First-order Boundary Condition Research Articles

Walkin’ Robin: Walk on Stars with Robin Boundary Conditions

Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes the walk on stars (WoSt) algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt produces bounded-variance estimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as the walk on boundary algorithm. We also develop bidirectional and boundary value caching strategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.

Modelling internal erosion using 2D smoothed particle hydrodynamics (SPH)

This paper presents a stabilised multi-phase smoothed particle hydrodynamics (SPH) model applicable to seepage-induced internal erosion and the resulting deformation in soils. Based on the continuum mixture theory, a new single-layer SPH model in the u-w-p formulation is derived for the mathematical description of the erodible porous material. A new modified constitutive model is formulated to account for the influence of erosion on the mechanical behaviour of soils. Extensions of a first-order consistent boundary condition as well as the diffusion algorithm for hydro-mechanical coupling in SPH are also proposed to accommodate the analysis of erosion-transport process. A novel viscous dissipation term is designed to mitigate the numerical instability commonly encountered in coupled problems. In contrast to existing stabilisation terms in the SPH literature, which operate on both the bulk component and shear component, this new stabilisation term offers the possibility to increase the dissipation of unphysical oscillation due to the shear deformation. The proposed method is validated through a series of benchmark tests. The results demonstrate the effectiveness of the proposed approach in addressing the coupled problem in porous materials involving multi-phase flow, phase interaction/transfer, and large deformation with enhanced accuracy, stability, and robustness.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

Viscous dissipation effects on slip flow heat transfer in rhombic microchannels

This study aimed to numerically investigate the viscous dissipation effect on forced convection in rhombic microchannels for gases in a slip flow regime. The numerical analysis was carried out by assuming a 2D steady-state flow. The solution of governing equations was obtained by adopting the finite element method and assuming that the fluid is Newtonian, with constant thermophysical properties, and in a fully developed laminar flow regime. The solution of the momentum equation is obtained by considering a first-order boundary condition, while the solution of the energy balance equation is obtained by assuming a constant wall heat flux (H2 boundary condition) and taking into account the wall temperature jump. The validation of the numerical model was carried out using the data available in the scientific literature. The numerical outcomes obtained for several values of the acute angle of the rhombus, the Knudsen number (i.e., rarefaction effects), and the Brinkman number (i.e., viscous dissipation effects) reveal that viscous forces play an opposite role with respect to rarefaction and significantly affect the convective heat transfer coefficient, especially for low Knudsen numbers and for high values of the acute angle. In particular, it was observed that Nu is significantly affected by the Brinkman number for acute angles higher than 50°.

Solving multiscale elliptic problems by sparse radial basis function neural networks

Machine learning has been successfully applied to various fields of scientific computing in recent years. For problems with multiscale features such as flows in porous media and mechanical properties of composite materials, however, it remains difficult for machine learning methods to resolve such multiscale features, especially when the small scale information is present. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem into a first-order system and employ multiple radial basis function neural networks (RBFNNs) to approximate unknown functions in the system. To avoid the overfitting and improve the performance of RBFNN, an additional regularization is introduced in the loss function. Thus the loss function contains two parts: the L2 loss for the residual of the first-order system and boundary conditions, and the ℓ1 regularization term for the weights of radial basis functions (RBFs). An algorithm for optimizing the specific loss function is introduced to accelerate the training process. The accuracy and effectiveness of the proposed method are demonstrated through a collection of multiscale problems with scale separation, discontinuity and multiple scales from one to three dimensions. Notably, the ℓ1 regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like O(ε−nτ), where ε is the smallest scale, n is the dimensionality, and τ is typically smaller than 1. It is worth mentioning that the proposed method not only has the numerical convergence and thus provides a reliable numerical solution in three dimensions when a classical method is typically not affordable, but also outperforms most other available machine learning methods in terms of accuracy and robustness. Executable files are available at https://github.com/wangzhiwensuda/SRBFNN-for-multiscale-elliptic-problems. The program was implemented in PyTorch and was run on Linux.

