Second-order Spatial Derivatives Research Articles

Turbocharger flow computations with the Space–Time Isogeometric Analysis (ST-IGA)

Abstract We focus on turbocharger computational flow analysis with a method that possesses higher accuracy in spatial and temporal representations. In the method we have developed for this purpose, we use a combination of (i) the Space–Time Variational Multiscale (ST-VMS) method, which is a stabilized formulation that also serves as a turbulence model, (ii) the ST Slip Interface (ST-SI) method, which maintains high-resolution representation of the boundary layers near spinning solid surfaces by allowing in a consistent fashion slip at the interface between the mesh covering a spinning surface and the mesh covering the rest of the domain, and (iii) the Isogeometric Analysis (IGA), where we use NURBS basis functions in space and time. The basis functions are spatially higher-order in all representations, and temporally higher-order in representation of the solid-surface and mesh motions. The ST nature of the method gives us higher-order accuracy in the flow solver, and when combined with temporally higher-order basis functions, a more accurate representation of the surface motion, and a mesh motion consistent with that. The spatially higher-order basis functions give us again higher-order accuracy in the flow solver, a more accurate, in some parts exact, representation of the surface geometry, and better representation in evaluating the second-order spatial derivatives. Using NURBS basis functions with a complex geometry is not trivial, however, once we generate the mesh, the computational efficiency is substantially increased. We focus on the turbine part of a turbocharger, but our method can also be applied to the compressor part and thus can be extended to the full turbocharger.

On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions

Purpose – The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions. Design/methodology/approach – Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained. Findings – It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation. Originality/value – This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

스플라인 정칙자를 사용한 투과 단층촬영을 위한 벌점우도 영상재구성

Recently, model-based iterative reconstruction (MBIR) has played an important role in transmission tomography by significantly improving the quality of reconstructed images for low-dose scans. MBIR is based on the penalized-likelihood (PL) approach, where the penalty term (also known as the regularizer) stabilizes the unstable likelihood term, thereby suppressing the noise. In this work we further improve MBIR by using a more expressive regularizer which can restore the underlying image more accurately. Here we used a spline regularizer derived from a linear combination of the two-dimensional splines with first- and second-order spatial derivatives and applied it to a non-quadratic convex penalty function. To derive a PL algorithm with the spline regularizer, we used a separable paraboloidal surrogates algorithm for convex optimization. The experimental results demonstrate that our regularization method improves reconstruction accuracy in terms of both regional percentage error and contrast recovery coefficient by restoring smooth edges as well as sharp edges more accurately.

A Metric for Reducing False Positives in the Computer-Aided Detection of Breast Cancer from Dynamic Contrast-Enhanced Magnetic Resonance Imaging Based Screening Examinations of High-Risk Women.

Magnetic resonance imaging (MRI)-enabled cancer screening has been shown to be a highly sensitive method for the early detection of breast cancer. Computer-aided detection systems have the potential to improve the screening process by standardizing radiologists to a high level of diagnostic accuracy. This retrospective study was approved by the institutional review board of Sunnybrook Health Sciences Centre. This study compares the performance of a proposed method for computer-aided detection (based on the second-order spatial derivative of the relative signal intensity) with the signal enhancement ratio (SER) on MRI-based breast screening examinations. Comparison is performed using receiver operating characteristic (ROC) curve analysis as well as free-response receiver operating characteristic (FROC) curve analysis. A modified computer-aided detection system combining the proposed approach with the SER method is also presented. The proposed method provides improvements in the rates of false positive markings over the SER method in the detection of breast cancer (as assessed by FROC analysis). The modified computer-aided detection system that incorporates both the proposed method and the SER method yields ROC results equal to that produced by SER while simultaneously providing improvements over the SER method in terms of false positives per noncancerous exam. The proposed method for identifying malignancies outperforms the SER method in terms of false positives on a challenging dataset containing many small lesions and may play a useful role in breast cancer screening by MRI as part of a computer-aided detection system.

Dispersion relation equation preserving FDTD method for nonlinear cubic Schrödinger equation

In this study we aim to solve the cubic nonlinear Schrödinger (CNLS) equation by the method of fractional steps. Over a time step from tn to tn+1, the linear part of the Schrödinger equation is solved firstly through four time integration steps. In this part of the simulation, the explicit symplectic scheme of fourth order accuracy is adopted to approximate the time derivative term. The second-order spatial derivative term in the linear Schrödinger equation is approximated by centered scheme. The resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation. In the second part of the simulation, the solution of the nonlinear equation is computed exactly thanks to the embedded invariant nature within each time increment. The proposed semi-discretized difference scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution. Several problems have been solved to show that application of this new finite difference scheme for the calculation of one- and two-dimensional Schrödinger equations can deemed conserve Hamiltonian quantities and preserve dispersion relation equation (DRE).

