Neumann Scheme Research Articles

A Robin‐Neumann scheme with quasi‐Newton acceleration for partitioned fluid‐structure interaction

AbstractThe Dirichlet–Neumann scheme is the most common partitioned algorithm for fluid‐structure interaction (FSI) and offers high flexibility concerning the solvers employed for the two subproblems. Nevertheless, it is not without shortcomings: to begin with, the inherent added‐mass effect often destabilizes the numerical solution severely. Moreover, the Dirichlet–Neumann scheme cannot be applied to FSI problems in which an incompressible fluid is fully enclosed by Dirichlet boundaries, as it is incapable of satisfying the volume constraint. In the last decade, interface quasi‐Newton methods have proven to control the added‐mass effect and substantially speed up convergence by adding a Newton‐like update step to the Dirichlet–Neumann coupling. They are, however, without effect on the incompressibility dilemma. As an alternative, the Robin‐Neumann scheme generalizes the fluid's boundary condition to a Robin condition by including the Cauchy stresses. While this modification in fact successfully tackles both drawbacks of the Dirichlet–Neumann approach, the price to be paid is a strong dependency on the Robin weighting parameter, with very limited a priori knowledge about good choices. This work proposes a strategy to merge these two ideas and benefit from their combined strengths. The resulting quasi‐Newton‐accelerated Robin‐Neumann scheme is compared to both Robin‐ and Dirichlet–Neumann variants. The numerical tests demonstrate that it does not only provide faster convergence, but also massively reduces the influence of the Robin parameter, mitigating the main drawback of the Robin‐Neumann algorithm.

An improvised collocation algorithm to solve generalized Burgers’–Huxley equation

In the proposed work, an improvised collocation technique with cubic B-spline as basis functions is applied to obtain the numerical solution of non-linear generalized Burgers’–Huxley equation, which has application in the soliton theory. In this technique, posteriori corrections are made to the cubic B-spline interpolant and its higher-order derivatives, which leads to enhancement in the order of convergence in the spatial domain. The temporal domain is discretized using a weighted finite difference scheme, to obtain the solution at each time level and the spatial domain is discretized using the improvised cubic B-spline collocation method. The stability analysis is carried out using the von Neumann scheme and the technique is found to be unconditionally stable. The theoretical proof of the order of convergence is discussed in detail using Green’s function approach. The L_{2}, L_{infty }, and absolute error norms are calculated as well as compared with the results available in the literature, which shows the improvement and efficacy of the proposed technique over the existing ones.

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

In the present work, the numerical solution of the Kuramoto–Sivashinsky (KS) equation is obtained using an improvised quintic B‐spline extrapolated collocation technique. This equation helps in the study of wave production in dissipative medium, investigates the hydrodynamic uncertainty in laminar flames, describes the turbulence of diverse physical processes, and so on. In this work, splines are used due to their higher smoothness property and the sparse nature of the matrices corresponding to the B‐splines. The improvised B‐splines are formed by forcing the quintic B‐spline interpolant to satisfy the interpolatory and some special end conditions. These basis functions are used in the collocation method for space integration and a weighted finite difference scheme is used for temporal domain integration. The stability of the technique is analyzed using the von Neumann scheme, and it is found to be unconditionally stable. The proposed method is found to be sixth‐order convergent in space and second‐order convergent in the time direction. The theoretical order of convergence is matching perfectly with the numerical one. The efficiency of this scheme is demonstrated by applying it to a number of examples. The L2, L∞, and global relative errors are calculated and compared with the previous work, especially with those where quintic B‐splines are used as basis functions. Also, the behavior of some KS equations is discussed for which the exact solution is not available. The aim of the paper is to show that such an improvised technique can be implemented to solve partial differential equations like the KS equation.

Multistate Magnetic Domain Wall Devices for Neuromorphic Computing

In recent years, neuromorphic computing has been intensively investigated, to take over the conventional or von Neumann scheme. Herein, the advantages of memristors as neurons and synapses are discussed. After a brief introduction to biological neurons and synapses, focus is put on spin‐based devices, including magnetic tunnel junction (MTJ) and domain wall devices. Certain materials and device designs aim at mimicking synapses’ functionality, whereas others gather both neurons and synapses. The advancements in spin‐based memory applications are of great advantage for neuromorphic computing and their implementation is presented herein.

