Fictitious Points Research Articles

The method of fundamental solutions for Brinkman flows. Part I. Exterior domains

The method of fundamental solutions (MFS) is developed for solving numerically the Brinkman flow in the porous medium outside obstacles of known or unknown shapes. The MFS uses the fundamental solution of the Brinkman equation as in the boundary element method (BEM), but the single-layer representation is desingularized by moving the boundary sources to fictitious points outside the solution domain. In the case of unbounded flow past obstacles, these source points are placed in the domain inside the obstacle on a contracted fictitious pseudo-boundary. When the obstacle is known, then the fluid flow in porous media problem is direct, linear and well-posed. In the case of Brinkman flow in the porous medium outside an infinitely long circular cylinder, the MFS numerical solution is found to be in very good agreement with the available analytical solution. However, when the obstacle is unknown and has to be determined from fluid velocity measurements at some points inside the fluid, the resulting problem becomes inverse, nonlinear and ill-posed. The \(\hbox {MATLAB}^{\copyright }\) optimization toolbox routine lsqnonlin is employed for minimizing the least-squares gap between the computed and measured fluid velocity which is further penalized with extra smoothness regularization terms in order to overcome the instability of the solution. For proper choices of the regularization parameters involved, accurate and stable numerical reconstructions are achieved for various star-shaped obstacles.

ВРАЩЕНИЯ ТОРОВ В СТРУКТУРЕ ЖИДКОГО КРИСТАЛЛА

Liquid crystals are a collection of flattened molecules. On the one hand, they have a welldefined structure, on the other hand, this structure is deformable, since its elements can change their position in space. In columnar structures, consisting of disks, plates, and tori, the fluid lines and cords are distinguished. In the case of tori and circular disks, a hexagonal structure of the arrangement of molecules is observed in a combination of the cord. The aim of this work is to obtain the stability of tori positions in an elementary fragment of a liquid crystal and to analyze their rotations in a self-consistent field of the surrounding toroidal molecules. To solve this problem, the method of mathematical modeling was used, based on classical models of molecular dynamics. The calculation is carried out on the basis of the model of cross-atom-atom interactions for molecular tori. The minimal fragment of the cord is selected, which makes it possible to determine the characteristic dynamic state of the central torus in the fragment. To describe the motion of the molecular tori, the equations of motion for their centers of mass and the Euler equations for their angular displacements are used. The minimal fragment of the material contains twenty-nine tori. The equations for displacements of the centers of mass of the tori are initially represented as ordinary differential equations of the second order. However, by introducing fictitious points into the consideration of velocities, they can be reduced to a system of first-order equations with a doubled number of lower-order equations. The resulting system of the first-order differential equations is integrated numerically using a highorder accuracy step-by-step scheme. All calculations are performed with a constant time step. The accuracy of the obtained numerical results is verified in terms of the balance of total energy of the system. Calculations show that the central torus of the presented fragment executes angular oscillations around its main axis with amplitude of more than one revolution. Thus, the performed calculations show that a representative fragment of the liquid crystal structure of molecular tori can be used as a generator of high-frequency mechanical vibrations.

Coupling of turbulence wall models and immersed boundaries on Cartesian grids

An improved coupling of immersed boundary method and turbulence wall models on Cartesian grids is proposed, for producing smooth wall surface pressure and skin friction at high Reynolds numbers. Spurious oscillations are frequently observed on these quantities with most immersed boundary wall modeling methods, especially for the skin friction which is found to be very sensitive to the solid surface's position and orientation against the Cartesian grids. The problem originates from the irregularity of the wall distance on the stair-step grid boundaries where the immersed boundary conditions are applied. To reduce this directional error, several modifications are presented to enhance the near wall solution. First, the commonly used interpolation for the flow velocity is replaced by one for the friction velocity, which has much less variation near wall. The concept of using a fictitious point to retrieve flow fields in the wall normal direction is abandoned and the interpolation is performed in the wall parallel plane with existing fluid points. Secondly, the velocity gradients at the approximated boundary are computed with advanced schemes and the normal gradient of the tangential velocity is reconstructed from the wall laws. To further protect the near wall solution, the normal velocity gradient and the working viscosity from the Spalart-Allmaras turbulence model are enforced by their theoretical solutions in the interior fluid close to the wall. Additionally, various post-processing algorithms for reconstructing wall surface quantities and force integrations are investigated. Other related factors are also discussed for their effects on the results. The validity of present method has been demonstrated through numerical benchmark tests on a flat plate at zero pressure gradient, both aligned and inclined with respect to the grid, as well as aerodynamic cases of NACA 23012 airfoil and NASA trap wing.

