A C1-conforming hp finite element method for fourth order singularly perturbed boundary value problems

The inlet region of a plane-parallel pressure flow is the subject of this study. The patterns of variations in the hydrodynamic inlet region under unsteady plane-parallel pressure flow of a viscous fluid are examined in this research. Based on the boundary layer equation and flow characteristics, the boundary conditions of the problem were determined and a boundary value problem was formulated. The problem's boundary conditions were established and a boundary value problem was developed based on the boundary layer equation and flow characteristics. In order to find patterns of velocity change over time and over the length of the inlet region under general boundary conditions, a method for integrating the boundary value conditions was created. Solutions for scenarios with a constant and parabolic velocity distribution in the inlet region were derived from the general solutions. Regularities of pressure and velocity change were found along the entire hydrodynamic inlet region. Using computer analysis, graphs of velocity changes over time at various points along the entire length of the inlet region are constructed. The patterns of velocity distribution along the entire length of the inlet region depending on time can be seen using graphs. This allows one to estimate the length of the hydrodynamic inlet region and calculate the fluid flow velocity at any point in this region. The findings enable revealing the essence of the processes running in an hydropneumatic automation system's transition sections. Based on the revealed regularities of the hydrodynamic parameters of viscous incompressible liquid during unsteady flows, it is possible to correctly design of the automatic systems' channels of regulating units ensuring their smooth and accurate operation

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.

In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the [Formula: see text] norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic.

A branch-price-and-cut algorithm is developed for a complex maritime inventory-routing problem with varying storage capacities and production/consumption rates at facilities. The resulting mixed-integer pricing problem is solved exactly and efficiently using a dynamic program that exploits certain “extremal” characteristics of the pricing problem. The formulation is tightened by using the problem's boundary conditions in preprocessing and to restrict the set of columns that are produced by the pricing problem. Branching schemes and cuts are introduced that can be implemented efficiently and that preserve the structure of the pricing problem. Some of the cuts are inspired by the capacity cuts known for the vehicle-routing problem, whereas others specifically target fractional solutions brought about by individual vessels “competing” for limited inventory at load ports and limited storage capacity at discharge ports. The branch-price-and-cut approach solves practically sized problems motivated by the operations of an oil company to optimality, and it provides reasonable bounds for larger instances.

Approximation of slow moving interface phase change problems using a generalized Fourier series and the CVBEM

The scaled boundary finite element method (SBFEM) is a semi-analytical method, whose versatility, accuracy, and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using an efficient high-order polynomial element in the SBFEM to form the approximation in the circumferential direction. The governing equations are formulated from the classical linear elasticity theory via the SBFEM technique. The scaled boundary finite element equations are formulated within a general framework integrating the influence of the distributed body source, mixed boundary conditions, contributions the side face with either prescribed surface load or prescribed displacement. The position of scaling center is considered for modeling problem. The proposed method is evaluated by solving two-dimensional linear problem. A selected set of results is reported to demonstrate the accuracy and convergence of the proposed method for solving problems in general boundary conditions.

The finite difference method that is the finite element method is used to solve the plane continuous problems. In this article, the theory and method of the finite difference method, as well as the application on the boundary problem are introduced. By analyzing the potential flew field equation and liquid diffusion equation, they are discreted using the difference method and the numerical analysis under certain boundary condition is conducted. In air pollution, the smoke in the diffusion is typical planar continuous problems. In this paper, the finite difference method is used to analyse and simulate the spread of the smoke.

Hydrodynamic interactions in Brownian dynamics: I. Periodic boundary conditions for computer simulations

Applications of magnetic-resonant wireless power transfer technology have been extensively researched because of its high efficiency, security and convenience. To recharge batteries, the induced ac voltage in the receiver coil of wireless systems must be rectified to dc, hence the power transfer systems are inevitably nonlinear. In this paper, a novel approach based on time-domain finite element method (FEM) is presented to analyze the nonlinear system in order to obtain frequency-sweeping solutions quickly. At the first step of the proposed method, the wireless power transfer system, excluding the rectifier circuit, will be equivalent to a voltage source and a general impedance at different frequencies. The equivalent circuit is then connected to the rectifier circuit. Numerical example being reported in this paper indicates that the results obtained using the proposed methodology are consistent with those obtained using traditional time-domain FEM method. If solutions at different operating frequencies are required, the proposed method can save a lot of computation time, because one just needs to solve the field problem using the time-domain FEM twice. From the numerical example on the performance analysis of the wireless power transfer system being studied, the computing time of the proposed time-domain method is only 2% of that required if traditional time-domain FEM method is used.