A PDF projection method: A pressure algorithm for stand-alone transported PDFs

In this paper, a new formulation of the projection approach is introduced for stand-alone probability density function (PDF) methods. The method is suitable for applications in low-Mach number transient turbulent reacting flows. The method is based on a fractional step method in which first the advection–diffusion–reaction equations are modelled and solved within a particle-based PDF method to predict an intermediate velocity field. Then the mean velocity field is projected onto a space where the continuity for the mean velocity is satisfied. In this approach, a Poisson equation is solved on the Eulerian grid to obtain the mean pressure field. Then the mean pressure is interpolated at the location of each stochastic Lagrangian particle. The formulation of the Poisson equation avoids the time derivatives of the density (due to convection) as well as second-order spatial derivatives. This in turn eliminates the major sources of instability in the presence of stochastic noise that are inherent in particle-based PDF methods. The convergence of the algorithm (in the non-turbulent case) is investigated first by the method of manufactured solutions. Then the algorithm is applied to a one-dimensional turbulent premixed flame in order to assess the accuracy and convergence of the method in the case of turbulent combustion. As a part of this work, we also apply the algorithm to a more realistic flow, namely a transient turbulent reacting jet, in order to assess the performance of the method.

Use of ANCF Surface Geometry in the Rigid Body Contact Problems: Application to Railroad Vehicle Dynamics

In the analysis of multibody system (MBS) dynamics, contact between two arbitrary rigid bodies is a fundamental feature in a variety of models. Many procedures have been proposed to solve the rigid body contact problem, most of which belong to one of the two categories: offline and online contact search methods. This investigation will focus on the development of a contact surface model for the rigid body contact problem in the case where an online three-dimensional nonconformal contact evaluation procedure, such as the elastic contact formulation—algebraic equations (ECF-A), is used. It is shown that the contact surface must have continuity in the second-order spatial derivatives when used in conjunction with ECF-A. Many of the existing surface models rely on direct linear interpolation of profile curves, which leads to first-order spatial derivative discontinuities. This, in turn, leads to erroneous spikes in the prediction of contact forces. To this end, an absolute nodal coordinate formulation (ANCF) thin plate surface model is developed in order to ensure second-order spatial derivative continuity to satisfy the requirements of the contact formulation used. A simple example of a railroad vehicle negotiating a turnout, which includes a variable cross-section rail, is tested for the cases of the new ANCF thin plate element surface, an existing ANCF thin plate element surface with first-order spatial derivative continuity, and the direct linear profile interpolation method. A comparison of the numerical results reveals the benefits of using the new ANCF surface geometry developed in this investigation.

Numerical solution of the Schrödinger equations via a reliable algorithm

In this paper, a reliable algorithm for solving Schrodinger equations is established. By second-order central difference scheme, the second-order spatial partial derivative of the Schrodinger equations are reduced to a system of first-order ordinary differential equations, that are solved by an efficient algorithm. The comparison of the numerical solution and the exact solution for some test cases shows that the given algorithm is easy and practical for extracting good approximate solutions of Schrodinger equations.

Statistics of spatial derivatives of velocity and pressure in turbulent channel flow

Statistical profiles of the first- and second-order spatial derivatives of velocity and pressure are reported for turbulent channel flow at Reτ = 590. The statistics were extracted from a high-resolution direct numerical simulation. To quantify the anisotropic behavior of fine-scale structures, the variances of the derivatives are compared with the theoretical values for isotropic turbulence. It is shown that appropriate combinations of first- and second-order velocity derivatives lead to (directional) viscous length scales without explicit occurrence of the viscosity in the definitions. To quantify the non-Gaussian and intermittent behavior of fine-scale structures, higher-order moments and probability density functions of spatial derivatives are reported. Absolute skewnesses and flatnesses of several spatial derivatives display high peaks in the near wall region. In the logarithmic and central regions of the channel flow, all first-order derivatives appear to be significantly more intermittent than in isotropic turbulence at the same Taylor Reynolds number. Since the nine variances of first-order velocity derivatives are the distinct elements of the turbulence dissipation, the budgets of these nine variances are shown, together with the budget of the turbulence dissipation. The comparison of the budgets in the near-wall region indicates that the normal derivative of the fluctuating streamwise velocity (∂u′/∂y) plays a more important role than other components of the fluctuating velocity gradient. The small-scale generation term formed by triple correlations of fluctuations of first-order velocity derivatives is analyzed. A typical mechanism of small-scale generation near the wall (around y+ = 1), the intensification of positive ∂u′/∂y by local strain fluctuation (compression in normal and stretching in spanwise direction), is illustrated and discussed.