A fictitious points one–step MPS–MFS technique

The method of fundamental solutions (MFS) is a simple and efficient numerical technique for solving certain homogenous partial differential equations (PDEs) which can be extended to solving inhomogeneous equations through the method of particular solutions (MPS). In this paper, radial basis functions (RBFs) are considered as the basis functions for the construction of a particular solution of the inhomogeneous equation. A hybrid method coupling these two methods using both fundamental solutions and RBFs as basis functions has been effective for solving a large class of PDEs. In this paper, we propose an improved fictitious points method in which the centres of the RBFs are distributed inside and outside the physical domain of the problem and which considerably improves the performance of the MPS–MFS. We also describe various techniques to deal with the several parameters present in the proposed method, such as the location of the fictitious points, the source location in the MFS, and the estimation of a good value of the RBF shape parameter. Five numerical examples in 2D/3D and for second/fourth–order PDEs are presented and the performance of the proposed method is compared with that of the traditional MPS–MFS.

The Use of Dijkstra's Algorithm in Assessing the Correctness of Imaging Brittle Damage in Concrete Beams by Means of Ultrasonic Transmission Tomography.

The accuracy of transmission ultrasonic tomography for the detection of brittle damage in concrete beams can be effectively supported by the graph theory and, in particular, by Dijkstra’s algorithm. It allows determining real paths of the fastest ultrasonic wave propagation in concrete containing localized elastically degraded zones at any stage of their evolution. This work confronts this type of approach with results that can be obtained from non-local isotropic damage mechanics. On this basis, the authors developed a method of reducing errors in tomographic reconstruction of longitudinal wave velocity maps which are caused by using the simplifying assumptions of straightness of the fastest wave propagation paths. The method is based on the appropriate elongation of measured propagation times of the wave transmitted between opposite sending-receiving transducers if the actual propagation paths deviate from straight lines. Thanks to this, the mathematical apparatus used typically in the tomography, in which the straightness of the fastest paths is assumed, can be still used. The work considers also the aspect of using fictitious wave sending-receiving points in ultrasonic tomography for which wave propagation times are calculated by interpolation of measured ones. The considerations are supported by experimental research conducted on laboratory reinforced concrete (RC) beams in the test of three-point bending and a prefabricated damaged RC beam.

Nonlinear dynamic analysis of shell structures by the formulation based on a discrete shear gap

In this work, a shear locking-free three-node shell element based on Mindlin–Reissner theory is presented for a nonlinear dynamic analysis including sheet metal forming. In the present formulation, only displacement and rotational degrees of freedom are utilized, and the discrete shear gap of each field node is deduced using rotations of all three field nodes in a local coordinate system by the integral from the fictitious central point to the corresponding field node, which makes it alleviate the shear locking phenomenon. In order to validate the availability in solving nonlinear dynamic problems, several benchmark problems and sheet metal forming applications are employed. The results show potentiality in application to practical problems because of the simple implementation.

Fractional-Differential Model of Heat Conductivity Process in Ferroelectrics under the Intensive Heating Conditions