An efficient numerical scheme for the simulation of time-fractional nonhomogeneous Benjamin-Bona-Mahony-Burger model

The Benjamin-Bona-Mahony-Burger (BBM-Burger) equation is important for explaining the unidirectional propagation of long waves in nonlinear dispersion systems. This manuscript proposes an algorithm based on cubic B-spline basis functions to study the nonhomogeneous time fractional model of BBM-Burger via Caputo derivative. The discretization of fractional derivative is achieved by L1 formula, while the temporal and spatial derivatives are interpolated by means of Crank-Nicolson and forward finite difference scheme together with B-spline basis functions. The performance of the Cubic B-spline scheme (CBS) is examined by three test problems with homogeneous initial and boundary conditions. The obtained results are found to be in good agreement with the exact solutions. The behaviour of travelling wave is studied and presented in the form of tables and graphics for various values of α and t. A linear stability analysis, based on the von Neumann scheme, shows that the CBS is unconditionally stable. Moreover, the accuracy of the scheme is quantified by computing error norms.

A weakly compressible hybridizable discontinuous Galerkin formulation for fluid–structure interaction problems

A scheme for the solution of fluid–structure interaction (FSI) problems with weakly compressible flows is proposed in this work. A novel hybridizable discontinuous Galerkin (HDG) method is derived for the discretization of the fluid equations, while the standard continuous Galerkin (CG) approach is adopted for the structural problem. The chosen HDG solver combines robustness of discontinuous Galerkin (DG) approaches in advection-dominated flows with higher-order accuracy and efficient implementations. Two coupling strategies are examined in this contribution, namely a partitioned Dirichlet–Neumann scheme in the context of hybrid HDG–CG discretizations and a monolithic approach based on Nitsche’s method, exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions. Numerical experiments show optimal convergence of the HDG and CG primal and mixed variables and superconvergence of the postprocessed fluid velocity. The robustness and the efficiency of the proposed weakly compressible formulation, in comparison to a fully incompressible one, are also highlighted on a selection of two and three dimensional FSI benchmark problems.

An approximation to the solution of time fractional modified Burgers’ equation using extended cubic B-spline method

This paper aims to investigate numerical solution of time fractional modified Burgers’ equation via Caputo fractional derivative. Extended cubic B-spline collocation scheme which reduces the nonlinear equation to a system of linear equation in the matrix form has been used for this investigation. The nonlinear part in fractional partial differential equation has been linearized by modified form of the existing method. The validity of proposed scheme has been examined on three test problems and effect of viscosity $$\nu $$ and $$\alpha \ \epsilon \ [0, 1]$$ variation displayed in 2D and 3D graphics. Moreover, the working of proposed scheme has also been explained through algorithm and stability of proposed scheme has been analyzed by von Neumann scheme and has proved to be unconditionally stable. To quantify the accuracy of suggested scheme, error norms have been computed.

Transient 3D conjugate heat transfer simulation of a rectangular GOX–GCH4 rocket combustion chamber and validation

This paper aims to predict the transient temperature fields of rocket combustion chambers. This is important for design and lifetime estimation. A thermally coupled simulation (conjugate heat transfer) model was developed and validated using a combustion chamber experiment with rectangular cross-section and the propellant combination gaseous oxygen–methane. The simulation is based on a thermal finite element model of the structure, Reynolds-averaged Navier–Stokes simulations of nine cooling channel flows and a Reynolds-averaged Navier–Stokes simulation with flamelet extension of the hot-gas flow. A pseudo-transient, implicit Dirichlet–Neumann scheme is used for calculating the equilibrium of structure and fluids. A comparison with experiment shows good agreement with the axial, vertical and transverse temperature distribution in the structure, the transient behavior and the wall heat flux through the cooling channels. It is shown that wall functions in the coolant domains can efficiently decrease the computational effort.

