Spherical Hole Research Articles

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

AbstractIn this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of $$\mathbb {R}^n$$ R n . Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite $$(n-q)$$ ( n - q ) -moment, where $$1<q<n$$ 1 < q < n is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

Topological derivative based sensitivity analysis for three-dimensional discrete variable topology optimization

This study introduces a novel Topological Derivative-based Sensitivity Analysis (TDSA) methodology for three-dimensional (3D) discrete variable topology optimization. Recently, the authors pointed out that the discrete variable sensitivity can be related to the topological derivative, and thus can be rationally approximated by the specially customized topological derivative for plane stress problems. However, the 3D discrete variable sensitivity requires the 3D topological derivative with element-shaped (e.g., unit cube) domain perturbation, whose analytical solution is not available in the literature. This paper proposes a unified parameter fitting framework to calculate the 3D topological derivative with arbitrary shape 3D domain perturbation. The formula for spherical hole perturbation obtained through this parameter fitting framework is identical to the well-known analytical topological derivative formula. Further, the 3D discrete variable sensitivity can be easily obtained through element-shaped domain perturbation. With the recently proposed Sequential Approximate Integer Programming (SAIP), numerical examples demonstrate that TDSA is precise not only for the self-adjoint 3D minimum compliance problems but also for the non-adjoint 3D compliant mechanism design problems. In sum, numerical examples have been conducted by feeding the derived sensitivity to the SAIP optimization algorithms, and the correctness of the sensitivity information has been well demonstrated. This research further solidifies the foundation of rational discrete variable sensitivity analysis in 3D topology optimization.

Enhancement of ionospheric heating effect by chemical release

The ionosphere can be artificially modified by employing ground-based high-power high-frequency electromagnetic waves to irradiate the ionosphere. This modification is achieved through the nonlinear interaction between the electromagnetic waves and the ionospheric plasma, leading to changes in the physical properties and structure of the ionosphere. The degree of artificial modification of the ionosphere is closely related to the heating energy density of high-frequency pump waves. Due to the high density of neutral constituents in the lower ionosphere and the high frequency of electron-neutral collisions, the energy of heating pump waves will be absorbed and attenuated during the penetration of the low ionosphere, seriously affecting the heating effect. This paper proposes a method to reduce the absorption of ionospheric heating pump waves by releasing electron attachment chemicals into low ionosphere to form a large-scale electron density hole. A model for mitigating pump waves absorption based on SF6 release is established, and the absorption at different frequencies is quantitatively calculated. The propagation characteristics of high-frequency signals in ionospheric holes are studied using a three-dimensional ray tracing method, and the results demonstrate that the chemical release method not only reduces the absorption attenuation of heating pump waves but also forms spherical electron density holes, which exhibit a focusing effect on the heating beam and enhance the heating effect. The results are of great significance for understanding the nonlinear interaction between electromagnetic wave and ionospheric plasma and improving the ionospheric heating efficiency.

Additive manufacturing of metallic metamaterials with enhanced mechanical properties enabled by microstructural and structural design

The emergence of additive manufacturing (AM) facilitates the fabrication of lightweight mechanical metamaterials characterized by intricate geometrical features. Here, we focus on the contributions of microstructural and structural design to the significant performance enhancement of metamaterials. Cubic plate-lattices featuring spherical holes were produced using laser powder bed fusion. Different from commonly used optimization of AM parameters to change the thermal histories and the resulting properties, we employ a simple strategy inspired by the crystallographic and AM features—tilting the build orientation. Compared to the normal build orientation, the tilted build orientation converts the printed microstructure of the plate-lattices from (100)-dominated to (111)- and (101)-dominated crystallographic texture and significantly refines the grain size, leading to remarkable 30% and 10% increases in the compressive strength and strain of the printed plate-lattices, respectively. For further tailoring the performance of metamaterials, we integrate a wavy plate topology design to improve the isotropy of properties and increase the impact attenuation. Our work paves the way to optimize additively manufactured metamaterials by combining microstructural and structural designs.

