Different Types Of Boundary Conditions Research Articles

Perfectly Matched Layers and Characteristic Boundaries in Lattice Boltzmann: Accuracy vs Cost

Artificial boundary conditions (BCs) play a ubiquitous role in numerical simulations of transport phenomena in several diverse fields, such as fluid dynamics, electromagnetism, acoustics, geophysics, and many more. They are essential for accurately capturing the behavior of physical systems whenever the simulation domain is truncated for computational efficiency purposes. Ideally, an artificial BC would allow relevant information to enter or leave the computational domain without introducing artifacts or unphysical effects. Boundary conditions designed to control spurious wave reflections are referred to as nonreflective boundary conditions (NRBCs). Another approach is given by the perfectly matched layers (PMLs), in which the computational domain is extended with multiple dampening layers, where outgoing waves are absorbed exponentially in time. In this work, the definition of PML is revised in the context of the lattice Boltzmann method. The impact of adopting different types of BCs at the edge of the dampening zone is evaluated and compared, in terms of both accuracy and computational costs. It is shown that for sufficiently large buffer zones, PMLs allow stable and accurate simulations even when using a simple zeroth-order extrapolation BC. Moreover, employing PMLs in combination with NRBCs potentially offers significant gains in accuracy at a modest computational overhead, provided the parameters of the BC are properly tuned to match the properties of the underlying fluid flow.

An assessment of the impact of boundary conditions in dynamical downscaling techniques for fetch-limited waves

ABSTRACT The handling of boundary conditions (BCs) is a key issue for the development of dynamical downscaling techniques in which nearshore wave models require the definition of the sea conditions at the offshore boundaries. This study addresses the implications of using simplified BCs represented by bulk or partitioned wave parameters with respect to using the complete information from directional wave spectra. As a large-scale model, we used WAVEWATCH III (WW3) running over the Mediterranean Sea providing the wave BCs for the small-scale SWAN model. One year of measured directional wave spectra from three deepwater buoys allowed the assessment of the WW3 model accuracy. The nested SWAN model simulations propagated the wave field over three nearshore areas. The accuracy of SWAN simulations was assessed through the comparison of computed results with wave measurements from nearshore buoys. Being the wave conditions dominated by unimodal seas, modest differences of error metrics are found between the datasets forced by different types of BCs. Conversely, focusing the attention on multimodal seas, an accurate definition of wave BCs achieved by the use of the directional wave spectra or, in alternative, by the reconstruction of wave spectra from partitioned wave parameters becomes crucial for nearshore wave predictions.

Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Diffusive wave model in a finite length channel with a concentrated lateral inflow subject to different types of boundary conditions

The diffusive wave model is one of the simplified forms of Saint-Venant equations, and it is often used instead of the full model. In this paper, we present an analytical solution for the linearized diffusive wave model represented by a simultaneous system of two first-order partial differential equations focused on spatial variation of a lateral inflow in a finite channel. A concentrated lateral inflow from a small-width tributary is considered through the Dirac delta function. We use the Laplace transform method to solve these equations analytically. Two types of upstream boundaries are considered here in the form of a flow-discharge hydrograph and a flow-depth hydrograph, while keeping a flow-depth hydrograph as the downstream boundary. Using unit-step responses of the lateral inflow, the effect of different boundaries on the flow-depth responses and the flow-discharge responses is studied for different values of the Peclet number (Pe). The flow depth is observed to be more sensitive to the downstream boundary and other parameters used in this work. Consideration of the flow depth as the upstream boundary reflects the effect of all the parameters on the unit-step responses presented. These responses are compared with the available semi-infinite channel responses, which are found to be an inappropriate substitute for the finite channel responses for Pe<5 which implies that the downstream boundary cannot be ignored for these cases. However, for the case Pe>5, although the semi-infinite channel responses are found to satisfactorily estimate the discharge along the entire channel, they can approximate the flow depth at the locations closer to the upstream boundary only.

