Fictitious Boundary Research Articles

A directional ghost-cell immersed boundary method for incompressible flows

This paper describes an efficient ghost-cell immersed boundary method for incompressible flow simulations. The method employs a locally directional extrapolation scheme along all discretization directions for the ghost values. Additionally, it involves fictitious discrete boundary forcing terms instead of the ghost values in the governing equations. In this way, the boundary is represented more accurately than in prior IBMs and it is possible to fulfill the divergence-free condition. When combined with high-order spatial discretization schemes, the IBM order is reduced locally near the immersed boundary in a step-wise manner. In this way, the method delivers more compact stencils and is able to deal with sharp interfaces. By handling the distance from the ghost point to the boundary carefully, the proposed method delivers lower truncation errors than standard IBM, with clean (persistent) convergence rate and enhanced stability. The parallel implementation of this approach is straightforward. Its accuracy has been checked by considering a variety of test cases, including irregular, three-dimensional, and moving boundaries. The local accuracy for the proposed method is formally second order, and is measured to be close to this value using numerical tests.

Two-phase flow simulation of scour beneath a vibrating pipeline during the tunnel erosion stage

A new numerical model is developed to simulate and investigate scour beneath a vibrating pipe during the tunnel erosion stage. This study is motivated by the fact that existing numerical models are not able to properly simulate scour under a vibrating pipeline, and the underlying physical mechanisms are not well understood due to the complex fluid-structure-sediment interaction. The present model incorporates the hybrid fictitious domain-immersed boundary method into a recently developed rheology-based two-phase model. The present model is validated against published experiment results of flow beneath a vibrating pipeline near a rigid boundary and scour beneath a fixed pipe. The flow velocity at the gap and the scour profile beneath the pipe are generally well produced by the model. Subsequently, the proposed model is applied to simulate scour under a vibrating pipe with different vibration amplitudes and frequencies. Among other things, it is found that maximum pipe acceleration has a dominant effect on the underlying physics that induce scour, irrespective of the combination of the vibration amplitude and frequency. An explanation for this finding is proposed based on various quantitative simulated results.

Numerical investigations for a chain of particles settling in a channel

The present work deals with the numerical investigation for a chain-like formation of suspended particles of circular shape in fluid flow. An explicit volume integral approach for the calculation of hydrodynamic forces acting on the particle’s surface is used. A collision strategy proposed by Glowinski, Singh, Joseph and coauthors for the treatment of colliding or approaching particles is used. The direct simulation technique (fictitious boundary method) achieves the solution of incompressible flow along with moving rigid bodies. The Eulerian based approach which is independent of particle size, particle shape and number of particles is adopted to solve the fluid-particle phenomena in the computational domain. The fluid domain consisting of a multigrid finite element background is considered and the whole numerical process is carried out through open source code FEATFLOW. Numerical experiments for settling particles which are close together forming chain like formations are performed and analyzed. The study is carried out using different configurations of particle positions resulting in different chain patterns. Consequences of such patterns on particles’ motion and on the fluid-particle system is discussed.

Method of fundamental solutions without fictitious boundary for three dimensional elasticity problems based on force-balance desingularization

An improved non-singular method of fundamental solutions (INMFS), where the sources are located on the domain boundary, is developed for 3D linear elasticity problems. To circumvent the singularities of the kernel in Fundamental Solution (FS) these are substituted by the volume integral of the FS over a small sphere. The normalized volume integral is used when the sources and collocation points are coincident. The desingularization of the fundamental traction is achieved by assuming the balance of the forces for complying with the mechanical equilibrium, calculated through meshless boundary patches coinciding with the boundary nodes. This improved approach avoids any need to solve the problem three times, as in the recently developed Non-singular Method of Fundamental Solutions (NMFS) by Liu and Šarler in 2018. The INMFS, NMFS and MFS solutions as well as the analytical solutions for a pair of single and one bi-material elasticity problems are employed to evaluate the viability and correctness of the new method in 3D. The INMFS results are reasonably accurate and converge uniformly to the analytical solution. The absence of an artificial boundary, trivial coding, and the straightforward use of the INMFS in problems with different materials in contact are demonstrated in this paper.