Ferroelectrics, due a number of characteristics, behave as hereditary materials with fractal structure. To model mathematically the systems with so-called memory effects one can use the fractional time-derivatives. The pyro-electric properties of ferroelectrics arouse interest in developing the fractional-differential approach to simulating heat conductivity process.The present study deals with development and numerical implementation of fractal heat conductivity model for hereditary materials using the concepts of fractional-differential calculus applied to the simulation of intensive heating processes in ferroelectrics.The paper proposes a mathematical model governed through mixed initial-boundary value problem for partial differential equation containing a fractional time-derivative as well as nonlinear temperature dependence on the heat capacity. To solve the problem the computational algorithm was designed which is based on an analog of the Crank – Nicolson finite difference scheme combining with the Grunwald – Letnikov formula for fractional time-derivative approximation. The approximation of Neumann boundary condition is included into the finite difference problem statement using scheme of fictitious mesh points. The total system of linear algebraic equations is solved by sweep method.The designed application program allows one to perform the computer simulation of heat conductivity process in hereditary materials. The model verification was performed for numerical solving test problem with known analytical solution. The results of computational experiments are demonstrated for the example of estimating heat distribution in a typical ferroelectric crystal of TGS (triglycine sulfate) near the temperature of phase transition. The fractional derivative order was approximately evaluated to be ~0.7 at variation of this parameter. We applied the comparison of fractal model implementation results with experimental data related to the time when the ferroelectric crystal is heated to Curie temperature. These findings demonstrate that one needs to use the modified models at the analysis of the field effects arising in hereditary materials.

Fault Location Method in Power Network by Applying Accurate Information of Arrival Time Differences of Modal Traveling Waves

Faults may be generated in power networks as a result of bad weather, human activities, or some other factors. The complex structure of power network adds to the difficulty of fault location (FL). This paper presents a novel FL scheme by utilizing the information of the time differences of arrival (TDOAs) of modal traveling waves (MTWs) asynchronously sampled in the network. First, the fault area is determined by searching the minimum TDOA of MTWs. Then, by applying the fictitious fault point method in the fault area, the minimum accumulated difference is used to detect the fault line. Finally, the accurate FL is estimated by solving the objective function of the absolute distance between the FL and multiple FL candidates. Various simulations are carried out in the IEEE 30-bus system. The calculation results demonstrate that the method is not affected by fault impedance, inception angle, location, and noises to a certain extent.

A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations

Novel compact schemes based on half-step discretization are developed to solve two dimensional fourth order quasi-linear parabolic equations. These discretizations use a single computational cell and the boundary conditions are incorporated in a natural way without the use of any fictitious points or discretization. The schemes have second order accuracy in time and fourth order accuracy in space. The proposed methods are directly applicable to problems constituting singular terms. We also present operator splitting method for the solution of linear parabolic equations which allows the use of the one-dimensional tri-diagonal solver multiple times. It is shown that the operator splitting method is unconditionally stable. The numerical schemes have been successfully applied to the two-dimensional Extended Fisher–Kolmogorov equation, Sivashinsky equation, Boussinesq equation and vibrations problem. The illustrative results confirm the theoretical order of magnitude and accuracy of the proposed methods.

Modal analysis of the input impedance of wind instruments. Application to the sound synthesis of a clarinet

This paper investigates the modal analysis of wind instruments as seen from the input of their air column. Beside the treatment of analytical models, a particular emphasis is given to the analysis of measured input impedances. This requires special care because the measurements cover only a limited frequency band and are affected by some unknown errors. This paper describes how the Prony analysis and the Least Squares Complex Exponential (LSCE) classical techniques can be used in this context and how the main pitfalls can be avoided in their application. A physically acceptable method of reconstruction of the low frequency band is proposed. A technique using fictitious points in the high frequency range is described in order to ensure the passivity of the resonator in the whole frequency band. The principles of a real-time synthesis of clarinet sounds based on the modal representation of the resonator is given as an application, with a method to efficiently handle the modal representation during the transition between fingerings.

Free vibration analysis of dowelled rectangular isotropic thin plate on a Modified Vlasov soil type by using discrete singular convolution method