A Finite Element Model for the Simulation of the UL-94 Burning Test

The tendency of the polymers to melt and drip when they are exposed to external heat source play a very important role in the ignition and the spread of fire. Numerical simulation is a promising methodology for predicting this behaviour. In this paper, a computational procedure that aims at analyzing the combustion, melting and flame spread of polymer is presented. The method models the polymer using a Lagrangian framework adopting the particle finite element method framework while the surrounding air is solved on a fixed Eulerian mesh. This approach allows to treat naturally the polymer shape deformations and to solve the thermo-mechanical problem in a staggered fashion. The problems are coupled using an embedded Dirichlet–Neumann scheme. A simple combustion model and a radiation modeling strategy are included in the air domain. With this strategy the burning of a polypropylene specimen under UL-94 vertical test conditions is simulated. Input parameters for the modelling (density, specific heat, conductivity and viscosity) and results for the validation of the numerical model has been obtained from different literature sources and by IMDEA burning a specimen of dimensions of $$148 \times 13 \times 3.2\,{\mathrm {mm}}^3$$ . Temperature measurements in the polymer have been recorder by means of three thermocouples exceeding the 1000 K. Simultaneously a digital camera was used to record the burning process. In addition, thermal decomposition of the material (Arrhenius coefficient $${\mathrm {A}}=7.14 \times 10^{16}\,{\mathrm {min}}^{-1}$$ and activation energy $${\mathrm {E}}=240.67\,{\mathrm {kJ/mol}}$$ ) as and changes in viscosity ( $$\mu $$ ) as a function of temperature were obtained. Finally, a good agreement between the experimental and the numerical can be seen in terms of shape of the polymer as well as in the temperature evolution inside the polymer.

Methods of Reverberation Mapping. I. Time-lag Determination by Measures of Randomness

Abstract A class of methods for measuring time delays between astronomical time series is introduced in the context of quasar reverberation mapping, which is based on measures of randomness or complexity of the data. Several distinct statistical estimators are considered that do not rely on polynomial interpolations of the light curves nor on their stochastic modeling, and do not require binning in correlation space. Methods based on von Neumann’s mean-square successive-difference estimator are found to be superior to those using other estimators. An optimized von Neumann scheme is formulated, which better handles sparsely sampled data and outperforms current implementations of discrete correlation function methods. This scheme is applied to existing reverberation data of varying quality, and consistency with previously reported time delays is found. In particular, the size–luminosity relation of the broad-line region in quasars is recovered with a scatter comparable to that obtained by other works, yet with fewer assumptions made concerning the process underlying the variability. The proposed method for time-lag determination is particularly relevant for irregularly sampled time series, and in cases where the process underlying the variability cannot be adequately modeled.

Numerical modelling of physical processes in a ballistic laboratory setup with a tapered adapter and plastic piston used for obtaining high muzzle velocities

Numerical modelling of a ballistic setup with a tapered adapter and plastic piston is considered. The processes in the firing chamber are described within the framework of quasi- one-dimensional gas dynamics and a geometrical law of propellant burn by means of Lagrangian mass coordinates. The deformable piston is considered to be an ideal liquid with specific equations of state. The numerical solution is obtained by means of a modified explicit von Neumann scheme. The calculation results given show that the ballistic setup with a tapered adapter and plastic piston produces increased shell muzzle velocities by a factor of more than 1.5-2.

Two different methods for numerical solution of the modified Burgers' equation.

A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.

Negative Entropy and Black Hole Information

Based on negative entropy in entanglement, it is shown that a single-system Copenhagen measurement protocol is equivalent to the two-system von Neumann scheme with the memory filling up the system with negative information similar to the Dirac sea of negative energy. After equating the two quantum measurement protocols, we then apply this equivalence to the black hole radiation. That is, the black hole evaporation corresponds to the quantum measurement process and the two evaporation approaches, the observable-based single-system and the two-system entanglement-based protocols, can be made equivalent using quantum memory. In particular, the measurement choice, \theta, with the memory state inside the horizon in the entanglement-based scheme is shown to correspond to the observable of the measurement choice, \theta, outside the horizon in the single-system protocol, that is, O_{\theta}^{out} = Q_{\theta}^{in}. This indicates that the black hole as quantum memory is filling up with negative information outside the horizon, and its entropy corresponds to the logarithm of a number of equally probable measurement choices. This shows that the black hole radiation is no different than ordinary quantum theory.