B-Spline FEM for Time-Harmonic Acoustic Scattering and Propagation

We study the application of a B-splines Finite Element Method (FEM) to time-harmonic scattering acoustic problems. The infinite space is truncated by a fictitious boundary and second-order Absorbing Boundary Conditions (ABCs) are applied. The truncation error is included in the exact solution so that the reported error is an indicator of the performance of the numerical method, in particular of the size of the pollution error. Numerical results performed with high-order basis functions (third or fourth order) showed no visible pollution error even for very high frequencies. To prove the ability of the method to increase its accuracy in the high frequency regime, we show how to implement a high-order Padé-type ABC on the fictitious outer boundary. The above-mentioned properties combined with exact geometrical representation make B-Spline FEM a very promising platform to solve high-frequency acoustic problems.

Investigation of borehole effects on azimuthal resistivity measurements using novel pseudo‐analytic formulas

ABSTRACTDownhole resistivity measurements provide valuable information for geosteering and formation evaluation. It is important to understand and correct the environmental effects, such as the borehole, the tool eccentricity and the resistivity anisotropy effects, of the measurements based on fast and accurate modelling methods. A new pseudo‐analytic solution for eccentric coaxial and tilted‐coil antennas in the cylindrically multilayered medium with transverse isotropic conductivity was developed. This method can take the eccentric tool, the borehole, the mud invasion and the resistivity anisotropy into account. These formulas enable us to simulate the responses of the downhole induction logging tools, including induction wireline and azimuthal propagation logging‐while‐drilling measurements. Instead of using the prior fictitious boundary, the generalized reflection coefficients are proposed to construct the linear matrix functions to deal with the tool eccentricity. The cylindrical functions are reorganized and presented in forms of ratios. Thus, the proposed formulas obviate the overflow issue in the computation and are more stable and efficient. The proposed approach is compared and validated with other well‐established methods. Simulations and case studies show that: (1) the responses of azimuthal resistivity logging‐while‐drilling tool are affected by the tool's eccentricity in both isotropic and anisotropic medium; (2) sine curve behaviours exist for the phase difference and amplitude ratio of azimuthal resistivity measurements along with different eccentricity azimuth.

Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations

This paper makes a first attempt to use a new localized method of fundamental solutions (LMFS) to accurately and stably solve the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations in complex geometries. The LMFS firstly divides the whole physical domain into several small overlapping subdomains, and then employs the traditional method of fundamental solutions (MFS) formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary. After that, a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under-specified boundary, and by using the given boundary conditions for the nodes on over-specified boundary. Finally, the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system. Compared with the traditional MFS with the “global” boundary discretization, the LMFS requires less computational cost, which may make the LMFS suitable for solving large-scale problems. Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data.

Dynamic analysis of multi-crack problems by the spline fictitious boundary element method based on Erdogan fundamental solutions

The appearance of multiple cracks on engineering structures has a large influence on the structural dynamic characteristics. The Erdogan fundamental solutions for static analysis of an infinite single cracked plate are introduced in this paper. Using the multi-domain coupling technique, the multi-crack problem is transformed into several single crack problems. For each single crack problem, based on the above fundamental solutions, dynamic fracture analysis is presented with the mathematical formulation and computational implementation of the spline fictitious boundary element method (SFBEM). Here, the stress boundary conditions on the crack surface are automatically satisfied and the singular behaviour at the crack tip is naturally captured. The angular frequencies and mode shapes of the multi-crack problem are obtained using the proposed method. The dynamic stress intensity factors (DSIFs) of the multi-crack problem are also obtained. Numerical examples are given to demonstrate the accuracy of the proposed method in comparison to the finite element method.