Flow of viscous reacting fluid through a porous filler in a flat channel

The flow and heat transfer of a viscous chemically reacting liquid in a flat channel during the molding of composite products are studied. The main assumptions in the task definition were made on the basis of the high viscosity of the liquid and its low thermal diffusivity. The Brinkman equation is used as a motion equation. The flow is accompanied by a chemical reaction, resulting in a sharp increase in viscosity. The viscosity is considered to depend on the temperature and the degree of conversion. This, in turn, led to the inclusion of the kinetic equation of a chemical reaction in the mathematical model. The energy equation is denoted using a single-temperature model and includes dissipative heat emissions. The problem is solved for temperature first-order boundary conditions. The calculations are given for the Nusselt number distribution and the velocity profile transformation. The task was solved numerically by the finite difference method using iterations.

Modeling Electric Vehicle’s Battery Module using Computational Homogenization Approach

Numerical simulations of heterogeneous structures like battery modules of electric vehicles are challenging due to the various length scales involved in it. Even with the latest computing technology, it is impossible to simulate the crash scene of the full vehicle resolving all length scales. Such hurdles have prevented manufacturers to understand the mechanical response of battery packs in vehicle crash scenarios. In this work, the problem of multiple length scales was solved using the RVE technique based on homogenization theory. An appropriate representative volume element was identified, and a 3D FE model was developed. Classical first-order boundary conditions were used in this research work. The RVE was subjected to several macroscopic deformations, and its response was obtained. The homogenized material properties were computed from the obtained responses, and a material model available in LS-Dyna’s material library was selected and calibrated to describe the nonlinear multiaxial behavior of the homogenized battery module at the macroscale. For validation, the USABC Crush and Drop Tests were simulated for the detailed and homogenized battery modules. The main output of this research is a robust and computationally efficient tool enabling satisfactory integration of a battery pack model to the vehicle for crash simulations, eliminating the need to simulate micro details at macro length scales. This approach significantly reduced the computational cost. For example, for a Drop Test simulation, the homogenized model reduced the simulation time from 40 hours (detailed model) to about 3 minutes, while maintaining a high precision ( R 2 = 0.9871 ) in predicting the load-displacement response. The system level modeling will enable the stakeholders to perform efficient optimization and safety evaluation for full-scale crashworthiness of electric vehicles.

3D MCSEM modeling based on domain-decomposition finite-element method

Domain decomposition approach is effective in converting large modeling problems into many small ones, so that the parallel computation can be easily facilitated and the memory consumption can be largely reduced. In this paper, we developed a domain-decomposition method for 3D time-domain marine controlled-source (MCSEM) modeling based on Dual-Primal Finite-Element Tearing and Interconnecting technique (FETI-DP). We use tetrahedral grids to do the spatial discretization first and then partition the mesh into subdomains. In each subdomain, the finite-element equations are independently constructed using the first-order vector shape functions and boundary conditions. To build the interconnections among these subdomains, we establish the interface equation for the Lagrange multipliers that are used as the boundary conditions for the subdomains. After solving the equations, the electric field for all subdomains can be obtained from the Lagrange multipliers. For the time discretization, we use unconditionally stable backward Euler scheme. We validate our algorithm by comparing with a semi-analytical solutions for a uniform half-space model under the ocean. Numerical experiments show that compared to the conventional finite-element method, our domain-decomposed method can save large amount of memory, render it possible to run 3D MCSEM modeling in small computational equipment.

Who is likely to hide knowledge after peer ostracism? An exchange-based perspective of contact quality and need to belong

PurposeThe purpose of this study is to uncover how peer ostracism (POS) elicits knowledge hiding directed towards ostracizing peers through the intervening role of peer contact quality (PCQ). Moreover, the authors aim to highlight the role of the need to belong (NTB) as a first-order boundary condition in direct and indirect hypothesized paths.Design/methodology/approachThe research opted for a three-wave time-lagged survey design. The data were obtained from the 234 teaching and non-teaching employees working in Higher Educational Sector in Pakistan through random sampling. Mediation and moderated mediation analysis was done by using PROCESS Models 4 and 7.FindingsThe results embraced the mediation, moderation and moderated mediation hypotheses. It was noted that POS creates negative exchange relationships. As a result, the ostracized employees withhold knowledge from the predating peer. NTB served as a buffering agent between POS and PCQ, as well as, in the indirect POS, PCQ and peer-directed knowledge hiding relationship.Practical implicationsThis research serves as a guideline for management and faculty of Higher Educational Institutions for minimization of POS to promote effective collegial contact quality and curb knowledge hiding.Originality/valueAlthough the research in workplace ostracism and knowledge hiding is not new, yet how this association emerges from the viewpoint of peers is not known. This study has added to the literature by answering who is more likely to reciprocate ostracism from peers by having poor quality contact and directing knowledge hiding towards the predator. By this, the authors have added to the limited stream of moderated mediation mechanisms underlying ostracism and knowledge hiding behaviour. In addition, the authors have drawn attention to the importance of peer relationships in higher educational settings.

