Types Of Boundary Conditions Research Articles

Effects of boundary conditions on the transmitral pressure gradient for numerical simulation of flow in a left heart model

The transmitral pressure gradient is a crucial clinical indicator for diagnosing mitral regurgitation and stenosis. Numerical simulation of specific patients can obtain the complete pressure distribution in the left heart system, which is an important approach to evaluate cardiac function. However, the results are sensitive to the boundary conditions used in the numerical simulations. In the present work, we constructed a left heart model and assessed the effects of four typical boundary conditions on the transmitral pressure gradient, which is computed based on the Bernoulli equation and the line probe, respectively. The results show that the transmitral pressure gradient obtained by the line probe is sensitive to the boundary conditions. The sensitivity is closely related to the pressure in the atrium but has negligible effect on the pressure in the ventricle. This study sheds light on evaluating the pressure gradient of patient-specific treatments based on the numerical simulation of a left heart model.

Exact Second-Order Stiffness Matrix of Axial-Loaded Beam–Columns on Winkler Foundation

This paper establishes the exact second-order stiffness matrix of axial-loaded beam–columns on Winkler foundation. This Winkler foundation model has been widely used in engineering applications. The equilibrium analysis of an axial-loaded beam–column on Winkler foundation is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. Two dimensionless factors for the axial force and foundation spring stiffness are defined, respectively. The exact second-order element stiffness matrix of the axial-loaded beam–column on Winkler foundation is then formulated, showing the relationship between the element-end deformations (translation and rotation angle) and the corresponding forces (shear force and bending moment). Such an exact second-order stiffness matrix may be used for the buckling and second-order solutions with allowance for the use of one element per member for the exact solution. The classical buckling problem of axial-loaded beam–columns on Winkler foundation with typical element-end boundary conditions is then analyzed by the developed matrix stiffness method. By comparing with classical buckling criterion expressions of axial-loaded beam–columns on Winkler foundation with various boundary conditions, it is demonstrated that the derived exact second-order stiffness matrix can be used to obtain the exact buckling solutions systematically.

Numerical solution of source identification multi-point problem of parabolic partial differential equation with Neumann type boundary condition

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.

Multiscale dynamic behavior of imperfect hybrid matrix/fiber nanocomposite nested conical shells with elastic interlayer

This investigation delves into the free vibration characteristics of coupled nested conical shells (CNCSs) made of porous composite materials. These two conical shells are connected by a mid-layer of elastic springs. The composite materials used in the shells consist of epoxy, nanofillers, and fibers. Two types of nanofillers are considered: Graphene Nanoplatelets (GNPs) and Carbon Nanotubes (CNTs), while E-glass fiber is used as the fiber. The nanofillers are distributed in four different patterns within the shell section. Porosity is uniformly distributed along the shell section and characterized by a coefficient. The rule of mixtures is employed to ascertain the equivalent material properties of the hybrid materials, while the Chamis approach is utilized for three-phase materials. First-order shear deformation theory (FSDT) and Donnell's theory are utilized for modeling the conical shells. The governing equations of motion are established through Hamilton's principle are solved using the generalized differential quadrature method (GDQM). Seven different boundary conditions (BCs) are considered to encompass the full range of BCs for CNCSs and four type of BCs for single truncated conical shell (STCS). The solution's accuracy is verified, and the effects of various parameters on the natural frequency parameter (NFP) of the shell are investigated, such as BCs, circumferential wave number (n), nanofillers pattern, semi-vertex angle, nanofillers angle, and mid-layer stiffness. Initially, a comprehensive investigation into the vibration behavior of a STCS is presented, followed by an analysis of the NFP of the CNCSs. The results demonstrate that the stiffness of the elastic mid-layer significantly influences the NFP of the system. The orientation of the nanofillers in the shell can increase or decrease the NFP. Additionally, the relationship between mode number and n depends on the type of BCs of the shells.

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.

On the existence and Lipschitz-like property of the solution map to parametric equilibrium problems governed by trifunction with applications

In this paper, we study a class of parametric equilibrium problems governed by a trifunction. We apply the Fan-KKM theorem under new type of monotonicity and coercivity conditions for trifunctions to obtain the existence result of a solution for the equilibrium problem. For the solution map of parametric equilibrium problems, we obtain the Hölder-type Lipschitz-like property relative to a closed and convex set, which generalizes some existing results in the literature. We illustrate the application of our abstract results in the study of a new model of a fluid flow problem with slip and leak boundary conditions of frictional type.

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Numerical Modelling of Phase Transformations of Water Droplets for Efficient Heat Recovery from Biofuel Flue Gas

The phase transformations of water droplets in the humid air flow for water injection into the flue gas in biofuel combustion technology under typical boundary conditions have been simulated. The numerical study was performed in two stages. The first one defines the influence of radiation on the thermal state of the water droplets and on the cycle of phase transformation modes. It is shown that the influence of radiation in a flue gas flow of 150℃→200℃ temperature on the thermal state of quantitatively injected water droplets is not significant, but it induces qualitative changes in the temperature field and influences phase transformations cycle of the droplets. In the second step, the cycling of the droplet phase transformation modes for water injection into the exhaust flue gas prior to a condensing economiser under typical boundary conditions was numerically investigated. It was justified that for efficient cooling and humidification of the flue gas before the condensing economiser it is necessary to inject water heated above the dew-point temperature by dispersing it into fine droplets.