Numerical schemes for the Iterative Nonlinear Contrast Source method

Nonlinear acoustics is gaining importance for medical acoustical imaging and high intensity focused ultrasound. With the latter one, high-amplitude acoustic wave fields are used to damage or kill cancer cells. For accurate treatment planning, a full-wave method which can model the propagation and scattering of the nonlinear field in attenuative, heterogeneous media is required. The Iterative Nonlinear Contrast Source (INCS) method is a full-wave method originally developed for homogeneous medium. It recasts the Westervelt equation into an integral equation which can be solved using a Neumann scheme. For heterogeneous media, the same approach results in additional contrast source terms. When these additional contrast sources become strong, convergence of the Neumann scheme may become an issue. To overcome this problem, the Westervelt equation may be linearized and the resulting linear integral equation may be solved with more advanced schemes such as Bi-CGSTAB. Restart strategies may be applied to eliminate systematic errors in the higher harmonics caused by the linearization. However, for realistic wave speed contrasts convergence remains problematic. To overcome these limitations, schemes such as steepest descent may be applied on the original nonlinear integral equation. In the present talk, the different schemes and their pros and cons will be discussed.

Efficient Generation of Random Bits From Finite State Markov Chains

The problem of random number generation from an uncorrelated random source (of unknown probability distribution) dates back to von Neumann's 1951 work. Elias (1972) generalized von Neumann's scheme and showed how to achieve optimal efficiency in unbiased random bits generation. Hence, a natural question is what if the sources are correlated? Both Elias and Samuelson proposed methods for generating unbiased random bits in the case of correlated sources (of unknown probability distribution), specifically, they considered finite Markov chains. However, their proposed methods are not efficient or have implementation difficulties. Blum (1986) devised an algorithm for efficiently generating random bits from degree-2 finite Markov chains in expected linear time, however, his beautiful method is still far from optimality on information-efficiency. In this paper, we generalize Blum's algorithm to arbitrary degree finite Markov chains and combine it with Elias's method for efficient generation of unbiased bits. As a result, we provide the first known algorithm that generates unbiased random bits from an arbitrary finite Markov chain, operates in expected linear time and achieves the information-theoretic upper bound on efficiency.

A full wave method for modeling nonlinear acoustic wave fields in media with inhomogeneous wave speed, attenuation, and parameter of nonlinearity

The iterative nonlinear contrast source (INCS) method is a numerical method that has originally been developed for modeling transient nonlinear acoustic wave fields in homogeneous (i.e., spatially independent) lossless or lossy media. Starting from the Westervelt equation, the INCS method considers the nonlinear term of the latter as a nonlinear contrast source in an otherwise linear wave equation and iteratively solves the corresponding nonlinear integral equation using a Neumann scheme. By adding an attenuative contrast source, the method has recently been extended to deal with spatially dependent attenuation, and a spatially dependent parameter of nonlinearity has also been introduced. This presentation is about the introduction of an additional contrast source that account for a spatially dependent wave speed. In this case, convergence problems arise with the Neumann scheme. This problem is solved by replacing the Neumann scheme with a conjugate gradient scheme in which the error functional is based on the complete nonlinear integral equation. Results show that this approach yields very accurate results for the nonlinear wave field (tested up to the fifth harmonic) in media with a spatially dependent wave speed, attenuation, and parameter of nonlinearity, as encountered in realistic medical ultrasound applications.