Fully resolved simulations of thermal convective suspensions of elliptic particles using a multigrid fictitious boundary method

In particulate flows, particle-particle collisions play a very important role in determining the flow behavior of the fluid-particle two-phase systems. Thus, it is very important in the numerical simulations of particulate flows to treat the particle-particle interactions with a felicitous method. In this paper, a recently developed direct numerical simulation (DNS) technique, the Finite Element Fictitious Boundary Method (FEM-FBM), for thermal convective particulate flows is used for the simulations of dense particulate suspensions of elliptic shaped particles. The momentum and temperature flow fields are coupled with the aid of Boussinesq approximation. The thermal and momentum interactions between solid and fluid phases are handled by using the Fictitious boundary method (FBM). The continuity, momentum, and energy equations are solved on a fixed Eulerian mesh which is independent of flow features by using a multi-grid finite element scheme. A modified collision model is proposed that can handle not only the interactions between the circular particles but also effectively treat the collisions between the elliptic shaped particles. Firstly, we validate the newly developed collision model for isothermal circular particles together with a comparative study of “drafting, kissing and tumbling” (DKT) motion of both elliptic and circular shaped isothermal particles. Then we investigated the DKT motions of the hot and cold elliptic particles with energy exchange and studied the lateral behavior of the particles in numerous settings. Further, we studied the DKT motion of catalyst particles and investigate the particles behavior at different Grashof numbers. Numerical tests are performed to show that the present method is robust and provides an efficient approach for the simulations of particulate flows with a large number of elliptic particles.

Three-Dimensional Thermo-Elasto-Hydrodynamic Computational Fluid Dynamics Model of a Tilting Pad Journal Bearing—Part I: Static Response

Abstract This paper presents the first simulation model of a tilting pad journal bearing (TPJB) using three-dimensional (3D) computational fluid dynamics (CFD), including multiphase flow, thermal-fluid, transitional turbulence, and thermal deformation of the shaft and pads employing two-way fluid–structure interaction (FSI). Part I presents a modeling method for the static performance. The model includes flow between pads BP, which eliminates the use of an uncertain, mixing coefficient (MC) in Reynold's equation approaches. The CFD model is benchmarked with Reynold's model with a 3D thermal-film, when the CFD model boundary conditions are consistent with the Reynolds boundary conditions. The Reynolds model employs an oversimplified MC representation of the three-dimensional mixing effect of the BP flow and heat transfer, and it also employs simplifying assumptions for the flow and heat transfer within the thin film between the journal and bearing. This manufactured comparison shows good agreement between the CFD and Reynold's equation models. The CFD model is generalized by removing these fictitious boundary conditions on pad inlets and outlets and instead models the flow and temperature between pads. The results show that Reynold's model MC approach can lead to significant differences with the CFD model including detailed flow and thermal modeling between pads. Thus, the CFD approach provides increased reliability of predictions. The paper provides an instructive methodology including detailed steps for properly applying CFD to tilt pad bearing modeling. Parts I and II focus on predicting static and dynamic response characteristic responses, respectively.

ЧИСЕЛЬНИЙ АНАЛІЗ РОЗВ’ЯЗКІВ ДВОВИМІРНИХ ЗАДАЧ ТЕПЛОПРОВІДНОСТІ ЗА БЕЗСІТКОВОЮ СХЕМОЮ З ВИКОРИСТАННЯМ ФУНДАМЕНТАЛЬНИХ І ЗАГАЛЬНИХ РОЗВ’ЯЗКІВ

У статті наводиться аналіз чисельних розв’язків двовимірних задач теплопровідності за безсітковою схемою, які отримані з використанням фундаментальних і загальних розв’язків модифікованого рівняння Гельмгольца. У роботі використовувався безсітковий метод, який ґрунтується на комбінації методу подвійного заміщення з використанням радіальних базисних функцій і методу частинних розв’язків. Порівняльний аналіз розв’язків крайових задач продемонстрований на прикладі двох тестових задач. Були отримані чисельні розв’язки двовимірних нестаціонарних задач теплопровідності з використанням фундаментального і загального розв’язків для різного числа інтерполяційних вузлів. Були визначені середньоквадратичні похибки розв’язків розглянутих задач, а також побудовані порівняльні графіки залежності середньоквадратичної похибки від числа інтерполяційних вузлів.