Computational methods for pore-scale simulation of rarefied gas flow

Direct simulation at the pore-scale is crucial to unravel rarefaction effect on gas transport in tight porous media. To satisfy the dual demands on modeling accuracy and computation effort, an appropriate method must be chosen. This work, therefore, evaluates four numerical methods for pore-scale rarefied gas flows in a two-dimensional (2D) model porous media over a wide range of rarefaction. These methods are the incompressible Navier-Stokes equation with the first-order velocity-slip boundary condition, and three gas-kinetic solvers i.e. a finite-difference (FD) iterative solver for the linearized Bhatnagar, Gross and Crook (BGK) model kinetic equation, a finite-volume (FV) solver for the non-linear Shakhov model, and an open-source direct simulation Monte Carlos (DSMC) solver. The benchmark cases cover the characteristic Knudsen number ranging from 0.0231 to 4.62 while the characteristic Reynolds numbers are kept to be less than 1.0. All the solvers are developed in OpenFOAM, except the FD solver, allowing us to investigate the effect of local grid refinements using the automatic Cartesian grid generator in OpenFOAM. The flow fields and the apparent permeabilities predicted by all the solvers have been compared in detail. Besides, the computational time of these solvers is measured and analyzed to demonstrate the relative cost of the three kinetic solvers. It is found that the FD solver is the most efficient one and gives accurate results over the whole range of Knudsen number. Finally, this study also evaluates the feasibility of the recently developed discrete unified gas-kinetic scheme (DUGKS), the algorithm in the aforementioned non-linear Shakhov model equation solver, for pore-scale rarefied gas flow. It is found the results predicted by this solver agree well with the other two kinetic solvers.

ON A METHOD FOR CALCULATING HEAT TRANSFER IN A MOVING FLUID TAKING INTO ACCOUNT ENERGY DISSIPATION

An approximate analytical solution to a heat transfer problem for a moving fluid in a cylindrical channel is obtained using an additional new function and additional boundary conditions in the heat balance integral method and taking into account energy dissipation under a first-order boundary condition that varies along the longitudinal coordinate. The use of an additional new function that determines the temperature change along the longitudinal variable in the center of the channel makes it possible to reduce the solution of the partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are found in such a way that their satisfaction for the new solution is equivalent to the satisfaction of the differential equation at boundary points.

A real time methodology of cluster-system theory-based reliability estimation using k-means clustering

With rapidly increase in the design and application of complex system engineering, life and reliability analysis methods for these engineering have received much attention. A novel real time reliability analysis methodology is proposed based on the cluster-system theory and k-means clustering. Firstly, the system consisting of three or more identical or similar sub-systems facing the same load or task is defined as cluster-system. By analyzing the key performance parameters, sub-systems with similar performance are divided into a family-system, so that each sub-system can be used as reference sample for other sub-systems in the identical family-system. The cubic spline interpolation method under first-order boundary conditions is used to fit the average variation of key performance parameters of family-systems. The residual life of family-system can be obtained by determining the threshold of key performance parameters of sub-system through experiments or experience. Then reliability of the whole system is estimated by the contribution of each sub-system in cluster-system to solve the problem that there is no fault data and reference samples in the reliability analysis of complex system. Application in the Five-hundred-meter Aperture Spherical Telescope (FAST) demonstrates the effectiveness of the proposed method, compared to traditional reliability analysis methods based on experimental statistics.