Large time behavior for the incompressible magnetohydrodynamic equations in half-spaces

The weighted Lr-asymptotic behavior of the strong solution and its first-order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half-space. Further, the L∞-decay rates of the second-order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.

Retrieval of the atomic displacements in the crystal from the coherent X-ray diffraction pattern.

The retrieval of spatially resolved atomic displacements is investigated via the phases of the direct(real)-space image reconstructed from the strained crystal's coherent X-ray diffraction pattern. It is demonstrated that limiting the spatial variation of the first- and second-order spatial displacement derivatives improves convergence of the iterative phase-retrieval algorithm for displacements reconstructions to the true solution. This approach is exploited to retrieve the displacement in a periodic array of silicon lines isolated by silicon dioxide filled trenches.

Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform meshes. The nonuniform meshes are obtained by equidistributing a positive monitor function, which depends on the second-order spatial derivative of the singular component of the solution. The truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. Semilinear IBVPs are also solved. Numerical experiments are carried out to support the theoretical results.

Distributed proportional plus second-order spatial derivative control for distributed parameter systems subject to spatiotemporal uncertainties

In this paper, a robust distributed control design based on proportional plus second-order spatial derivative (P-sD\(^2\)) is proposed for exponential stabilization and minimization of spatial variation of a class of distributed parameter systems (DPSs) with spatiotemporal uncertainties, whose model is represented by parabolic partial differential equations with spatially varying coefficients. Based on the Lyapunov’s direct method, a robust distributed P-sD\(^2\) controller is developed to not only exponentially stabilize the DPS for all admissible spatiotemporal uncertainties but also minimize the spatial variation of the process. The outcome of the robust distributed P-sD\(^2\) control problem is formulated as a spatial differential bilinear matrix inequality (SDBMI) problem. A local optimization algorithm that the SDBMI is treated as a double spatial differential linear matrix inequality (SDLMI) is presented to solve this SDBMI problem. Furthermore, the SDLMI optimization problem can be approximately solved via the finite difference method and the existing convex optimization techniques. Finally, the proposed design method is successfully applied to feedback control problem of the FitzHugh–Nagumo equation.

Beam-patterns of a collocated unit of three orthogonally oriented second-order directional sensors

A microphone or a hydrophone measures the incident acoustic wavefield as a scalar field of pressure. Underlying this pressure scalar is the particle-velocity vector, which constitutes the spatial gradient of the wavefield. Each Cartesian component of this gradient vector may be directly measured (without computing any spatial derivative) by a particle-velocity sensor aligned in parallel to that Cartesian coordinate. The entire 3 × 1 gradient vector may be measured at any point in space, using three such particle-velocity sensors, collocated but orthogonally oriented among themselves. Such a collocated triad has an array manifold independent of signal frequency, thus uncoupling azimuth-elevation beamforming from the frequency-dimension. This triad’s beam-pattern has already been investigated by Wong and Chi ["Beam patterns of an underwater acoustic vector hydrophone located away from any reflecting boundary,” IEEE J. Ocean. Eng. 27(3), 628–637 (2002)]. While the particle-velocity sensor measures the first-order spatial derivative of the pressure field, a second-order spatial derivative of the pressure-field could likewise be defined and be measured by a “second-order directional sensor” that would have sharper directivity. This paper follows up that investigation, but now for second-order directional sensors.

Simulation of second-order wave interaction with fixed and floating structures in time domain

The application of a time-domain second-order method in the numerical simulation of the nonlinear wave interaction with surface piercing fixed and floating circular cylinders is described. In this approach, Taylor series expansions are applied to the boundary conditions on the instantaneous free water surface and body surface, and Stokes perturbation procedure is then used to establish corresponding boundary value problems at the first and second order of wave steepness with respect to a time-independent fluid domain. A boundary element method based on a B-spline function expansion is adopted to calculate the wave field at each time step, and the time stepping scheme is implemented to predict the boundary conditions at the next time step. The combined diffraction–radiation problem is solved when the wave interaction with a floating body is considered, unlike treating them separately in the conventional frequency domain method. Additionally, a mathematical transformation is derived to remove the second-order spatial derivative appearing in the body boundary condition that may lead to the potential loss of accuracy. As an illustration, numerical results of the wave diffraction around a bottom-mounted circular cylinder, wave radiation by a circular cylinder undergoing specified motions and wave interaction with a freely and moored floating circular cylinder are presented. Comparisons of the wave forces on the fixed and floating structures with the second-order frequency domain and fully nonlinear solutions indicate that the present numerical method is accurate, efficient and stable.