Fine boundary regularity for the singular fractional p-Laplacian

We study the boundary weighted regularity of weak solutions u to a s-fractional p-Laplacian equation in a bounded C1,1 domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with s∈(0,1). We prove optimal up-to-the-boundary regularity of u, which is Cs(Ω‾) for any p>1 and, in the singular case p∈(1,2), that u/dΩs has a Hölder continuous extension to the closure of Ω, dΩ(x) meaning the distance of x from the complement of Ω. This last result is the singular counterpart of the one in [30], where the degenerate case p⩾2 is considered.

Analysis of energy dissipation in hyperbolic problems influenced by internal and boundary control mechanisms

ABSTRACT This article primarily focuses on the rational stabilisation of the wave equation, supplied with a second-order dynamical boundary condition of hyperbolic type, while considering an additional internal damping mechanism within the specified ring. To achieve rational decay rates of the associated energy, it is imperative to exponentially stabilise a portion of the domain using a global Ingham's-type estimate. This paper will subsequently illustrate how this partially localised exponential stabilisation, combined with a Bessel analysis, leads to a rational decrease in the overall energy of the system considered.

Adaptive local boundary conditions to improve deformable image registration

Objective. In medical imaging, it is often crucial to accurately assess and correct movement during image-guided therapy. Deformable image registration (DIR) consists in estimating the required spatial transformation to align a moving image with a fixed one. However, it is acknowledged that for DIR methods, boundary conditions applied to the solution are critical in preventing mis-registration. This poses an issue particularly when areas of interest are located near the image border. Despite the extensive research on registration techniques, relatively few have addressed the issue of boundary conditions in the context of medical DIR. Our aim is a step towards customizing boundary conditions to suit the diverse registration tasks at hand. Approach. We analyze the behavior of two typical global boundary conditions: homogeneous Dirichlet and homogeneous Neumann. We propose a generic, locally adaptive, Robin-type condition enabling to balance between Dirichlet and Neumann boundary conditions, depending on incoming/outgoing flow fields on the image boundaries. The proposed framework is entirely automatized through the determination of a reduced set of hyperparameters optimized via energy minimization. Main results. The proposed approach was tested on a mono-modal computed tomography (CT) thorax registration task and an abdominal CT-to-MRI registration task. For the first task, we observed a relative improvement in terms of target registration error of up to 12% (mean 4%), compared to homogeneous Dirichlet and homogeneous Neumann. For the second task, the automatic framework provides results close to the best achievable. Significance. This study underscores the importance of tailoring the registration problem at the image boundaries. In this research, we introduce a novel method to adapt the boundary conditions on a voxel-by-voxel basis, yielding optimized results in two distinct tasks: mono-modal CT thorax registration and abdominal CT-to-MRI registration. The proposed framework enables optimized boundary conditions in image registration without prior assumptions regarding the images or the motion.

A singular perturbation result for a class of periodic-parabolic BVPs

Abstract In this article, we obtain a very sharp version of some singular perturbation results going back to Dancer and Hess [Behaviour of a semilinear periodic-parabolic problem when a parameter is small, Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and Daners and López-Gómez [The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions, J. Dynam. Differential Equations 6 (1994), 659–670] valid for a general class of semilinear periodic-parabolic problems of logistic type under general boundary conditions of mixed type. The results of Dancer and Hess [Behaviour of a semilinear periodic-parabolic problem when a parameter is small, Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and [The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions, J. Dynam. Differential Equations 6 (1994), 659–670] were found, respectively, for Neumann and Dirichlet boundary conditions with L = − Δ {\mathfrak{L}}=-\Delta . In this article, L {\mathfrak{L}} stands for a general second-order elliptic operator.

Analysis of fractional Euler-Bernoulli bending beams using Green’s function method

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green’s function for the highest-order term 1<α≤2, which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green’s function kernel. The theory and properties of fractional Green’s function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam’s characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number.

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Stabilised finite element method for Stokes problem with nonlinear slip condition

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier representing boundary traction and the stabilised formulation is based on simultaneously stabilising both the pressure and the traction. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.

Countably Many Positive Symmetric Solutions For Sturm Liouville Type Boundary Conditions Of Second Order Iterative System

In this paper we consider second order iterative boundary value problem with Sturm Liouville type boundary conditions and establish the existence of countably many positive symmetric solutions by using Krasnoselskii’s fixed point theorem for operator on a cone.

On well-posedness of Navier–Stokes variational inequalities

In this paper, well-posedness is carried out on variational inequalities in a general form for the stationary Navier–Stokes equations modeling viscous incompressible fluid flows with a nonsmooth slip boundary condition of friction type. For a special form of the variational inequality commonly studied in existing literature, our proof technique leads to a substantial improvement on the well-posedness condition.

Error estimates for completely discrete FEM in energy‐type and weaker norms

AbstractThe paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.

On a Stokes hemivariational inequality for incompressible fluid flows with damping

In this paper, a Stokes hemivariational inequality is studied for incompressible fluid flows with the damping effect. The hemivariational inequality feature is caused by the presence of a nonsmooth slip boundary condition of friction type. Well-posedness of the Stokes hemivariational inequality is established through the consideration of a minimization problem. Mixed finite element methods are introduced to solve the Stokes hemivariational inequality and error estimates are derived for the mixed finite element solutions. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is introduced to solve the mixed finite element system. Numerical results are reported on the performance of the proposed algorithm and the numerical convergence orders of the finite element solutions.

A unified mixed finite element method for fourth-order time-dependent problems using biorthogonal systems

This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper.