Nonlinear Iterative Scheme Research Articles

Wall-Modeled Large-Eddy Simulation Method for Unstructured-Grid Navier–Stokes Solvers

This paper reports on the implementation and assessment of a wall-modeled large-eddy simulation (WMLES) methodology in an unstructured-grid, node-centered flow solver, FUN3D, that is developed and supported at the NASA Langley Research Center. Finite-volume (FV) and finite-element (FE) discretization schemes considered in the study provide formal second-order spatial accuracy. Large-eddy simulations (LES) resolve large-scale turbulent-flow features, and small-scale effects are modeled using the Vreman subgrid-scale model. At solid-wall boundaries, a shear-stress model is employed to provide a proper boundary-flux closure. The nonlinear equations are integrated in time using either an optimized backward difference formula or an implicit multistage Runge–Kutta temporal scheme. The implicit equations at each time step are solved by strong nonlinear iteration schemes. WMLES demonstrations are shown for two high-lift configurations, namely, the McDonnell Douglas 30P30N multi-element airfoil and a NASA High-Lift Common Research Model. Results show that the WMLES approaches implemented in the FV and FE discretization methods produce consistent solutions and are capable of capturing key aerodynamic characteristics and flow structures for high-lift configurations at a wide range of angles of attack, including maximum-lift conditions. In the 30P30N example, correct trends in the variations of integrated aerodynamic forces and moments, surface pressure distributions, and boundary-layer profiles are captured as the Reynolds number is increased.

Study of the Accuracy of the Ronen Method at Material Interfaces

The Ronen Method (RM) is a nonlinear iterative scheme that demands successive resolutions of the diffusion equation, where local diffusion constants are modified to reproduce more accurate estimates of the neutron currents by a transport operator. The methodology is currently formulated using the formalism of the collision probability method for evaluation of the current. The RM was recently tested on a complete suite of one-dimensional (1-D) multigroup benchmark problems. Small differences in the flux (less than 2%) were reported at material interfaces and close to the vacuum boundary with respect to the reference solution from transport. This work investigates first a possible numerical equivalence between transport and diffusion in some representative 1-D problems from the same benchmark test suite. The equivalence is sought with optimal diffusion coefficients computed using reference transport solutions that allow for adjusting the diffusion model. The RM, which attempts to obtain such equivalent diffusion coefficients without knowing the reference solution, is then compared to the optimal coefficients. The accuracy of the flux distribution at material interfaces is investigated for different approximations of the vacuum boundary and by decreasing progressively the RM convergence tolerance set in the iterative scheme. Using tighter convergence criteria, the RM calculates more accurate flux distributions at all material interfaces, regardless of the value of the diffusion coefficient and the extrapolated distance set at the beginning of the iterative scheme. Maximum flux deviations are remarkably reduced when the RM convergence tolerance is set to eight or more significant digits, leading to improvements in the flux deviation of two orders in magnitude and providing numerical proof for equivalence with transport in the tested configurations.

On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations

The aim of this work is to develop a conservative, positivity-preserving (PP), nonlinear finite volume (FV) scheme for the multi-term nonlocal Nagumo-type equations on distorted meshes. These equations involve anisotropic diffusion tensor coefficients, and the solution exhibits a weak singularity at t=0. To achieve this, we combine the L1 scheme on a graded mesh with a PP-FV scheme on randomly distorted meshes, resulting in a conservative, positivity-preserving nonlinear L1-PP-FV scheme. By employing a nonlinear-splitting technique, we derive a series of simple nonlinear discrete equations that can be solved implicitly. We also provide proof that both the proposed FV scheme and the nonlinear iteration (NI) scheme are capable of maintaining positivity. Finally, we present numerical results to validate our theoretical analysis.

Asymptotic analysis of transonic shocks in divergent nozzles with respect to the expanding angle

In this paper we study the asymptotic behavior of the transonic shock solutions in divergent nozzles as the expanding angle goes to zero. It is well-known that there exist infinite shock solutions for steady 1-D flows in a flat nozzle with the position of the shock front being arbitrary, while there exists a unique shock solution in an divergent nozzle as the pressure at the exit is given within an appropriate interval. By analyzing the asymptotic behavior of the shock solutions as the expanding goes to zero, we are also trying to figure out a criterion which may be used to select the physical one among all shock solutions in the flat nozzle. It finally turns out that the limit shock solution as the expanding angle goes to zero strongly depends on the asymptotic behavior of the receiver pressure, which is assumed to be a function of the expanding angle, imposed at the exit. In particular, as the Mach number of the flow at the entrance is larger than γ+32, the function of the receiver pressure can be set identical to the value of the pressure behind the shock front for the flat nozzle. Then the limit shock solution may be considered as the physical one for the flat nozzle, and the limit position of the shock front being the admissible position of the shock front. To show these results, one of the key steps is to establish quantitative estimates for the position of the shock front for the given pressure at the exit. To this end, a free boundary problem for the linearized Euler system will be proposed which gives an approximation of the shock solution, then a nonlinear iteration scheme could be carried out to approach the shock solution. Moreover, the error estimates between the exact shock solution and the approximation are also established, which give quantitative information on the position of the shock front.

Transonic shocks for 2D steady exothermically reacting Euler flows in a finite nozzle

This paper concerns the existence of transonic shocks for steady exothermically reacting Euler flows in an almost flat nozzle with the small rate of the exothermic reaction. One of the key points is to quantitatively determine the position of the shock front in the nozzle. We focus on contributions of the perturbation of the flat nozzle and the exothermic reaction in determining the position of the shock front by comparing the orders of σ and κ, where σ represents the scale of the perturbation of the flat nozzle and κ represents the rate of the exothermic reaction. To this end, a free boundary problem for the linearized reacting Euler system is proposed to catch an approximating position of the shock front and the associated approximating shock solution, with which the existence of a shock solution close to it can be established via a nonlinear iteration scheme. One of the key steps is to solve the nonlinear equation derived from the solvability condition for the elliptic sub-problem in the domain of the subsonic flow behind the shock front, which determines the free boundary of the proposed problem. Four typical cases are analyzed, which describe possible interactions between the geometry of the nozzle boundary and the exothermic reaction. The results also manifest that the exothermic reaction has a stabilization effect on transonic shocks in the nozzles.

The Energy-Efficient Resource Allocation of Multi-Modal Perception for Affective Brain-Computer Interactions Based on Non-Linear Iterative Prediction Scheme

<p>For the whole environmental settings in this research, the conventional affective brain-computer interactions can not build a good performance on energy-efficient resource of network’s forwarding ports and routing paths due to its poor allocation function of cognitive radio networks, based on the novel interactive networking architecture, the model of non-linear iterative prediction scheme in interaction was successively proposed. This research proposes a modified LSTM algorithm with a structure of non-linear iterative in complexity prediction, joins the multiple k modes selection and multi-agent systems, maximizes EERA of forwarding and routing while maintaining the communication quality. Firstly, considering whether this affective brain-computer interactions need the networking communication in system. Secondly, adjusting the forwarding and routing factors of energy-efficient resource allocation by selecting the best optimal energy-efficient resource for the links through the non-linear iterative prediction in a multi-modal perception. The simulation results show that compared with the other models and algorithms, the proposed scheme for affective brain-computer interactions, which has a nice performance on a higher EERA and channel utilization of a networking architecture of brain-computer interactions.</p> <p> </p>

Persistence of the steady planar normal shock structure in 3‐D unsteady potential flows

AbstractThis paper concerns the dynamic stability of the steady three‐dimensional (3‐D) wave structure of a planar normal shock front intersecting perpendicularly to a planar solid wall for unsteady potential flows. The stability problem can be formulated as a free boundary problem of a quasi‐linear hyperbolic equation of second order in a dihedral‐space domain between the shock front and the solid wall. The key difficulty is brought by the edge singularity of the space domain, the intersection curve between the shock front and the solid wall. