Equivalent Boundary Research Articles

Part-scale numerical simulation for laser-powder directed energy deposition: experimental calibration

Abstract To increase the environmental sustainability of products, the repair and refurbishment of damaged or obsolete components can be carried out using additive manufacturing (AM) processes. The Laser-powder directed energy deposition (LP-DED) process is an interesting candidate due to its many potential applications, particularly for the repair of metal parts. For a more thorough knowledge and understanding of the process, numerical approaches are necessary to avoid extensive and costly experimental campaigns. In this work, an original 3D thermal FEM model is proposed to predict the temperature during the LP-DED process at the macroscopic scale. The numerical simulation was based on the layer-by-layer method using an equivalent volumetric heat source and then using the superlayer method. The predicted temperature is compared with experimental temperature measurements during the deposition of stainless steel (316L) single-bead walls and multi-layer squares. The experimental results showed the influence of process parameters and deposition strategy on the temperature, which reached a maximum around 220 °C. The numerical simulation showed an overestimation of the temperature of about 120 to 200 °C compared to the experiments for all the tests carried out. A correction factor was applied to the numerical heat source which allowed to reduce the discrepancies to 50 °C. Finally, the superlayer approach led to a reduction of computational time at the expense of accuracy. Current limitations highlighted the importance of the definition of the equivalent heat source definition and boundary conditions. The chosen approach (layer-by-layer vs. superlayer) depends on the applications and the studied geometry.

Modeling and Pricing European‐Style Continuous‐Installment Option Under the Heston Stochastic Volatility Model: A PDE Approach

ABSTRACTInstallment options, as path‐dependent contingent claims, involve paying the premium discretely or continuously in installments, rather than as a lump sum at the time of purchase. In this paper, we applied the PDE approach to price European continuous‐installment option and consider Heston stochastic volatility model for the dynamics of the underlying asset. We proved the existence and uniqueness of the weak solution for our pricing problem based on the two‐dimensional finite element method. Due to the flexibility to continue or stop paying installments, installment options pricing can be modeled as an optimal stopping time problem. This problem is formulated as an equivalent free boundary problem and then as a linear complementarity problem (LCP). We wrote the resulted LCP in the form of a variational inequality and used the finite element method for the discretization. Then the resulting time‐dependent LCPs are solved by using a projected successive over relaxation iteration method. Finally, we implemented our numerical method. The numerical results verified the efficiency and accuracy of the proposed numerical method.

Residual Energy and Broken Symmetry in Reduced Magnetohydrodynamics

Alfvénic interactions that transfer energy from large to small spatial scales lie at the heart of magnetohydrodynamic turbulence. An important feature of the turbulence is the generation of negative residual energy—excess energy in magnetic fluctuations compared to velocity fluctuations. By contrast, an MHD Alfvén wave has equal amounts of energy in fluctuations of each type. Alfvénic quasi-modes that do not satisfy the Alfvén wave dispersion relation and exist only in the presence of a nonlinear term can contain either positive or negative residual energy, but until now, an intuitive physical explanation for why negative residual energy is preferred has remained elusive. This paper shows that the equations of reduced MHD are symmetric in that they have no intrinsic preference for one sign of the residual energy over the other. An initial state that is not an exact solution to the equations can break this symmetry in a way that leads to net-negative residual energy generation. Such a state leads to a solution with three distinct parts: nonresonant Alfvénic quasi-modes, normal modes produced to satisfy initial conditions, and resonant normal modes that grow in time. The latter two parts strongly depend on initial conditions; the resulting symmetry breaking leads to net-negative residual energy both in Alfvénic quasi-modes and ω = k ∥ V A = 0 modes. These modes have net-positive residual energy in the equivalent boundary value problem, suggesting that the initial value setup is a better match for solar wind turbulence.