A linearized contrast source method for full-wave modeling of nonlinear acoustic wave fields in media with strong and inhomogeneous attenuation

The iterative nonlinear contrast source (INCS) method is a numerical method originally developed for modeling transient nonlinear acoustic wave fields in homogeneous (i.e., spatially independent) lossless or lossy media. This method is based on the Westervelt equation and assumes that the nonlinear term describes a distributed contrast source in a linear background medium. Next, the equivalent nonlinear integral equation is iteratively solved by means of a Neumann scheme. In medical diagnostic ultrasound, the observed acoustic nonlinearity is weak, resulting in a weak nonlinear contrast source and a fast convergence of the scheme. To model spatially dependent attenuation, the Westervelt equation has recently been extended with an attenuative contrast source. This presentation deals with the situation of strong local attenuation. In that case, the attenuative contrast source is large and the Neumann scheme may fail to converge. By linearizing the nonlinear contrast source, the integral equation becomes linear and can be solved with more robust schemes, e.g., Bi-CGSTAB. Numerical simulations show that this approach yields convergent results for nonlinear acoustic fields in realistic media with strong local attenuation. Moreover, it is shown that in practical situations, the error due to the linearization remains sufficiently small for the first few harmonics.

A linearized contrast source method for full-wave modeling of nonlinear acoustic wave fields in homogeneous media

Many novel medical echography modalities, like super harmonic imaging (SHI), employ imaging of higher harmonics arising from nonlinear propagation. Design optimization of dedicated imaging probes for these modalities requires accurate computation of the higher harmonic beam profiles. The existing iterative nonlinear contrast source method can perform such simulations. With this method, the nonlinear term of the Westervelt equation is considered to represent a distributed contrast source in a linear background medium. The full transient nonlinear acoustic wave field follows from the Neumann iterative solution of the corresponding integral equation. Each iteration involves the spatiotemporal convolution of the background Green's function with an estimate of the contrast source, and appropriate filtering enables a discretization approaching two points per smallest wavelength or period. To further reduce the computational load and to anticipate extension of the method, in this presentation, the effect of linearizing the nonlinear contrast source is investigated. This enables the replacement of the Neumann scheme by more efficient schemes. Numerical simulations show that for design purposes the effect of linearization on the harmonic components involved with SHI remains sufficiently small. Moreover, it is shown that a significant decrease in computational costs may be achieved by using a Bi-CGSTAB scheme.

A contrast source method for nonlinear acoustic wave fields in media with spatially inhomogeneous attenuation

Experimental data reveals that attenuation is an important phenomenon in medical ultrasound. Attenuation is particularly important for medical applications based on nonlinear acoustics, since higher harmonics experience higher attenuation than the fundamental. Here, a method is presented to accurately solve the wave equation for nonlinear acoustic media with spatially inhomogeneous attenuation. Losses are modeled by a spatially dependent compliance relaxation function, which is included in the Westervelt equation. Introduction of absorption in the form of a causal relaxation function automatically results in the appearance of dispersion. The appearance of inhomogeneities implies the presence of a spatially inhomogeneous contrast source in the presented full-wave method leading to inclusion of forward and backward scattering. The contrast source problem is solved iteratively using a Neumann scheme, similar to the iterative nonlinear contrast source (INCS) method. The presented method is directionally independent and capable of dealing with weakly to moderately nonlinear, large scale, three-dimensional wave fields occurring in diagnostic ultrasound. Convergence of the method has been investigated and results for homogeneous, lossy, linear media show full agreement with the exact results. Moreover, the performance of the method is demonstrated through simulations involving steered and unsteered beams in nonlinear media with spatially homogeneous and inhomogeneous attenuation.

Modeling nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity.

The refraction and scattering of nonlinear acoustic waves play an important role in the realistic application of medical ultrasound. One cause of these effects is the tissue dependence of the nonlinear medium behavior. A method that is able to model those effects is essential for the design of transducers for novel ultrasound modalities. Starting from the Westervelt equation, nonlinear pressure wave fields can be modeled via a contrast source formulation, as has been done with the INCS method. An extension of this method will be presented that can handle inhomogeneities in the coefficient of nonlinearity. The contrast source formulation results in an integral equation, which is solved iteratively using a Neumann scheme. The convergence of this scheme has been investigated for relevant media (e.g., blood, brain, and liver). Further, as an example, the method has been applied to compute the 1-D nonlinear acoustic wave field in an inhomogeneous medium insonified by a 1 MHz Gaussian pulse propagating up to 100 mm. The results show that the method is able to predict the propagation and the scattering effects of nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity. This motivates a similar extension of the 3-D INCS method.