Thermomagnetic responses of a thermoelastic medium containing a spherical hole exposed to a timed laser pulse heat source

The current work aims to investigate the interaction between thermal, magnetic, and elastic phenomena in a homogeneous and unbounded material that includes a spherical cavity. A laser beam with a Gaussian distribution was used to expose the solid thermal medium, thereby bringing a source of heat into the system. The study is grounded in the expanded theory of thermoelasticity without energy dissipation, which considers the interconnection between thermal, elastic, and magnetic phenomena. This theoretical framework serves as a basis for understanding the interactions and behaviors of thermal and mechanical processes within an elastic material. The system was analyzed using a direct approach, employing the Laplace transformation method to provide solutions for the system's equations. This methodology enables the computation and presentation of precise equations for several domains of concern, such as displacement, temperature, and thermal stresses. The discussion and analysis focused on a copper-based material, and graphical representations were used to demonstrate the behavior of the researched physical fields. These graphical representations aid in comprehending the development of displacement, temperature, and thermal stresses within the material caused by the laser beam and the interplay between thermal, magnetic, and elastic processes. The results demonstrate that the laser pulse parameters have a significant impact on the temperature and thermal stress distribution, thereby influencing the thermo-mechanical reactions of the solid medium. Ultimately, this study enhances our comprehension of the intricate behavior and interplay of thermal, magnetic, and elastic phenomena in thermoelastic materials.

Fractional Heat Conduction with Heat Absorption in a Solid with a Spherical Cavity under Time-Harmonic Heat Flux

The central-symmetric time-fractional heat conduction equation with heat absorption is investigated in a solid with a spherical hole under time-harmonic heat flux at the boundary. The problem is solved using the auxiliary function, for which the Robin-type boundary condition with a prescribed value of a linear combination of a function and its normal derivative is fulfilled. The Laplace and Fourier sine–cosine integral transformations are employed. Graphical representations of numerical simulation results are given for typical values of the parameters.

Fast and Accurate Measurement of Hole Systems in Curved Surfaces

Curved surface structural parts with hole systems are widely used, and accurate measurement of the hole systems is crucial for assembly and functionality. This study presents a novel approach using machine vision and structural science principles to accurately measure spherical hole systems. We introduce key technologies, including measurement parameter definition, system design, and error modeling, in the paper. Our approach overcomes the limitations of existing methods, offering flexibility, precision, and automation measurement of the hole system. Experimental results demonstrate an accuracy of 0.348′ (arcminutes). This research contributes to the optical measurement of curved surface hole systems and improves their alignment and functionality.

Coron's problem for the critical Lane-Emden system

We address the solvability of the critical Lane-Emden system in a smooth bounded domain with a small spherical hole of radius ϵ>0. We prove that the system admits a family of positive solutions that concentrate around the center of the hole as ϵ→0, obtaining a concrete qualitative description of the solutions as well. To the best of our knowledge, this is the first existence result for the critical Lane-Emden system on a bounded domain, while the non-existence result on star-shaped bounded domains has been known since the early 1990s due to Mitidieri (1993) [30] and van der Vorst (1992) [36].

A monotonicity result for the first Steklov–Dirichlet Laplacian eigenvalue

AbstractIn this paper, we consider the first Steklov–Dirichlet eigenvalue of the Laplace operator in annular domains with a spherical hole. We prove a monotonicity result with respect to the hole, when the outer region is centrally symmetric.

An energy-stable finite element method for nonlinear Maxwell's equations

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell's equations in the optical media characterized by linear Lorentz dispersion, nonlinear Kerr effect and delayed Raman scattering. Using Crank-Nicolson time discretization and special treatment of the nonlinearity, we propose a second order accurate (in time) finite element method for the nonlinear problem. Fully discrete energy stability analysis is established to show that the scheme is unconditionally stable. The nonlinear system is solved by Newton-Krylov method. In order to efficiently solve the linearized algebraic system in each Newton sub-iteration, we further develop an effective preconditioner by using suitable approximations to the nonlinearity and the auxiliary space preconditioning technique. Numerical experiments are provided to examine the accuracy, the energy stability, the parallel scalability and the robustness in preconditioning of our proposed scheme. We then apply the scheme to simulate the spatial soliton propagation in 3D nonlinear glass. We also demonstrate the performance of the scheme in handling complex geometry through simulating the wave scattering by a spherical hole in the nonlinear optical glass.

An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $\Omega=\Omega_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset \Omega_0$, then the first Steklov-Dirichlet eigenvalue $\sigma_1(\Omega)$ has a maximum when $R_1$ and the measure of $\Omega$ are fixed. Moreover, if $\Omega_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.

Transient responses to an infinite solid with a spherical cavity according to the MGT thermo-diffusion model with fractional derivatives without nonsingular kernels

The main purpose of this paper is to describe the coupling mechanism between heat transfer and diffusion in thermoelastic materials. This will be investigated by deriving a new thermal diffusion model that allows waves to move at limited speeds, unlike the traditional thermodiffusion models. Based on the thermal relaxation concept, the fractional Moore-Gibson-Thompson equation can be used to describe the equations for heat conduction and mass diffusion. The system of equations is developed using the Green-Naghdi Type III theories and the Atangana–Baleanu-type (AB) fractional differential operator instead of the integer derivatives of time. Under the proposed concept of the fractional thermo-diffusion model, a problem of an unbounded elastic structure with a spherical hole is investigated. The surface within the bore is free of traction and exposed to thermal shock and chemical potential. The system of governing equations was combined into a sixth-order differential equation to determine the analytical results. For this purpose, the Laplace transform was used, and a numerical algorithm to calculate the inversions was applied. The numerical results demonstrate that the improved fractional operator considerably impacts the dynamics of all research fields.

Influence Analysis of Lubricant Recesses on the Working Capacity of the Bridge Span Spherical Bearing

The load on transport and logistics systems is increasing every year. This is due to car park growth around the world. Thus, increasing bridge structure durability is an urgent task for bridge-building companies. This study analyses the contact deformation of spherical bearing elements through an anti-friction polymer layer with different geometrical configurations of recesses for the lubricant, i.e., annular grooves and spherical holes. The material of the anti-friction layer (a modified polytetrafluoroethylene (PTFE)) is modelled within the framework of the deformation theory of plasticity. The procedure of automating the numerical model construction depends on the input parameters, including the thickness of the layer, the basic geometrical parameters of the recesses for the lubricant, and the distance between the rows of recesses, etc. The influence of the arrangement of filling sliding anti-friction layers on recesses for lubricants in the form of spherical holes on the contact deformation behaviour of bridge bearings has been considered. The reduction of lubricant volume in the sliding anti-friction layer with the geometry of recesses in the form of spherical holes ranges from 26 to 48.4%, depending on the filling scheme, has been found. In this case, structures with lubrication recesses in the form of spherical holes have several advantages, including a more uniform distribution of contact parameters in the interface areas of the steel plates with the anti-friction layer, reduction of the maximum level of the plastic deformation intensity, displacements along the normal relative to the free end of the sliding layer, and the settlement of the bearing.

Controllable Synthesis, Formation Process, and Luminescence Performances of Diverse Yttrium Compounds with Hollow Structures.

Hollow spherical Y2O3 and YBO3 have been prepared by a facile template-directed strategy using phenol-formaldehyde (PF) resin spheres as templates. The PF@Y(OH)CO3 precursor can be fabricated by a simple precipitation route. The Y2O3 hollow spheres are obtained via a direct annealing process, and the hollow spherical YBO3 are fabricated via a hydrothermal route followed by an annealing process at the expense of the same PF@Y(OH)CO3 precursor. The whole synthesis procedure is performed in aqueous solution without any surfactant or catalyst. Moreover, YVO4 quasi-octahedral microcrystals with spherical holes are obtained. The formation mechanisms of the yttrium compounds with different morphologies have been discussed. By incorporating proper rare earth activator ions into the Y2O3, YBO3, and YVO4 hosts, the as-synthesized luminescent materials can exhibit eminent performances with both down-conversion and up-conversion luminescence. Furthermore, the as-fabricated light-emitting diode (LED) devices can emit dazzling characteristic emission light, which reveals that the phosphors have application potential in lighting and displays. This simple synthesis strategy may provide a new idea for the fabrication of inorganic compounds with perfect hollow structures and excellent properties.

Influence of Compound Spherical Trenched Holes on Film Cooling Performance at the end of Combustor Simulator

The major effects of cylindrical and spherical trenched cooling holes with distance between the hole surface and the combustion chamber panel and the filler diameter on the spherical hole contact surface and the panel surface H=0.3,R=D/2=0.1, H=0.3, R=D/2=0.2 and H=0.3, R=D/2=0.3 cm at BR=3.18 on the film cooling effectiveness near the combustor end wall surface is an important subject to study in details. In this research, a three-dimensional representation of a Pratt and Whitney gas turbine engine was simulated and analysed with a commercial finite volume package FLUENT 6.2.26. The analyses were done with RANS turbulence model on internal cooling passages. The combustor simulator was combined with the interaction of two rows of dilution jets, which were staggered in the streamwise direction and aligned in the spanwise direction. In comparison with the baseline case, the application of trenched holes increased the effectiveness of film cooling up to 47% near the wall surface and an average of 35% in depth of combustor simulator .

Magneto-Thermoelastic Response in an Infinite Medium with a Spherical Hole in the Context of High Order Time-Derivatives and Triple-Phase-Lag Model.

The article presents the interactions of magneto-thermoelastic effects in an isotropic material with a spherical cavity. The spherical cavity is expected to be tractionless and subjected to both heat and magnetic fields. The motion equation contains the Lorentz force. Laplace’s transformation methodology is used with a refined multi-time-derivative triple-phase-lag thermoelasticity theory to develop the generalized magneto-thermoelastic coupled solution. Many results were obtained to serve as benchmarks for future comparisons. The effects of time, magnetic field, and electric permittivity under the thermal environment were investigated.

Generalized thermoelastic MGT model for a functionally graded heterogeneous unbounded medium containing a spherical hole

Generalized thermoelastic MGT model for a functionally graded heterogeneous unbounded medium containing a spherical hole

Possible connection between time-splitting parameter and surface tension in Casimir-type problems

We discuss four different, though related, fundamental topics related to the Casimir effect: (1) We suggest that the application of Casimir theory to real dielectric materials, thus implying the atomic spacing as a course-grained length parameter, makes it natural to assume that this parameter is of the same order of magnitude as the QFT time-splitting parameter multiplied with the velocity of light. (2) We show that application of Casimir theory to a thick fluid shell (apparently a closed mechanical system) leads actually to an unstable situation if no extra mechanical forces, typically surface tension forces, are brought into consideration. (3) We analyze how the presence of a radial Casimir repulsive pressure modifies the filling process of a spherical vacuum hole in an infinite fluid (the Reynolds problem), with the result that a bounce occurs at a finite though very small radius. (4) As a comment on an apparently similar situation in general relativity, we consider the gravitational collapse of a singular shell. It might seem natural to allow for the presence of a repulsive Casimir pressure in this case also, thereby obtaining a bounce-like situation again. However, we have to conclude that such a procedure implies an omission of the Casimir field’s gravitational energy, and is therefore hardly tenable, although it is in our opinion worth mentioning.

Magneto-Thermoelastic Response in an Unbounded Medium Containing a Spherical Hole via Multi-Time-Derivative Thermoelasticity Theories.

This article introduces magneto-thermoelastic exchanges in an unbounded medium with a spherical cavity. A refined multi-time-derivative dual-phase-lag thermoelasticity model is applied for this reason. The surface of the spherical hole is considered traction-free and under both constant heating and external magnetic field. A generalized magneto-thermoelastic coupled solution is developed utilizing Laplace’s transform. The field variables are shown graphically and examined to demonstrate the impacts of the magnetic field, phase-lags, and other parameters on the field quantities. The present theory is examined to assess its validity including comparison with the existing literature.

Convergence of the pressure in the homogenization of the Stokes equations in randomly perforated domains

We consider the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. This problem has been studied by the same authors in Giunti and Höfer (2019) [7] where convergence of the fluid velocity field towards the solution of the Brinkman equations has been established. In the present we consider the pressure associated to the solution of the Stokes equations in the perforated domain. We prove that it is possible to extend this pressure inside the holes and slightly modify it in a region of asymptotically negligible harmonic capacity such that it weakly converges to the pressure associated with the solution of the Brinkman equations.