Initial Boundary Value Problem for Partial Differential–Algebraic Equations With Parameter

ABSTRACTThe paper addresses an initial boundary value problem for a partial differential–algebraic equation involving a parameter. An integral condition with respect to the time derivative of the unknown function is provided as an additional condition to determine this parameter. The Dzhumabaev parameterization method is employed to solve the problem. The domain is subdivided, and functional parameters are defined as the values of the solution along the internal lines of the subdomains. This reformulates the original problem into an equivalent initial boundary value problem for a system of hyperbolic equations with parameters and associated functional relations. The paper develops algorithms to solve the problem, demonstrating their applicability. Furthermore, conditions for the existence and uniqueness of a solution to the initial boundary value problem, involving the partial differential–algebraic equation with a parameter and discrete memory, are established.

Variant of the Mathematical Theory of Non-Thin Multilayer Nonlinearly Elastic Plates of Symmetric Structures

A new version of the mathematical theory of non-thin multilayer nonlinearly elastic Kauderer plates of symmetric structures has been constructed. The components of the stress-strain state (SSS) and boundary conditions on the side surface of the plate are considered functions of three spatial coordinates. The variant is based on the development of components of the SSS of the plate along the transverse coordinate with the help of combinations of Legendre polynomials within each layer. The three-dimensional physically nonlinear boundary value problem for a multilayer non-thin plate is reduced to a two-dimensional one using the three-dimensional equations of the theory of elasticity and Reissner's variational principle. The main dependencies are obtained. Derived systems of high-order differential equations of equilibrium with partial derivatives and boundary conditions containing nonlinear terms from the SSS. The conjugation conditions at the boundary of adjacent layers (rigid connection) are exactly fulfilled for displacements, transverse tangential and normal stresses. The boundary conditions for stresses on the horizontal faces of the plate are also exactly fulfilled. The system of high-order differential equations of equilibrium is transformed into two systems that independently describe skew-symmetric and symmetric deformation relative to the median plane of the plate. In turn, each of these systems is transformed into differential equations that describe vortex boundary effects and equations that describe an internal SSS with a potential boundary layer-type boundary effect. The solution of systems of high-order equations by the developed method is reduced to the solution of second-order differential equations (Poisson and Helmholtz equations). General solutions of the system of differential equilibrium equations are obtained. The validity and accuracy of the variant of the mathematical theory is confirmed by comparisons with exact solutions of bending problems for linearly elastic plates with different mechanical and geometrical parameters. The newly constructed variant of the theory and the developed method provide a real opportunity to solve boundary problems in a spatial formulation with high accuracy for multilayer platens under various loads and boundary conditions on the lateral surface.

Resolving turbulence and drag over textured surfaces using texture-less simulations: the case of slip/no-slip textures

We study the effect of surface texture on an overlying turbulent flow for the case of textures made of an alternating slip/no-slip pattern, a common model for superhydrophobic surfaces, but also a particularly simple form of texture. For texture sizes $L^+ \gtrsim 25$ , we have previously reported that, even though the texture effectively imposes homogeneous slip boundary conditions on the overlying, background turbulence, this is not its sole effect. The effective conditions only produce an origin offset on the background turbulence, which remains otherwise smooth-wall-like. For actual textures, however, as their size increases from $L^+ \gtrsim 25$ the flow progressively departs from this smooth-wall-like regime, resulting in additional shear Reynolds stress and increased drag, in a non-homogeneous fashion that could not be reproduced by the effective boundary conditions. This paper focuses on the underlying physical mechanism of this phenomenon. We argue that it is caused by the nonlinear interaction of the texture-coherent flow, directly induced by the surface topology, and the background turbulence, as it acts directly on the latter and alters it. This does not occur at the boundary where effective conditions are imposed, but within the overlying flow itself, where the interaction acts as a forcing on the governing equations of the background turbulence, and takes the form of cross-advective terms between the latter and the texture-coherent flow. We show this by conducting simulations where we remove the texture and introduce additional, forcing terms in the Navier–Stokes equations, in addition to the equivalent homogeneous slip boundary conditions. The forcing terms capture the effect of the nonlinear interaction on the background turbulence without the need to resolve the surface texture. We show that, when the forcing terms are derived accounting for the amplitude modulation of the texture-coherent flow by the background turbulence, they quantitatively capture the changes in the flow up to texture sizes $L^+ \approx 70{-}100$ . This includes not just the roughness function but also the changes in the flow statistics and structure.