Development and Application of a Pre-Corrected Fast Fourier Transform Accelerated Multi-Layer Boundary Element Method for the Simulation of Shallow Water Acoustic Propagation

Because of the complexities associated with the domain geometry and environments, accurate prediction of acoustics propagation and scattering in realistic shallow water environments by direct numerical simulation is challenging. Based on the pre-corrected Fast Fourier Transform (PFFT) method, we accelerated the classical boundary element method (BEM) to predict the acoustic propagation in a multi-layer shallow water environment. The classical boundary element method formulate the acoustics propagation problem as a linear equation system in the form of [A]{x}={b}, where [A] is an N×N dense matrix composed of influence coefficients. Solving such linear equation system requires O(N2/N3) computational cost for iterative/direct methods. The developed method, PFFT-BEM, can effectively reduce the computational efforts for direct numerical simulations from O(N2~3) to O(Nlog N), where N is the total number of boundary unknowns. To numerically simulate the sound propagation in a shallow water environment, we applied the first-order non-reflecting boundary condition in the truncated numerical domain boundary to eliminate the errors due to reflected waves. Multi-layer coupled formulation was used to include the environment inhomogeneity in PFFT-BEM. Through multiple convergence tests on the number of layers and elements, we validated and quantified the accuracy of PFFT-BEM. To demonstrate the usefulness and capability of the developed PFFT-BEM, we simulated three-dimensional (3D) underwater sound propagation through 3D geometries to check the efficacy of the established classical method: the 3D Parabolic equation model. Finally, PFFT-BEM was employed to simulate sound propagation through a complex multi-layer shallow water environment with internal waves. The “3D+T” results obtained by PFFT-BEM compared well with the physical test, thereby proving the capability and correctness of this method.

The study of the heat transfer process in bodies with internal heat sources of variable power

An analytical solution to an unsteady heat conductivity problem based on the method of using a new function for a plate under first-order boundary conditions with time-varying internal heat sources was obtained. Obtaining an exact analytical solution for such problems has encountered serious mathematical difficulties. However, analytical solutions have significant advantages compared to numerical ones. For example, solutions obtained in an analytical form make it possible to perform the parametric analysis of the system under study, parametric identification, programming of measuring devices; controlling the production process, etc. Therefore, approximate analytical methods have been widely used. So, this paper is devoted to the development of such a method. The solutions obtained have a simple form of algebraic polynomials without special functions. This allows doing the research in the fields of isotherms and determining their velocities.

Thermo-Elastic Creep Analysis and Life Assessment of Thick Truncated Conical Shells with Variable Thickness

A semi-analytical method is presented to investigate time-dependent thermo-elastic creep behavior and life assessment of thick truncated conical shells with variable thickness subjected to internal pressure and thermal load. Based on the first-order shear deformation theory (FSDT), equilibrium equations and boundary conditions are derived using the minimum total potential energy principle. To the best of the researcher’s knowledge, in previous studies, thermo-elastic creep analysis of conical shell with variable thickness based on the FSDT has not been investigated. Norton’s law is assumed as the material creep constitutive model. The multilayered method is proposed to solve the resulting equations, which yields an accurate solution. Subsequently, the stresses at different creep times can be obtained by means of an iterative approach. Using Robinson’s linear life fraction damage rule, the creep damages of conical shells are determined and Larson–Miller parameter (LMP) is employed for assessing the remaining life. The results of the proposed approach are validated with those of the finite element method (FEM) and good agreement was found. The results indicate that the present analysis is accurate and computationally efficient.

Heat Transfer in Porous Microchannels with Second-Order Slipping Boundary Conditions

The paper presents results of a study of forced convection in a vertical flat and circular microchannels incorporating porous medium under slip boundary conditions of the first and second orders. The problem was solved analytically and compared with numerical simulations based on the lattice Boltzmann method. Effects of porosity and slip velocity on velocity and temperature profiles were investigated. Behavior of the normalized Nusselt number as a function of the Knudsen and Prandtl numbers, as well as the parameter M characterizing porosity of the medium in the microchannel, was also elucidated. Computations indicated that a decrease in porosity (an increase in the parameter M) causes a decrease in the velocity and temperature jumps on the wall, which contributes to the increase in the Nusselt number. Considering the effect of the second-order slip boundary conditions leads to the decrease in the velocity jump on the wall, when the coefficient A2 (this coefficient considers the second-order slip boundary conditions) changes from negative to positive values. The heat transfer rate at high Prandtl numbers increases with the increasing Knudsen number, because of the improved thermal interaction of the flow with the channel wall. Given the second-order boundary conditions, the effect of parameter A2 was not observed at small Prandtl numbers (Pr ≤ 1). For A2 0 the normalized Nusselt number decreases in comparison with the case of A2 = 0 (the first-order boundary conditions). The results by the analytical solution and the lattice Boltzmann method are in a good agreement with each other for A2 ≥ 0, with the differences ≤ 1%. For A2 < 0, the differences between two models are more noticeable. For A2 ≥ 0, predictions by the analytical model lie higher than the numerical results, whereas for A2 < 0, numerical simulations exceed the analytical solution.