Strong solutions to the 3D full compressible magnetohydrodynamic flows

In this paper, we deal with the strong solutions to the full compressible magnetohydrodynamic flows in a domain Ω⊆R3. We prove the local existence and unique of strong solution under three different types of boundary conditions provided that the initial data satisfies a natural compatibility condition. The initial density of such a strong solution is allowed to contain vacuum states. Compared to the classical compressible MHD, the key issue in the limit passage is to obtain the a priori estimates of the absolute temperature, some new estimate ideas are introduced to estimate these results.

Nonlinear deformations of size-dependent porous functionally graded plates in a temperature field

In this paper, the Variational Iteration Method (VIM) or Extended Kantorovich Method (EKM) is formulated for the first time for flexible porous functionally graded (PFGM) plates with different boundary conditions and geometric nonlinearity according to the theory of Theodore von Karman. Its accuracy and efficiency are demonstrated. The plate is subjected to a uniform transversal load and temperature field. The displacement field of the plate is approximated based on the classical plate theory (CTP) or Kirchhoff's plate theory. The governing equations are derived using Hamilton's principle. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. The material properties vary with thickness and are temperature dependent. Four porosity distribution patterns are considered in this study. Several examples are solved to demonstrate the proposed algorithm. The results obtained are compared with solutions obtained by the Bubnov-Galerkin Method (BGM) in higher approximations, the Finite Difference Method (FDM) of second order accuracy, as well as with results obtained by the Finite Element Method (FEM) of other authors. The results include an analysis of the effect of size dependent parameters, porosity type pattern, porosity index, functionally graded index, temperature field and different types of boundary conditions on the stress–strain state and bending deflection of plates.

A new C0 continuous refined zigzag {1,2} finite element formulation for flexural and free vibration analyses of laminated composite beams

This study presents a novel C0 continuous refined zigzag finite element formulation {1,2}, namely RZE{1,2}, for the bending and free vibration analyses of laminated composite beams. The Refined Zigzag Theory (RZT) effectively combines accuracy and computational efficiency, making it a robust approach for thin and thick laminated composite structures. The RZT eliminates the need for shear correction factors, thereby enhancing the overall streamline of the analysis process. The present RZE{1,2} formulation takes into account the transverse stretching by introducing quadratic through-thickness variations of deflection components. The governing equations of the RZT are derived by means of Hamilton’s principle. The through-the-thickness variations of the transverse shear and normal stresses are calculated by integrating the stress equilibrium equations in a post-processing step. Therefore, the Peridynamic Least Squares Minimization (PDLSM) approach is utilized to obtain precise derivatives of axial stresses in the stress equilibrium equations. A new finite element formulation is built up with 3-nodes with a total of 15 DOF. In order to study the influence of transverse stretching, different types of boundary conditions and material variations are applied for laminated composite beams for the bending and free vibration analyses. The distribution of displacements as well as the axial and transverse stresses of the thick beams are extensively examined. The outcomes of RZT are in good agreement with the reference solutions in the literature. The findings of the present study reveal that the presence of the transverse stretching produces more realistic predictions for thick beams where the shear deformations are influential.

On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

<abstract><p>In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.</p></abstract>

Stress-strain state of a porous flexible rectangular FGM size-dependent plate subjected to different types of transverse loading: Analysis and numerical solution using several alternative methods

In this study, a mathematical model of flexible porous functionally graded (PFG) Kirchhoff size-dependent plates on a rectangular plane is constructed. Hamilton's principle provides new size-dependent governing equations, taking into account geometrical nonlinearity according to the model of Theodor von Kármán, as well as boundary conditions and initial conditions for the displacement of the plates. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. Numerical investigations analysing the stress-strain state of flexible Kirchhoff functionally graded size-dependent porous plates on a rectangular plane are based on the application of the Variational Iteration Method (VIM) or the Extended Kantorovich Method (EKM). The efficiency and high accuracy of the VIM method are demonstrated. The validity and reliability of the solutions obtained by VIM are discussed and compared with solutions obtained by other methods, such as the BGM in higher approximations, FDM of second-order accuracy and the FEM. Additionally, the solutions are compared with results obtained by other authors in the study of porous functionally graded macro, micro, and nanoplates. The solutions obtained are considered accurate. The results include an analysis of the influence of size-dependent parameters, type of material porosity, porosity index, functionally graded index, and different types of boundary conditions on the stress-strain state of plates under the action of different types of transverse loading.

