaling property, first and second shift properties, the transformation of periodic functions, Heaviside function, and the transformation of Dirac’s delta function, and so on. There is a mathematical model that describes the cell population dynamics in the colonic crypt and colorectal cancer. We have applied the Jafari transform for solving this model.

Multi-material topology optimization of permanent magnet synchronous motors

This paper presents a multi-material topology optimization for the design of permanent magnet synchronous motors (PMSMs). Specifically, structural shapes of permanent magnet (PM) and iron core in a PMSM rotor are simultaneously designed together with the orientation of PM magnetization. For a co-design of PM and iron core, relative permeability and residual magnetic flux density are interpolated by the three-field density approach based on the Helmholtz filtering and regularized Heaviside step function. Here, the Helmholtz filtering aims to attain smooth border in design results, and the Heaviside function enables us to acquire a clear border (i.e. zero-one design) without intermediate densities. The optimization goal is set as maximizing the average torque of PMSMs. The average torque is calculated by Maxwell stress tensor (MST) method considering a maximum torque per ampere (MTPA) control. To validate the effectiveness of the proposed multi-material topology optimization approach, a PMSM rotor with 4 poles and 12 slots is designed. In addition, design results at various settings of input current amplitude and PM strength are compared and discussed. When the input current is stronger than the PM strength, rotor PM and iron core are designed for utilizing both PM and reluctance torque components like V-shape interior PMSMs. On the other hand, in the case of stronger PM strength, PM is designed near the air-gap, which utilizes only PM torque component like surface PMSMs.

Artificial Neural Network Activation Functions in Exact Analytical Form (Heaviside, ReLU, PReLU, ELU, SELU, ELiSH)

Activation functions are fundamental elements in artificial neural networks. The mathematical formulation of some activation functions (e.g. Heaviside function and Rectified Linear Unit function) are not expressed in an explicit closed form. This made them numerically unstable and computationally complex during estimation. This paper introduces a novel explicit analytic form for those activation functions. The proposed mathematical equations match exactly the original definition of the studied activation function. The proposed equations can be adapted better in optimization, forward and backward propagation algorithm employed in an artificial neural network.

Artificial Neural Network Activation Functions in Exact Analytical Form (Heaviside, ReLU, PReLU, ELU, SELU, ELiSH)

Activation functions are fundamental elements in artificial neural networks. The mathematical formulation of some activation functions (e.g. Heaviside function and Rectified Linear Unit function) are not expressed in an explicit closed form. This made them numerically unstable and computationally complex during estimation. This paper introduces a novel explicit analytic form for those activation functions. The proposed mathematical equations match exactly the original definition of the studied activation function. The proposed equations can be adapted better in optimization, forward and backward propagation algorithm employed in an artificial neural network.

Manufacturable casting parts design with topology optimization of structural assemblies

This paper presents a design method for manufacturable casting parts based on topology optimization of structural assemblies, which considers the geometry requirement and the manufacturing constraint of die-set material cost. The problem formulation follows the previous work presented in multi-component topology optimization for stamped sheet metal assemblies (MTO-S). Based on the vector method combined with Heaviside function, the moldability constraints for casting parts is formulated. As the base structure of component is easily misidentified as an undercut structure by the moldability constraints in the structural assemblies, the component baseline is proposed to realize the automatic filtering of the “fake” undercut structures which can be extended to the parting line to obtain the form of two-mold design. Several numerical examples on compliance minimization of single-mold and two-mold casting parts are conducted to verify the validity of the proposed method. The optimized results show that there is no interior void for each component and the component manufacturability has been improved obviously. The setting of minimum-area bounding box (MABB) area constraint limits and the number of components will have a significant effect on the performance of the optimized structure. Users can achieve the desirable solution based on their actual demand by making trade-offs between the structural performance and manufacturing cost.

Characterizing Network Anomaly Traffic with Euclidean Distance-Based Multiscale Fuzzy Entropy

The prosperity of mobile networks and social networks brings revolutionary conveniences to our daily lives. However, due to the complexity and fragility of the network environment, network attacks are becoming more and more serious. Characterization of network traffic is commonly used to model and detect network anomalies and finally to raise the cybersecurity awareness capability of network administrators. As a tool to characterize system running status, entropy-based time-series complexity measurement methods such as Multiscale Entropy (MSE), Composite Multiscale Entropy (CMSE), and Fuzzy Approximate Entropy (FuzzyEn) have been widely used in anomaly detection. However, the existing methods calculate the distance between vectors solely using the two most different elements of the two vectors. Furthermore, the similarity of vectors is calculated using the Heaviside function, which has a problem of bouncing between 0 and 1. The Euclidean Distance-Based Multiscale Fuzzy Entropy (EDM-Fuzzy) algorithm was proposed to avoid the two disadvantages and to measure entropy values of system signals more precisely, accurately, and stably. In this paper, the EDM-Fuzzy is applied to analyze the characteristics of abnormal network traffic such as botnet network traffic and Distributed Denial of Service (DDoS) attack traffic. The experimental analysis shows that the EDM-Fuzzy entropy technology is able to characterize the differences between normal traffic and abnormal traffic. The EDM-Fuzzy entropy characteristics of ARP traffic discovered in this paper can be used to detect various types of network traffic anomalies including botnet and DDoS attacks.

Error modeling and singularity analysis of planar parallel mechanisms using optimized expression of joint clearances and link deformation

A fast error modeling method is proposed to analyze the pose deviation of 3-RPR planar parallel mechanisms with multiple revolute joint clearances. The pose error arisen from clearances and limb deformation is modeled from the point of inverse kinematics and estimated numerically. Manipulator poses are represented by elements of the Lie group SE(2) and the discontinuous deformation of limbs due to clearances are modeled by the Heaviside function, which is then approximated with the hyperbolic tangent function. After establishing the error model, an efficient error compensation method is introduced. The proposed error modeling and accuracy analysis method are validated by case study of a 3-RPR mechanism. The results show that the hyperbolic tangent function runs faster than the step function and it provides better convergence. Finally, the singularity of the 3-RPR mechanism with load is analyzed and the results reveal the effects of loads and clearances on mechanism singularities.