• There is a certain depth of soil beyond which frequency do not vary consistently. • Translational stiffness of Modified Vlasov soil is important in free vibration . • Rotational stiffness of Modified Vlasov soil is negligible in free vibration . • Dynamically activated depth affects frequency values for plates more than for beams. • Proposed method allows same calculation code for all boundary conditions by varying plate edge stiffness. The behaviour of plates resting on elastic foundations has wide interest because it is an important model for many engineering applications. Natural frequencies and corresponding mode shapes are guides to investigating the behaviour of a soil-structure system with time. Therefore, free vibration analysis constitutes a considerable part of research done on this topic. In this study, the effects of the stiffness of a surrounding Modified Vlasov soil type, the soil depth, and the aspect ratio on the free vibration of a thin rectangular dowelled plate on the Modified Vlasov soil type are investigated. We used a discrete singular convolution (DSC) with a Taylor series expansion method to apply the boundary conditions. The two formulas of the Taylor series expansion are used in the DSC to eliminate the fictitious points outside the physical domain. It is noticed that beyond a certain depth, neither the effective depth of the soil nor the, frequency varies consistently. Furthermore, the dynamically activated depth influences the frequency values for plates more than for beams. In addition, the effect of the elastic rotational stiffness of the surrounding soil is found to be negligible compared to that of the elastic vertical translational one. The results obtained can serve as a basis for the dynamic analysis of rigid pavements.

A New Method for Determining Optimal Regularization Parameter in Near‐Field Acoustic Holography

Tikhonov regularization method is effective in stabilizing reconstruction process of the near‐field acoustic holography (NAH) based on the equivalent source method (ESM), and the selection of the optimal regularization parameter is a key problem that determines the regularization effect. In this work, a new method for determining the optimal regularization parameter is proposed. The transfer matrix relating the source strengths of the equivalent sources to the measured pressures on the hologram surface is augmented by adding a fictitious point source with zero strength. The minimization of the norm of this fictitious point source strength is as the criterion for choosing the optimal regularization parameter since the reconstructed value should tend to zero. The original inverse problem in calculating the source strengths is converted into a univariate optimization problem which is solved by a one‐dimensional search technique. Two numerical simulations with a point driven simply supported plate and a pulsating sphere are investigated to validate the performance of the proposed method by comparison with the L‐curve method. The results demonstrate that the proposed method can determine the regularization parameter correctly and effectively for the reconstruction in NAH.

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.

Novel Discrete Singular Convolution for High-Frequency Vibration Analysis of Structural Elements

The prediction and control of high-frequency vibrations is of crucial importance to aerostructures. However, developing accurate and numerically stable methods for the analysis of high-frequency vibration is an extremely challenging task. In this paper, a novel discrete singular convolution algorithm is presented for efficient analysis of high-frequency vibration of structural elements. A nonregularized Lagrange’s delta sequence kernel based on harmonic functions is adopted. Taylor series expansion method is used to eliminate the degrees of freedom at fictitious points outside the physical domain, and thus various combinations of boundary conditions can be applied rigorously. Free vibration of beams and rectangular plates with and without two adjacent free edges is solved, and the results are compared with either exact solutions or results obtained by the differential quadrature method using much larger number of grid points. It is demonstrated that the performance of the proposed method for predicting high mode frequencies is excellent and better than existing algorithms. For the beam with both ends simply supported, all mode frequencies obtained by proposed method are very accurate. Such excellent feature of the algorithm was never reported before.

A Central Point-Based Discrete Shear Gap Method for Plates and Shells Analysis Using Triangular Elements

In this paper, discrete shear gap method (DSG)-based central point of the three-node triangular element (CP-DSG3) is presented for both linear and geometrically nonlinear analysis of plates and shells. A fictitious central point is introduced as the base point of the DSG to overcome the anisotropy of the formerly proposed formulation. In the CP-DSG3, the discrete shear gap at each field node is deduced using rotations of all three nodes in local coordinate system, which makes it an isotropic triangular plate and shell element. At the same time, the CP-DSG3 inherits the advantages of shear-locking-free from the DSG. Moreover, a series of numerical examples consisting of standard patch test, linear and geometrically nonlinear problems have been investigated and the results demonstrate the excellent performance and accuracy of the proposed CP-DSG3.

A New Fast Numerical Method Based on Off-Step Discretization for Two-Dimensional Quasilinear Hyperbolic Partial Differential Equations

We report a new 3-level implicit compact numerical method of order four in time and four in space based on off-step discretization for the solution of two-space-dimensional quasilinear hyperbolic equation [Formula: see text], [Formula: see text], [Formula: see text] defined in the region [Formula: see text], [Formula: see text], [Formula: see text]. We require only 19 grid points for the unknown variable [Formula: see text] and two extra off-step points each in [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to two-dimensional hyperbolic equations with singular coefficients, which is the main attraction of our work. We do not require any fictitious points for computation. The proposed method when applied to a two-dimensional damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve damped wave equation. Many benchmark problems are solved to confirm the fourth-order convergence of the proposed method.