Modeling of Knudsen Layer Effects in the Micro-Scale Backward-Facing Step in the Slip Flow Regime

The effect of the Knudsen layer in the thermal micro-scale gas flows has been investigated. The effective mean free path model has been implemented in the open source computational fluid dynamics (CFD) code, to extend its applicability up to slip and early transition flow regime. The conventional Navier-Stokes constitutive relations and the first-order non-equilibrium boundary conditions are modified based on the effective mean free path, which depends on the distance from the solid surface. The predictive capability of the standard ‘Maxwell velocity slip—Smoluchwoski temperature jump’ and hybrid boundary conditions ‘Langmuir Maxwell velocity slip—Langmuir Smoluchwoski temperature jump’ in conjunction with the Knudsen layer formulation has been evaluated in the present work. Simulations are carried out over a nano-/micro-scale backward facing step geometry in which flow experiences adverse pressure gradient, separation and re-attachment. Results are validated against the direct simulation Monte Carlo (DSMC) data, and have shown significant improvement over the existing CFD solvers. Non-equilibrium effects on the velocity and temperature of gas on the surface of the backward facing step channel are studied by varying the flow Knudsen number, inlet flow temperature, and wall temperature. Results show that the modified solver with hybrid Langmuir based boundary conditions gives the best predictions when the Knudsen layer is incorporated, and the standard Maxwell-Smoluchowski can accurately capture momentum and the thermal Knudsen layer when the temperature of the wall is higher than the fluid flow.

A stochastic Stefan-type problem under first-order boundary conditions

Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in economics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and nonlinear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which introduces additional challenges in the analysis. Working on the distribution space, the Itô–Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochastic evolution equations and stochastic maximal $L^{p}$-regularity, we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary interaction even when the coefficients of the original problem have linear growths.

On the Spectrum of Deflated Matrices with Applications to the Deflated Shifted Laplace Preconditioner for the Helmholtz Equation

The deflation technique for accelerating Krylov subspace iterative methods for the solution of linear systems has long been well established. The first landmark papers of Nicolaides and Dostál date back to the late 1980s, where deflation is used for Hermitian positive definite (HPD) linear systems. In the last decade deflation has been used and analyzed in combination with domain decomposition and multigrid methods, leading to very efficient algorithms. Examples are the multilevel Krylov methods introduced by Erlangga and Nabben, and the works of Sheikh et al. in which multilevel deflation techniques are presented. Although these algorithms work very well in practice for non-Hermitian problems, not many theoretical results are yet known for this class of problems. We study here the general case of deflation operators in arbitrary inner products, and give inclusion regions for the spectrum of an arbitrary deflated matrix based on the field of values. The new inclusion regions generalize previous results for HPD matrices. Moreover, we give a bound based on the field of values for the norms of the residuals of deflated GMRES. We apply our results to linear systems arising from the Helmholtz equation, focusing on the combination of the complex shifted Laplace (CSL) preconditioner with a two-level deflation preconditioner, which is the basis of a multilevel method introduced by Erlangga and Nabben. The analysis here only covers the case of exact inversion of the CSL preconditioner. It has been observed in numerical experiments that the eigenvalues of the deflated CSL-preconditioned system lie on the exact same disks as the CSL-preconditioned linear systems and are shifted away from zero. Here we are able to prove these surprising results for a class of problems with Dirichlet and first-order Sommerfeld boundary conditions. Our results help to explain the high performance of multilevel Krylov methods based on deflation for the Helmholtz equation.