Analytical solution of surrounding rock stress and plastic zone of rectangular roadway under non-uniform stress field

The shape and extent of the plastic zone can effectively reflect the degree of deformation and failure of the surrounding rock. By applying complex variable functions and the theory of elasticity, a mechanical model for the surrounding rock of a rectangular roadway is established. The analytical solution for the stress in the surrounding rock of a rectangular roadway under a non-uniform stress field is derived, and the spatial stress distribution pattern in the surrounding rock of the rectangular roadway is obtained. Furthermore, using the Mohr-Coulomb criterion, the boundary equation of the plastic zone in the surrounding rock of a rectangular roadway is derived, and the morphological characteristics of the plastic zone are obtained. The results indicate that the plastic zones on both sides of the rectangular roadway are arc-shaped. The asymmetry of the stress environment has a significant impact on the plastic zones in the roof and floor of the rectangular roadway, causing the shape of the plastic zones to resemble a “petal” pattern. The lateral pressure coefficient and the height-to-width ratio have a significant impact on the stress distribution and the morphological characteristics of the plastic zone in the surrounding rock of the rectangular roadway. By validating the computational accuracy of the analytical solution through numerical simulations and conducting an engineering applicability analysis based on a coal mine roadway, it has been demonstrated that the analytical calculation method is both rational and possesses good engineering applicability. This provides reliable theoretical guidance for the design calculations and construction planning of similar roadway projects.

Research on energy transfer and vibration suppression of nonlinear energy sink with nonlinear damping

The nonlinear energy sink is a nonlinear shock absorber that provides targeted energy transfer and plays a crucial role in structural vibration suppression. In this paper, the energy transfer and dynamic characteristics of nonlinear damping nonlinear energy sink systems are analyzed. Firstly, the theoretical model of the nonlinear damping NES system is described, and secondly, the equations of the system in the free state are derived by using the complex variable average method and the multi-scale method to obtain the energy transfer efficiency equation of the system. Through the study of the slow invariant manifold and energy transfer efficiency of the system, the influence of each system parameter on the energy transfer efficiency is analyzed. Then the equations of the system under harmonic excitation are processed in the same method to obtain the boundary equations of the system, through which the saddle-node bifurcation diagram of the system is plotted to study the influence of each system parameter on the saddle-node bifurcation characteristics of the system. Finally, the slow-varying equation of the system under harmonic excitation is derived, and the influence of each system parameter on the frequency detuning parameter interval of the strongly modulated response is obtained using the phase trajectory and a one-dimensional mapping diagram of the system, and the time-mileage diagram and Poincare section are used to prove the damping effect of the strongly modulated response.

Thermal characterization of vertical interface by scanning photothermal radiometry.

In this work, scanning photothermal radiometry is used for imaging and to characterize a submicron crack. From the thermal images, the evolution of the crack is mapped in the space with micrometer resolution. A vertical contact interface at the steel-steel junction is used to represent a micro-crack with a thickness less than 0.5μm. The thermal quadrupole approach is used to model the heat transfer within the semi-infinite vertical crack. Then, using the phase mapping and that calculated from the model, the estimation of both the equivalent thermal boundary resistance of the interface and the average interface thickness was done.