A class of challenging problems encountered in geomechanics pertains to the prediction of the transient response of elastic systems of infinite or semi-infinite extent subjected to time dependent loading and boundary conditions. The nature and complexity of such problems restrict the range of available closed form solutions to the ones of linear problems of highly simplified geometries and boundary conditions. Among the numerical techniques, the most popular and well established one is the Finite Element Method (FEM) which can routinely handle complex geometries, medium inhomogeneities, and material non-linearities, and is well suited for both frequency and time domain analysis. However, the FEM by nature encompasses the major disadvantage of modeling infinite, and semi-infinite domains by finite size meshes violating, thus, the radiation damping condition. The problem is confronted, to some extend, by the development of hybrid techniques, and transmitting boundaries, for frequency domain analysis. Yet, time domain FEM solutions involving unbounded 3-D solid media, if available at all, may increase the size of the problem to computationally prohibitive levels. An alternative to the FEM is the Boundary Element Method (BEM), which is based on integral equation formulations, developed in the beginning of the century. The first 3-D time domain BEM formulation, associated with soil-structure interaction problems, has been reported by Karabalis and Beskos [1]. In this formulation the Stokes fundamental solutions are developed on the basis of a step impulse forcing function.

Spatially resolved spectroscopy provides a means for measuring the optical properties of biological tissues, based on analytical solutions to diffusion approximation for semi-infinite media under the normal illumination of an infinitely small light beam. The method is, however, prone to error in measurement because the actual boundary condition and light beam often deviate from that used in deriving the analytical solutions. It is therefore important to quantify the effect of different boundary conditions and light beams on spatially resolved diffuse reflectance in order to improve the measurement accuracy of the technique. This research was aimed at using finite element method (FEM) to model light propagation in turbid media, subjected to normal illumination by a continuous-wave beam of infinitely small or finite size. Three types of boundary conditions [i.e., partial current (PCBC), extrapolated (EBC), and zero (ZBC)] were evaluated and compared against Monte Carlo (MC) simulations, since MC could provide accurate fluence rate and diffuse reflectance. The effect of beam size was also investigated. Overall results showed that FEM provided results as accurate as those of the analytical method when an appropriate boundary condition was applied. ZBC did not give satisfactory results in most cases. FEM-PCBC yielded a better fluence rate at the boundary than did FEM-EBC, while they were almost identical in predicting diffuse reflectance. Results further showed that FEM coupled with EBC effectively simulated spatially resolved diffuse reflectance under the illumination of a finite size beam. A large beam introduced more error, especially within the region of illumination. Research also confirmed an earlier finding that a light beam of less than 1 mm diameter should be used for estimation of optical parameters. FEM is effective for modeling light propagation in biological tissues and can be used for improving the optical property measurement by the spatially resolved technique.

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. All the derivations are done on continuous level, and the estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1, independently of the discretization method chosen. In particular, we make no use of the Galerkin orthogonality, which enables us to implement the estimator for measuring the error of the fictitious domain/penalty finite element method. The estimator is easily computable, and the only constant present in the estimator is the one from Friedrichs' inequality; the constant depends solely on the domain geometry, and the estimator is quite non‐sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

Finite element (FE) method is a form of mathematical modelling used to study the stress, strain and dynamics of a structure. The method has been used to analyse anatomical structures in the human body. In cardiac surgery it allows the quantifying of patient specific geometric features to both assess disease state and potentially aid in planning surgical intervention such as mitral valve repairs, prosthetic valve design or congenital reconstructive cardiac surgery. Important variables for a finite element model are model geometry, the type of mesh elements used, material properties and boundary condition. This review explains these variables, their limitations and describes their application in cardiac surgery.

In many engineering problems, the meshfree methods (MMs) have been dynamically projected and increasingly advanced in order to overwhelm some hitches in the predictable numerical methods. Over the past three decades in many different application area, MMs have found their way ranging from solid mechanics analysis, fluid problems, vibration analysis, heat transfer and optimization to numerical solutions of all kinds of (partial) differential equations. As every technique has shortcomings, the meshfree method also has drawbacks like higher computational cost and imposition of boundary condition which can be overruled by coupling it with the finite element method (FEM). In the past two decades, coupled MMs and FEM have appeared into a new session of computational methods with significant achievement. In addition, a noteworthy amount of growth has been made in addressing the major deficiencies that were present in the conventional methods and MMs at the premature phases. The objective of the present work is to provide a comprehensive review of various coupling techniques used for interface elements of MMs and FEM and general discussion on shape function formulation of FE and element free Galerkin method (EFGM). Key contribution of coupling techniques for coupled EFGM and FEM to structure mechanics application as primary concern.

Simulation of three-dimensional transient eddy current problems is important to numerous applications. The Finite Element Method (FEM) has proven be to be powerful numerical technique for solving the Partial Differential Equations (PDE) describing eddy currents. In order to solve the PDE, boundary conditions must be provided, and in many applications the boundary conditions are not known explicitly but can be provided by a Resistor-Inductor-Capacitor (RLC) circuit model. The emphasis of this paper is on an efficient and exact coupling of the RLC network equations with the FEM equations. The coupling is based on an exact linear algebra identity known as the Sherman-Morrison-Woodbury (SMW) formula. One advantage of this approach is that the FEM matrices are not modified. This is important if a fast 'black-box' solver is available for the FEM matrices, these solvers typically require that the matrices have certain mathematical properties and these properties are not modified by the SMW approach. A second advantage is that the SMW approach is valid for an arbitrary number of independent external circuits.