Free vibration analysis of angle-ply symmetric laminated plates with free boundary conditions by the discrete singular convolution

The discrete singular convolution (DSC) is used for free vibration analysis of three-layer angle-ply symmetric laminated plates with free boundaries, including laminated plates with two adjacent free edges. During formulating the weighting coefficients of derivatives having different orders, two Taylor series expansions with different orders are used to eliminate the degrees of freedom at fictitious points outside the physical domain. Thus, the difficulty in handling free boundary conditions by using the DSC is overcome. Results are presented and compared with either exact solutions or the ones obtained by the differential quadrature method (DQM). It is shown that the DSC with the novel way to apply the boundary conditions yields very accurate lower mode frequencies as well as relatively accurate higher mode frequencies. The excellent performance of the DSC on high mode frequencies of beams and isotropic plates is retained and also independent of boundary conditions.

Discrete singular convolution and Taylor series expansion method for free vibration analysis of beams and rectangular plates with free boundaries

Since the excellent performance of the discrete singular convolution (DSC) for high order frequency analysis has not been demonstrated for rectangular plates with free edges and corners thus far, this paper proposes a novel approach in applying the DSC for vibration analysis of beams and rectangular plates with arbitrary boundary conditions. The Taylor series expansion method is used to apply the boundary conditions. The existing difficulty encountered at plate corner points is overcome by the proposed method. The essential idea of the novel method is that two Taylor series expansions with different orders are used to eliminate the fictitious points outside the physical domain. Various beam and rectangular plate problems are solved by the proposed DSC and the results are compared with exact solutions, finite element data and results obtained by the differential quadrature method (DQM) using much higher number of grid points. It is demonstrated that the Taylor series expansion method to apply the boundary conditions is general and accurate. It is also found that the performance of the proposed DSC is excellent and independent of boundary conditions.

The Gravitational Attraction between Hemispherical Masses

This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved side to curved side, and flat side to curved side. Expressions for the second and third variation are derived from the first, with the use of superposition and the well-known gravitational behavior of a spherical mass as equivalent to a point mass at its center. The study covers two masses of equal diameter and of different diameters, such that one is four times that of the other. Calculations are done for separations from zero to fifty times the radius of the larger of the two, which is effectively the asymptotic limit. It is demonstrated that at any separation, the force can be expressed as if the two hemispheres were point masses separated by a certain distance. Expressions for that distance and the location of the (fictitious) point masses within each hemisphere are presented. Unlike the case of two spherical masses, the location within their respective hemisphere is not necessarily the same for each point and both are dependent upon the separation between the two hemispheres. The calculation for the first variation is done in two ways. The first is a “brute force” multi-dimensional integral with the help of Wolfram Mathematica. The second is an axial expansion for the potential modified for off-axis locations by Legendre polynomials. With only a few terms in the expansion, the results of the second method are in extremely good agreement with those of the first. Finally, an interesting application to a split earth is presented.

Globally robust explicit model predictive control of constrained systems exploiting SVM‐based approximation

SummaryThis paper presents a systematic method to address the reduction of online computational complexity and infeasibility problem of explicit model predictive control for constrained systems under external disturbance. In feasible state space, in order to avoid the expensive database searching procedure, support vector machine‐based approximation is proposed to yield a novel unified explicit optimal control law rather than a piecewise affine one developed by explicit model predictive control. In infeasible state space, through constructing finite maximum control invariant sets around fictitious equilibrium points, a reachable controller is devised to steer the infeasible state asymptotically to the feasible state space without violating the hard constraint. Consequently, global robustness is guaranteed by introducing a minimum robust positively invariant set by means of the tube‐based technique, despite the coexistence of external disturbance and training error. Finally, the performance of the presently proposed control law is evaluated through three groups of numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.