Relation between the propagator matrix of geodesic deviation and the second-order derivatives of the characteristic function

In the Finsler geometry, which is a generalization of the Riemann geometry, the metric tensor also depends on the direction of propagation. The basics of the Finsler geometry were formulated by William Rowan Hamilton in 1832. Hamilton’s formulation is based on the first-order partial differential Hamilton–Jacobi equations for the characteristic function which represents the distance between two points. The characteristic function and geodesics together with the geodesic deviation in the Finsler space can be calculated efficiently by Hamilton’s method. The Hamiltonian equations of geodesic deviation are considerably simpler than the Riemannian or Finslerian equations of geodesic deviation. The linear ordinary differential equations of geodesic deviation may serve to calculate geodesic deviations, amplitudes of waves and the second-order spatial derivatives of the characteristic function or action. The propagator matrix of geodesic deviation contains all the linearly independent solutions of the equations of geodesic deviation. In this paper, we use the Hamiltonian formulation to derive the relation between the propagator matrix of geodesic deviation and the second-order spatial derivatives of the characteristic function in the Finsler geometry. We assume that the Hamiltonian function is a positively homogeneous function of the second degree with respect to the spatial gradient of the characteristic function, which corresponds to the Riemannian or Finslerian equations of geodesics and of geodesic deviation. The derived equations, which represent the main result of this paper, are applicable to the Finsler geometry, the Riemann geometry, and their various applications such as general relativity or the high-frequency approximations of wave propagation.

Globally optimal finite-difference schemes based on least squares

Spatial finite-difference (FD) coefficients are usually determined by the Taylor-series expansion (TE) or optimization methods. The former can provide high accuracy on a smaller wavenumber or frequency zone, and the latter can give moderate accuracy on a larger zone. Present optimization methods applied to calculate FD coefficients are generally gradient-like or global optimization-like algorithms, and thus iterations are involved. They are more computationally expensive, and sometimes the global solution may not be found. I examined second-order spatial derivatives and computed the optimized spatial FD coefficients over the given wavenumber range using the least-squares (LS) method. The results indicated that the FD accuracy increased with increasing operator length and decreasing wavenumber range. Therefore, for the given error and operator length, globally optimal spatial FD coefficients can be easily obtained. Some optimal FD coefficients were given. I developed schemes to obtain optimized LS-based spatial FD coefficients by minimizing the relative error of space-domain dispersion relation for second-order derivatives and time-space-domain dispersion relation for the acoustic wave equation. I discovered that minimizing the relative error of the space-domain dispersion relation provides less phase velocity error for small wavenumbers, compared to minimizing the absolute error. I also found that minimizing the relative error of the time-space-domain dispersion relation can reduce relative errors of phase velocity. Accuracy analysis demonstrated the correctness and advantage of schemes. I gave three examples of 2D acoustic FD modeling for a homogeneous, a large velocity-contrast, and a heterogeneous model, respectively. LS-based spatial FD operators have variable lengths for different velocities. Modeling examples demonstrated that the proposed LS-based FD scheme can maintain the same modeling precision while using a shorter spatial FD operator length, thus reducing the computation cost relative to conventional TE-based FD schemes, particularly for the higher order.

The parameter uniform numerical method for singularly perturbed parabolic reaction–diffusion problems on equidistributed grids

This article presents the study of singularly perturbed parabolic reaction–diffusion problems with boundary layers. To solve these problems, we use a modified backward Euler finite difference scheme on layer adapted nonuniform meshes at each time level. The nonuniform meshes are obtained by equidistribution of a positive monitor function, which involves the second-order spatial derivative of the singular component of the solution. The equidistributing monitor function at each time level allows us to use this technique to non-linear parabolic problems. The truncation error and the stability analysis are obtained. Parameter–uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.

Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform grids. The nonuniform grids are obtained by equidistribution of a positive monitor function, which is a linear combination of a constant and the second-order spatial derivative of the singular component of the solution on every temporal level. Truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.

An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions

We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the enclosed volume of a vesicle are written as functionals of a continuous phase-field, which describes the interface in a smeared manner. Such functionals involve up to second-order spatial derivatives of the phase-field, leading to fourth-order Euler–Lagrange partial differential equations (PDE). The solutions develop sharp internal layers in the vicinity of the putative interface, and are nearly constant elsewhere. Thanks to the smoothness of the local maximum-entropy (max-ent) meshfree basis functions, we approximate numerically this high-order phase-field model with a direct Ritz–Galerkin method. The flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions. Thus, the proposed approach is more efficient than common tensor product methods (e.g. finite differences or spectral methods), and simpler than unstructured C0 finite element methods, applicable by reformulating the model as a system of second-order PDE. The proposed method, implemented here under the assumption of axisymmetry, allows us to show numerical evidence of convergence of the phase-field solutions to the sharp interface limit as the regularization parameter approaches zero. In a companion paper, we present a Lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid.