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Engineering structures usually operate in some specific frequency bands. An effective way to avoid resonance is to shift the structure’s natural frequencies out of these frequency bands. However, in the optimization procedure, which frequency orders will fall into these bands are not known a priori. This makes it difficult to use the existing frequency constraint formulations, which require prescribed orders. For solving this issue, a novel formulation of the frequency band constraint based on a modified Heaviside function is proposed in this paper. The new formulation is continuous and differentiable; thus, the sensitivity of the constraint function can be derived and used in a gradient-based optimization method. Topology optimization for maximizing the structural fundamental frequency while circumventing the natural frequencies located in the working frequency bands is studied. For eliminating the frequently happened numerical problems in the natural frequency topology optimization process, including mode switching, checkerboard phenomena, and gray elements, the “bound formulation” and “robust formulation” are applied. Three numerical examples, including 2D and 3D problems, are solved by the proposed method. Frequency band gaps of the optimized results are obtained by considering the frequency band constraints, which validates the effectiveness of the developed method.

This paper presents a density-based topology optimization approach to design structures under self-weight load. Such loads change their magnitude and/or location as the topology optimization advances and pose several unique challenges, e.g., non-monotonous behavior of compliance objective, parasitic effects of the low-stiffness elements, and tendency to lose constrained nature of the problems. The modified SIMP material scheme is employed with the three-field density representation technique (original, filtered, and projected design fields) to achieve optimized solutions close to 0–1. A novel mass density interpolation strategy is proposed using a smooth Heaviside function, which provides a continuous transition between solid and void states of elements and facilitates tuning of the non-monotonous behavior of the objective. A constraint that implicitly imposes a lower bound on the permitted volume is conceptualized using the maximum permitted mass and the current mass of the evolving design. Sensitivities of the objective and self-weight are evaluated using the adjoint-variable method. Compliance of the domain is minimized to achieve the optimized designs using the Method of Moving Asymptotes. The efficacy and robustness of the presented approach are demonstrated by designing various 2D and 3D structures involving self-weight. The proposed approach maintains the constrained nature of the optimization problems and provides smooth and rapid objective convergence.

To prevent numerical instabilities and ensure manufacturability, restrictions should be applied in topology optimization. In this paper, a volume preserving density filter based on Heaviside functions is presented. Different from earlier Heaviside density filters, this filter is volume preserving, which ensures efficiency and stability in optimization. The new filter is compared with four other filters through a compliance minimization problem.

This paper reviews recent progress in the measurement and modelling of stochastic electromagnetic fields, focusing on propagation approaches based on Wigner functions and the method of moments technique. The respective propagation methods are exemplified by application to measurements of electromagnetic emissions from a stirred, cavity-backed aperture. We discuss early elements of statistical electromagnetics in Heaviside's papers, driven mainly by an analogy of electromagnetic wave propagation with heat transfer. These ideas include concepts of momentum and directionality in the realm of propagation through confined media with irregular boundaries. We then review and extend concepts using Wigner functions to propagate the statistical properties of electromagnetic fields. We discuss in particular how to include polarization in this formalism leading to a Wigner tensor formulation and a relation to an averaged Poynting vector. This article is part of the theme issue 'Celebrating 125 years of Oliver Heaviside's 'Electromagnetic Theory''.

In this paper, a feasible feedback scheme is used to stabilize the multi-scroll attractors in the Jerk circuit, and the controller is realized using mixed Heaviside function. It is found that arbitrary number ( n = 2 , 3 , 4 , 5 , … ) of multi-scroll attractors can be selected from a controlled Jerk circuit, and these multi-scroll attractors can be reproduced using Pspice. The implementation of circuit and controller using Pspice is also presented. The potential mechanism could be that an external forcing in the Sine type is practical to generate a group of equilibrium points, and a linear controller composed of Heaviside function is effective to stabilize the n -scroll attractors in the chaotic systems.