Analyzing multi-mode and multi-position reliability of composite reinforced plate based on branch-bound method

To overcome the concurrency of multi-mode and multi-position failures of composite structures, this paper explores the application of the branch-bound method (BBM) to the multi-mode and multi-position reliability analysis and combines the Monte Carlo method to solve the multi-mode reliability. Firstly, "branch" is carried out by combining structural composition with loading state to identify all possible potential failure modes; "boundary" is carried out by combining structural characteristics with stress state to determine main failure modes and failure positions, thus establishing a multi-mode reliability analysis model. Secondly, by translating the unequal failure criterion to safe boundary equations, a framework for the reliability analysis of composite materials was established, and the Monte Carlo method was used to solve the reliability of multi-mode and multi-position failures. Finally, a reliability analysis case was used to validate the composite T-type reinforced plate under axial compressive load. The use of the BBM reduces the computational scale by 75%. The multi-mode reliability of the reinforced plate under axial compressive load is 0.998.

A Method of Boundary Equations for Nonlinear Poisson–Boltzmann Equation Arising in Biomolecular Systems

A Method of Boundary Equations for Nonlinear Poisson–Boltzmann Equation Arising in Biomolecular Systems

Radiation of a bunch in a waveguide having areas with deeply corrugated and smooth walls

We analyze radiation from a filamentary bunch that uniformly moves in a circular waveguide and crosses the boundary between two areas: the one has a corrugated wall and another has a smooth wall. It is assumed that the corrugation period is much less than the wavelengths under consideration. However, in difference with our previous works, the corrugation depth is of the same order as the wavelength. The influence of the corrugation on the electromagnetic field is investigated using the method of the equivalent boundary conditions. At first, we study a regular corrugated waveguide. Then two cases of the waveguides with boundaries between the corrugated and smooth regions is studied: first, when the bunch flies out of the corrugated part into the smooth one, and second, when the bunch flies into the corrugated part from the smooth one. We obtain and analyze the analytical expressions for the field components in both areas and reveal the main physical peculiarities of the bunch radiation.

Temperatures of AdS2 black holes and holography revisited

In a dilaton gravity model, we revisit the calculation of the temperature of an evaporating black hole that is initially formed by a shock wave, taking into account the quantum backreaction. Based on the holographic principle, along with the assumption of a boundary equation of motion, we show that the black hole energy is maintained for a while during the early stage of evaporation. Gradually, it decreases as time goes on and eventually vanishes. Thus, the Stefan–Boltzmann law tells us that the black hole temperature, defined by the emission rate of the black hole energy, starts from zero temperature and reaches a maximum value at a critical time, and finally vanishes. It is also shown that the maximum temperature of the evaporating black hole never exceeds the Hawking temperature of the eternal AdS2 black hole. We discuss physical implications of the initial zero temperature of the evaporating black hole.

Combining statistical analyses and GIS-based approach for modeling the sanitary boundary of drinking water wells in Yaoundé, Cameroon

In slums and urban areas with unplanned housing such as in the city of Yaounde, Cameroon, poor water quality and inadequate sanitation pose significant health risks. The absence of locally-defined sanitary boundaries, tailored to hydrogeological conditions, hinders effective zoning and land use planning, exacerbating environmental degradation and health hazards. In this study, the sanitary boundary between drinking water wells and sources of pollution in the city of Yaoundé was defined using statistical analysis techniques and Geographic Information Systems (GIS). The Groundwater Quality Index (GWQI) and certain significant parameters affecting water quality notably the transmissivity of the aquifer and well depth were used to establish the sanitary boundary equation which was then interpolated in a GIS environment to obtain the sanitary boundary map of the study area. The linear equation deduced from the significant factors was defined to map the health boundaries between wells and pollution sources for a nominal value of the GWQI (GWQI = 25). Of the 112 wells analysed, 37 % had an excellent GWQI, 16 % were good while the remaining 47 % were poor. Statistical analysis showed a strong correlation between the GWQI and significant factors of groundwater pollution, such as the distance between well and pit latrines (r = −0.753), the aquifer transmissivity of the formation (r = 0.671) and the depth of the wells (r = - 0.855) but no correlation with elevation (r = 0.017) and well age (r = 0.090). Linear regression analysis confirmed the association of the GWQI with the main factors of pollution (p ≤ 0.05). A coefficient of determination of R2 = 0.85 was obtained when validating the linear regression plot based on independent data between measured and predicted GWQI. The sanitary boundary map shows that the wells in our study area should be located between 39 m and 370 m, with an average value of 215 m. New regulations on the distance between well and pit latrines are essential to prevent groundwater pollution.

Identifying the dominant operating variables to evaluate the cascading failure potential in the power system by theory of mutual information

In this study, the potential of cascading failure as a result of transmission line outages through large power systems is evaluated. Transmission lines are more dispose of the outage resulting from different fault occurrences and extend the uncontrolled failures compared to other power system devices. In this case, power system dynamic security must evaluate the possibility of cascading failures using a power system control center (PSCC) in order to limit the spread of outages caused by line failures. For this issue, using the PSCC online information consisting of power system operating variables, it is possible to present a variables-based scheme to estimate power system cascading failures concerning fault events. In order to cover the issue, based on developing a set of dominant operating variables (DOVs), this paper presents an online scheme of identifying cascading failures caused by line outages in the presence of line operational diversities and communication restrictions. In this scenario, during each time interval, by evaluating the proposed DOVs, the line information consisting of dynamic securities and sensitivities to cascading failure are investigated which the lines with the potential of cascading failures are identified online. Proper DOVs are determined for this issue by studying a variety of mathematical formulations according to theory of mutual information (MI) and measuring the entropy among the variables. In a real-time environment, by using the proposed DOVs and substituting them through online boundary equations based on the Least Square Error (LSE) method, the potentials of power system cascading failures without acting any line outage are provided. The effectiveness of the suggested technique is evaluated using a typical IEEE-39 bus and IEEE-118 bus, and proper results with high accuracy are achieved by evaluating the system potential concerning large cascading failures.

Infrared radiation characteristics simulation of exhaust suppression type ships

An equivalent calculation method for the infrared-suppressed ship’s infrared radiation is carried out, which couples thermal characteristics of the ship surface and the exhaust system. Firstly, a refined physical model for the infrared suppressed exhaust system is proposed, which considers the processes of convective heat transfer and radiative heat transfer. The model generates the equivalent heat flow boundary conditions of the internal heat source through the numerical calculation of Computational Fluid Dynamics (CFD), and combines with the energy balance physical model. The energy balance model on the ship surface is developed, which considers external thermal environment conditions including itself radiation, solar radiation, atmospheric radiation, and reflective radiation among surface elements. The temperature calculation model for the ship with a jet exhaust system is formed. Subsequently, an exhaust-suppressed ship infrared radiation calculation model is formed based on the infrared radiation rendering of solid targets and the narrow spectral principle of exhaust plumes, which considers the spontaneous radiation and environmental radiation of ships. Finally, the infrared simulation results are compared and verified based on experimental data. The results indicate that the infrared radiation error on the ship’s typical parts is less than 30 %. Additionally, 90.1 % of the cabin walls using exhaust suppression systems have a temperature difference of less than 3 K under different operating conditions. It suggests that the exhaust suppression system is effective in reducing the temperature of the cabin wall. The calculation method for ship infrared characteristics constructed in this paper has reasonable results, and it can be used for infrared characteristic analysis of exhaust suppression ships.

Construction and kinematic performance analysis of a suspension support for wind tunnel tests of spinning projectiles based on wire-driven parallel robot with kinematic redundancy

This paper presents a novel suspension support tailored for wind tunnel tests of spinning projectiles based on Wire-Driven Parallel Robot (WDPR), uniquely characterized by an SPM (Spinning Projectile Model)-centered mobile platform. First, an SPM-centered mobile platform, featuring two redundant and another unconstrained Degree of Freedom (DOF), and its suspension support mechanism are designed together, collectively constructing a WDPR endowed with kinematic redundancy. Afterward, the kinematics of the mechanism, boundary equations for the redundant DOFs, and relevant kinematic performance indices are then proposed and formulated. The results from both prototype experiments and numerical assessments are presented. The capability of the support mechanism to replicate the complex coupled motions of the SPM is verified by the experimental results, while the proposed kinematics and boundary equations are also validated. Furthermore, it is revealed by numerical assessments that the redundant DOFs of the mobile platform exert a minimal impact on the kinematic performance of the suspension support. Finally, the optimal global attitude performance is obtained when these DOFs are set to zero if they are restricted to constants. However, local attitude performance can be further improved by the variable values.

Stagnation point slip flow of Al2O3/γ-Al2O3 nanofluids subject to Lorentz force and nonlinear thermal radiation over a stretching sheet

The flow analysis of Al 2 O 3 and γ ‐Al 2 O 3 nanofluids near a stagnating point over a stretching sheet having velocity slip at the solid–liquid interface is presented. There is a quadratic surface temperature and a linear deformation acting on the wall boundary. Due to the uniformly applied transverse magnetic flux, the Lorentz force has an impact on the flow. The effect of nonlinear TR is integrated into the heat equation. Further, water H2O and EG C2H6O2 are base fluids. The governing boundary equations are translated into dimensionless ODEs using relevant similarity variables and solved numerically using the Runge-Kutta algorithm scheme (fourth-order) and the shooting algorithm. The impacts of the parameters, that is, velocity ratio ε , velocity slip δ , nonlinear TR Rd, the temperature ratio parameter θ w , and magnetohydrodynamics M are observed for the velocity h 2 ≤ y ≤ 0 and temperature θ ( η ) fields graphically. The results of local skin friction and local Nusselt number are shown in the table. A comparative analysis of the present results and the results of previous studies has also been made in tabular form. The research indicates that water as a base fluid gives better thermal conductivity than ethylene glycol. The consequence of TR enhances the heat transfer rate of both Al 2 O 3 / γ ‐Al 2 O 3 with H2O and C2H6O2.

Discovering physics-informed neural networks model for solving partial differential equations through evolutionary computation

In recent years, the researches about solving partial differential equations (PDEs) based on artificial neural network have attracted considerable attention. In these researches, the neural network models are usually designed depend on human experience or trial and error. Despite the emergence of several model searching methods, these methods primarily concentrate on optimizing the hyperparameters of fully connected neural network model based on the framework of physics-informed neural networks (PINNs), and the corresponding search spaces are relatively restricted, thereby limiting the exploration of superior models. This article proposes an evolutionary computation method aimed at discovering the PINNs model with higher approximation accuracy and faster convergence rate. In addition to searching the numbers of layers and neurons per hidden layer, this method concurrently explores the optimal shortcut connections between the layers and the novel parametric activation functions expressed by the binary trees. In evolution, the strategy about dynamic population size and training epochs (DPSTE) is adopted, which significantly increases the number of models to be explored and facilitates the discovery of models with fast convergence rate. In experiments, the performance of different models that are searched through Bayesian optimization, random search and evolution is compared in solving Klein–Gordon, Burgers, and Lamé equations. The experimental results affirm that the models discovered by the proposed evolutionary computation method generally exhibit superior approximation accuracy and convergence rate, and these models also show commendable generalization performance with respect to the source term, initial and boundary conditions, equation coefficient and computational domain. The corresponding code is available at https://github.com/MathBon/Discover-PINNs-Model.

Electromagnetic Characterization of Permanent Magnet Eddy Current Structures Based on Backplane Distance Adjustment

To address the problem of problematic spray design inside mining anchor-digging equipment, a switching seal using a permanent magnet eddy current drive is initially presented here. The layer model of the permanent magnet eddy current structure is established, the subdomain analysis model is introduced, the permanent magnet eddy current structure is divided into six regions along the axial direction, and the boundary equations are established at the interfaces of each region. The vector magnetic potential equations in each region are deduced, along with the electromagnetic torque and axial force equations. The computational results are compared and analyzed with the results of finite element simulation, verifying the accuracy of the theoretical model. The design of experiments is used to verify the feasibility of the switching seal using the permanent magnet eddy current structure.