Dynamics of elliptic particle sedimentation with thermal convection

In this paper, a recently developed direct numerical simulation technique, the Finite Element Fictitious Boundary Method (FEM-FBM) [K. Walayat et al., “An efficient multi-grid finite element fictitious boundary method for particulate flows with thermal convection,” Int. J. Heat Mass Transfer 126, 452–465 (2018)], is used to simulate sedimentation of an elliptic particle with thermal convection. The momentum and temperature flow fields are coupled with the aid of Boussinesq approximation. The thermal and momentum interactions between solid and fluid phases are handled by using the fictitious boundary method (FBM). The continuity, momentum, and energy equations are solved on a fixed Eulerian mesh which is independent of flow features by using a multi-grid finite element scheme. Two validation tests are conducted to show the accuracy of the present method, and then the effects of thermal properties of fluid on the sedimentation of an elliptic particle are studied. It is demonstrated that the dynamics of hot elliptic particle sedimentation depend on the thermal diffusivity and thermal expansion of the fluid. A comparative study of the forces and torque acting on the hot, cold, and isothermal particle is reported. Moreover, different sedimentation modes of hot and cold elliptic particles are identified in an infinitely long channel. The mechanism of transitions of particle settling modes from tumbling to inclined and then to the horizontal mode is discovered. Also, we discovered a new sedimentation mode of the hot elliptic particle in cold fluid, i.e., the vertical mode. Furthermore, buoyancy effects for the catalyst particle are studied at different initial orientations.

Spline Fictitious Boundary Element Alternating Method for Edge Crack Problems with Mixed Boundary Conditions

Spline Fictitious Boundary Element Alternating Method for Edge Crack Problems with Mixed Boundary Conditions

A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems

• Physical essence of the fictitious boundary is explained. • The influence law of the fictitious boundary on numerical results is revealed. • A dual-level method of fundamental solutions is proposed. • A new strategy to solve the ill-conditioning of the radial basis functions methods is provided. • The Helmholtz problems with up to non-dimensional wavenumber of 600 is simulated. Physical essence of the fictitious boundary of the method of fundamental solutions has been a mystery for a long time. In this study, we attempt to explain the reason why fictitious boundary has such a dramatic effect on numerical results. The influence law of the fictitious boundary on numerical results is revealed. Based on this understanding, a dual-level method of fundamental solutions with self-adaptive adjustment coefficients is proposed. The competitive attributes of the method are that it inherits the high numerical efficiency of the method of fundamental solutions, and it improves numerical stability significantly. The effect of the fictitious boundary on numerical results is eliminated by introducing the concept of equivalent slope. It should be noted that the dual-level method of fundamental solutions can simulate exterior high frequency acoustic problems under the lowest sampling frequency allowed by the Shannon's sampling theorem. Numerical experiment with up to non-dimensional wavenumber of 600 has successfully been conducted on a single laptop when one uses 100,000 degrees of freedom.

Evaluation of the T-stress and stress intensity factor for multi-crack problem using spline fictitious boundary element alternating method

In this paper, the T-stress and stress intensity factor (SIF) of multiple cracks with arbitrary position in a finite plate is evaluated by the spline fictitious boundary element alternating method. The multi-crack problem is firstly divided into a simple model without crack which can be solved by the spline fictitious boundary element method and several infinite domains with one crack which can be solved by the fundamental solution of an infinite domain with a crack, namely Muskhelishvili's fundamental solutions. The technique is superior as no meshing is needed near crack face and the analytical solution for solving infinite domains with one crack is accurate and efficient. Then, instead of using the asymptotic expansion, the closed-form expression for calculating the T-stress in multi-crack problem is derived directly, which makes it convenient and accurate for calculating the T-stress. Besides, the SIF can be calculated using the analytical SIF expression in Muskhelishvili's fundamental solutions. Finally, T-stresses and SIFs in a numerical example with double cracks are computed to validate the accuracy of the presented method, And the other two examples with three cracks are further studied to investigate the influence of lengths and locations of multiple cracks on their T-stresses and SIFs.

Generalized method of fundamental solutions (GMFS) for boundary value problems

In order to cope with the instability of the method of fundamental solutions (MFS), which caused by source offset, source location, or a fictitious boundary, a generalized method of fundamental solutions (GMFS) is proposed. The crucial part of the GMFS is using a generalized fundamental solution approximation (GFSA), which adopts a bilinear combination of fundamental solutions to approximate, rather than the linear combination of the MFS. Then the numerical solution of the GMFS is decided by a group of offsets corresponding to an intervention-point diffusion (IPD), instead of the MFS’ offset of a single source. To demonstrate the effectiveness of the proposed approach, five numerical examples are given. The results have shown that the GMFS is more accurate, stable, and has a better convergence rate than the traditional MFS.

Analysis of solid particles falling down and interacting in a channel with sedimentation using fictitious boundary method

We have examined the behavior of solid particles in particulate flows. The interaction of particles with each other and with the fluid is analyzed. Solid particles can move freely through a fixed computational mesh using an Eulerian approach. Fictitious boundary method (FBM) is used for treating the interaction between particles and the fluid. Hydrodynamic forces acting on the particle’s surface are calculated using an explicit volume integral approach. A collision model proposed by Glowinski, Singh, Joseph and coauthors is used to handle particle-wall and particle-particle interactions. The particulate flow is computed using multigrid finite element solver FEATFLOW. Numerical experiments are performed considering two particles falling and colliding and sedimentation of many particles while interacting with each other. Results for these experiments are presented and compared with the reference values. Effects of the particle-particle interaction on the motion of the particles and on the physical behavior of the fluid-particle system has been analyzed.

An efficient multi-grid finite element fictitious boundary method for particulate flows with thermal convection

This paper presents a direct numerical simulation (DNS) technique, the Finite Element Fictitious Boundary Method (FEM-FBM) for the simulation of fluid-solid two-phase flows with heat transfer. The heat transfer equation is introduced to study thermal convection in fluid-solid two-phase flows. The Boussinesq approximation is considered for the coupling of momentum and temperature flow fields. Multi-grid finite element solver is used to compute flow equations of mass, momentum, and energy on a fixed Eulerian mesh which is independent of time and the solid particles are allowed to move freely in the whole computational domain. Fictitious boundary method (FBM) is used to treat the particles inside the fluid, FBM takes account of the thermal and momentum interaction between the fluid and the particles. The accuracy and stability of presented method are validated by comparing our test cases results with the results reported in the available literature. Numerical tests are performed to show that this method is potentially powerful and provides an efficient approach to simulate complex thermal convective particulate flows with a large number of particles.

Analysis of multi-crack problems by the spline fictitious boundary element method based on Erdogan fundamental solutions

The Erdogan fundamental solutions are derived from an infinite plane containing a crack. When they are used in the formulation of the boundary element method (BEM), the stress boundary conditions on the crack surface are automatically satisfied and the singular behavior at the crack tip can be naturally reflected. Using the multi-domain technique, the multi-crack problem can be transformed into a series of single-crack problems involving displacement continuity conditions along common boundaries. In this paper, the displacements which are expressed in terms of the integral of a complex function in the Erdogan fundamental solutions are derived in closed-form expressions. Then, the multi-domain spline fictitious boundary element method (SFBEM) based on the above fundamental solutions is proposed and formulated for analyzing multi-crack problems. The computational accuracy and stability of the proposed method are verified by comparing the stress intensity factor (SIF) results of a double-inner crack problem with different inclined angles and crack lengths against those calculated by the finite element method. Also, the SIF results of a double-edge crack problem with different crack lengths are compared with those obtained from studies. Finally, the proposed method is applied to the analysis of the triple-crack problem, in which the shielding effects of multi-crack and stress contours are studied with different crack lengths and locations.

Boundary mesh free method with distributed sources for 2D elasticity problems

This paper presents development Boundary Distributed Source method for 2D elasticity problems, in which sources are located at domain boundary, i.e., without fictitious boundary. Singularities of the kernel are totally removed by replacing the concentrated point sources with area-distributed sources–circular disc centered at the source point in this study–providing new kernels resulting from analytical integration of the original singular fundamental solution and its derivative on the area of the source disc. All the coefficients for Dirichlet and Neumann boundary conditions in the systems of equations can be determined analytically except diagonal coefficients for Neumann boundary conditions; they are calculated indirectly by the consideration of two reference solutions of rigid body motion in the two directions.