A highly accurate indirect boundary integral equation solution for three dimensional elastic scattering problem

In this paper, we propose a numerical formula for three dimensional elastic wave scattering problem. Different from the classical boundary integral equation method, the three dimensional elastic scattering problem is transformed into a coupled boundary value problem including a scalar Helmholtz equation and a vector Helmholtz equation by the decomposition of the displacement. The coupled boundary value problem can be solved by the boundary integral equations based on the Helmholtz equation. A denseness result is given to support this algorithm. A comparative study with the classical method of fundamental solutions getting from the boundary integral equation method is given to check the accuracy with different types of boundary conditions. From the numerical results, the present method can give more accurate results than the classical method of fundamental solutions method.

Stability analysis of a vertical excavation reinforced with horizontal nails using three-dimensional finite elements limit analysis

Three-dimensional stability analysis of a vertical excavation, reinforced with grouted horizontal nails, has been carried out by using the three dimensional lower and upper bounds based finite elements limit analysis (FELA) using Optum G3. The nail along with the surrounding grouted cementitious material is assumed to behave as a single monolithic cylindrical block which was assumed to be either rigid or an elasto-plastic material following the Von-Mises yield criterion. The excavation is assumed to fail on account of either an interface shear slippage between nails and surrounding soil mass or on an account of axial tensile rupture of the composite nail material for a few typical cases as well. The soil unit weight has been varied to induce the ultimate shear failure of the soil nailed excavation. The unit weight of the soil mass at which the failure of the nailed wall occurs is termed as γcr. Two different types of boundary conditions along the base of the excavation have been considered. In the first case, the in-situ soil mass behind the excavation is extended below the base as well (Case 1) and in the second case, the excavation is assumed to simply lie on a hard stratum (Case 2). The results have been provided in the form of a non-dimensional stability number N=γcrHc as a function of soil friction angle φ, Sv/H, SH/H, d/H, L/H and m; where H refers to height of the wall, d is diameter of the grouted nail, Sv and SH imply centre to centre vertical and horizontal spacing of the nails, and m is the mobilization factor between grouted nails and adjoining soil mass. The results from the analysis have been found to compare well with the data reported in literature. All the results have been presented in the form of non-dimensional stability charts, which will be useful for design for any given values of height H and cohesion c. From the observed failure mechanisms, an analysis has also been carried out to determine the optimized lengths of nails. The optimized lengths of nails for both Cases 1 and 2 have been also provided. The type of failure mechanism, base or face failure, dictates the optimized lengths of the nails.

A 2.5D hybrid SBM-MFS methodology for the evaluation of free-field vibrations induced by underground railway infrastructures

In this paper, a new meshless methodology to predict underground railway-induced vibrations is presented. In contrast to previous approaches, the proposed method uses meshless methods to model both the railway tunnel structure and the wave propagation in the surrounding soil. The method has been evaluated in the framework of an example of railway-induced vibrations using a shell with a square cross-section, embedded in a homogeneous full-space. The method is verified against an analytical solution considering different types of boundary conditions and compared with the 2.5D FEM-SBM and the 2.5D FEM-BEM approaches in terms of accuracy and computational efficiency. The presented comparisons illustrate that the proposed approach is a suitable strategy for predicting underground railway-induced vibrations, both in terms of accuracy and computational efficiency. Moreover, the use of meshless approaches for modelling both tunnel and soil not only simplifies the implementation of the approach but also makes it easier to use. Consequently, the proposed approach is found to be a suitable alternative tool for the prediction of railway-induced vibrations.