A new interactive truss layout optimization web-app has been developed for educational use. This has been designed to be used on a range of devices, from mobile phones to desktop PCs. Truss designs are first generated via numerical layout optimization and then rationalized via geometry optimization. It is then shown that these designs can be simplified using a computationally inexpensive process that allows the user to control the trade-off between complexity and structural volume. The process involves the use of smooth Heaviside representations of member existence variables, with nodal slack forces employed that allow unstable intermediate truss structures. Full details of the web-app are provided in this contribution, from underlying formulation to cloud computing implementation. A range of numerical examples are used to demonstrate the efficacy of the web-app, and to show how it can potentially be used in educational and practical engineering settings.

This study presents a novel reformulation of the Klein–Gordon (KG) equation by embedding it within a system of first-order Maxwell–Heaviside (MH)-like equations, enabling its numerical solution using the finite-difference time-domain method based on the Yee algorithm. This approach extends the scalar KG field into a pair of fictitious Maxwellian vector fields. This reformulation not only provides an efficient computational framework, capable of handling nonlinearity and inhomogeneity, but also introduces a first-order structure with symmetric field dynamics. Plane-wave quantization of these fields reveals a conserved, non-negative quantity, forming what is termed Conserved Maxwellian Fields (CMFs), that addresses the longstanding issue of negative probability density in the conventional KG theory. Furthermore, the resulting CMFs exhibit deep structural analogies with Dirac spinors, particularly in three spatial dimensions, where only two CMF modes exist with monopole-like divergence. These findings bridge the gap between scalar field dynamics and electromagnetic field theory, offering both computational utility and potential insight into hidden structures in quantum field theory.

Conventional definitions of 'near fields' set bounds that describe where near fields may be found. These definitions tell us nothing about what near fields are, why they exist or how they work. In 1893, Heaviside derived the electromagnetic energy velocity for plane waves. Subsequent work demonstrated that although energy moves in synchronicity with radiated electromagnetic fields at the speed of light, in reactive fields the energy velocity slows down, converging to zero in the case of static fields. Combining Heaviside's energy velocity relation with the field Lagrangian yields a simple parametrization for the reactivity of electromagnetic fields that provides profound insights to the behaviour of electromagnetic systems. Fields guide energy. As waves interfere, they guide energy along paths that may be substantially different from the trajectories of the waves themselves. The results of this paper not only resolve the long-standing paradox of runaway acceleration from radiation reaction, but also make clear that pilot wave theory is the natural and logical consequence of the need for quantum mechanics correspond to the macroscopic results of the classical electromagnetic theory. This article is part of the theme issue 'Celebrating 125 years of Oliver Heaviside's 'Electromagnetic Theory''.

The challenge for practical application of frame structural optimization had previously been investigated by many works, while the mechanical performance requirements such as the displacement, stress, and stability requirements, were often considered separately within optimization, hindering their practical applications. For this purpose, an integrated topology and size optimization strategy of frame structures, in which the structural weight is taken as the objective with the constraints regarding the displacement, stress, as well as stability, is presented in this paper. Different from former researches, each beam is assigned with a topology variable representing the presence of the beam and a size variable correspond to the cross-sectional geometric properties. To achieve an optimized design with standard members, by cooperating the ordered multi-material SIMP (solid isotropic material with penalization) interpolation with the normalized Heaviside functions, the continuous size design variables are projected onto the discrete standard sizes conformed to standard library. Moreover, the comprehensive measure, including the stress relaxation, the pseudobuckling mode treatment scheme, the aggregation constraint, and varying constraint limit schemes, is employed to deal with the multiple constraints in the optimization model. Then, the sensitivities of the objective and constraint functions with respect to topology and size design variables are derived, respectively, and the proposed integrated optimization problem is solved by a nested optimization algorithm. Finally, several numerical examples are presented to demonstrate the feasibility of the proposed approach.

It is so important to predict electric fields in advance before designing electrical devices or electronic products. In general, an electric field for an electric device to be designed is usually calculated using a commercial numerical analysis program. However, numerical analysis programs are very expensive, require high specifications for computers, and are also insufficient to have a physical understanding of the trends in electric fields. Therefore, in this paper, the electric field is obtained using the derived analytical equation using Heaviside unit-step function about the discontinuity of dielectric constants especially in the structure of the inclined dielectric media because of the discontinuity about the dielectric constants at the interface where two dielectric materials meet. Finally, the analytical method-based results are compared with the numerical analysis results and verified.