Dirac solitons and topological edge states in the β-Fermi-Pasta-Ulam-Tsingou dimer lattice.

We consider a dimer lattice of the Fermi-Pasta-Ulam-Tsingou (FPUT) type, where alternating linear couplings have a controllably small difference and the cubic nonlinearity (β-FPUT) is the same for all interaction pairs. We use a weakly nonlinear formal reduction within the lattice band gap to obtain a continuum, nonlinear Dirac-type system. We derive the Dirac soliton profiles and the model's conservation laws analytically. We then examine the cases of the semi-infinite and the finite domains and illustrate how the soliton solutions of the bulk problem can be glued to the boundaries for different types of boundary conditions. We thus explain the existence of various kinds of nonlinear edge states in the system, of which only one leads to the standard topological edge states observed in the linear limit. We finally examine the stability of bulk and edge states and verify them through direct numerical simulations, in which we observe a solitonlike wave setting into motion due to the instability.

Theoretical analysis for free vibration of submerged conical motor stator

The conical rim-driven motor stator is a structure with variable thickness and non-standard boundaries, which can be treated as an orthotropic conical shell. To prevent resonance and reduce electromagnetic noise, a modal analysis of the stator is necessary. This paper presents an analytical model for the free vibration of the conical stator in both onshore and underwater environments. Due to the relatively large thickness of the motor stator, the first-order shear deformation theory (FSDT) is employed. The fluid loading is calculated by discretizing the conical shell into a series of narrow cylindrical shells featuring diverse radii and thickness. The resulting equations are solved employing the transfer matrix method (TMM), which is capable of different types of boundary conditions (BCs). The convergence and accuracy of the proposed method are validated through a comparative analysis involving numerical simulations and outcomes derived from the thin shell theory. The importance of considering shear deformation and rotatory inertia in determining the natural frequency of motor stators is demonstrated. Finally, the impact of fluid loading on the vibration behavior of stators with various geometric parameters and boundary conditions are investigated and discussed. The proposed model presents a precise and convenient approach for analyzing the free vibration of submerged conical motor stators, and can be extended to conventional cylindrical motor stators through appropriate degradation procedures.

Validation of Ablation and Thermal Response Model for Three-Dimensional Multifunctional Ablator

A semi-empirical physics-based ablation and thermal response model was developed for the 3-Dimensional Multifunctional Ablative Thermal Protection System (3D-MAT) material. Model validation was achieved through comparison between computation and available data obtained in the arc-jet test series, which were conducted at NASA Ames Research Center from 2014 to 2020. The charring ablator simulations for arc-jet test models presented in this work were computed by the Two-Dimensional Implicit Thermal Response and Ablation (TITAN) code, and the Data-Parallel Line Relaxation Method (DPLR) code was used for arc-jet flow simulations to estimate the arc-jet stream total enthalpy and define the aerothermal boundary conditions over the test model surface for TITAN simulations. Because of low surface catalytic efficiency of 3D-MAT char, the exact surface heating could not be determined. Thus, three different types of boundary conditions, including 1) fully catalytic surface heating, 2) noncatalytic surface heating, and 3) surface temperature and recession, were used in the TITAN simulation for model validation. The predicted surface and in-depth temperature history for arc-jet test models were compared with pyrometer and thermocouple data, and the predicted test model surface recession and char depth were compared against the posttest measurements.

Modeling Water Flow in Variably Saturated Porous Soils and Alluvial Sediments

The sustainable exploitation of groundwater resources is a multifaceted and complex problem, which is controlled, among many other factors and processes, by water flow in porous soils and sediments. Modeling water flow in unsaturated, non-deformable porous media is commonly based on a partial differential equation, which translates the mass conservation principle into mathematical terms. Such an equation assumes that the variation of the volumetric water content (θ) in the medium is balanced by the net flux of water flow, i.e., the divergence of specific discharge, if source/sink terms are negligible. Specific discharge is in turn related to the matric potential (h), through the non-linear Darcy–Buckingham law. The resulting equation can be rewritten in different ways, in order to express it as a partial differential equation where a single physical quantity is considered to be a dependent variable. Namely, the most common instances are the Fokker–Planck Equation (for θ), and the Richards Equation (for h). The other two forms can be given for generalized matric flux potential (Φ) and for hydraulic conductivity (K). The latter two cases are shown to limit the non-linearity to multiplicative terms for an exponential K-to-h relationship. Different types of boundary conditions are examined for the four different formalisms. Moreover, remarks given on the physico-mathematical properties of the relationships between K, h, and θ could be useful for further theoretical and practical studies.

Achieving stable convergence of neural networks for estimating acoustic field in uniform ducts

The ability of the neural networks as function approximators can be exploited to solve several governing differential equations. In this work, 1-D Helmholz equation is solved to predict the acoustic pressure in a uniform duct. Solving the Helmholtz equation across a range of frequencies, especially at higher frequencies is challenging as the loss function destabilizes the training process, thereby preventing it from converging to the true solution with the desired accuracy. To overcome this issue, a dynamic learning rate technique is proposed that helps to stabilize the training process and improve overall accuracy of the network. The efficiency of the method is demonstrated by comparing the results with a static learning rate method and the analytical solutions. A good agreement is observed between the predicted solution with dynamic learning rate and the analytical solution up to 2000 Hz. Without dynamic learning rate, the relative errors are observed tobe 2% and 58% at 500 and 2000 Hz, respectively, whereas they reduced to 0.6% and 0.1%, respectively, with the dynamic learning rate at the same frequencies. The proposed dynamic learning rate method is found to be effective for different types of boundary conditions.

The Weighted Least-Squares Collocation Method for Elastic Wave Obstacle Scattering Problems

Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is [Formula: see text] for Sobolev kernel with kernel smoothness [Formula: see text]. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles.

Homogenization assumptions for the two-scale analysis of first-order shear deformable shells

This contribution presents a multiscale approach for the analysis of shell structures using Reissner–Mindlin kinematics. A distinctive feature is that the thickness of the representative volume element (RVE) corresponds to the shell thickness. The main focus of this paper is on the choice of correct boundary conditions for the RVE. Three different types of boundary conditions, which fulfil the Hill–Mandel condition, are presented to bridge the two scales. A common feature is the application of zero-traction boundary conditions at the top and bottom surfaces of the RVE. Furthermore, an internal constraint is used to reduce the dependency of the stiffness components on the RVE size. The introduced boundary conditions differ mainly in the application of shear strains and their symmetry requirements on the RVE. The characteristic features are compared by means of linear-elastic benchmark tests. It is shown that the stress resultants and tangent stiffness components are obtained correctly. Moreover, the presented approach is verified using different macroscopic shell structures and different mesostructures. Both, linear and nonlinear small strain examples are compared to analytical values or full-scale solutions and demonstrate a wide applicability of the present formulation.

Semi-analytical modelling of mode jumping phenomena in pressurized fuselage panels

In this paper, a novel semi-analytical model is developed to capture the mode jumping phenomenon in composite and metallic pressurized fuselage panels. The analytical formulation is derived using the Ritz-based approach and the resulting system of nonlinear equations are solved numerically by homotopy methods. Panels with two different aspect ratios and three different types of boundary conditions are investigated. The panels are loaded in two sequential steps. In the first step the panel is loaded by pressure resulting in a primary mode shape. In the second step, the panel is loaded by an in-plane axial compressive load which causes a sudden change in the primary mode shape leading to a mode jumping event. For verification purposes, finite element models are developed and analysed using the commercial software Abaqus. It is demonstrated that the semi-analytical model is able to successfully capture the mode jumping phenomenon and predict the corresponding load. New findings concerning the mode jumping problem in pressurized fuselage panels are analysed and interpreted.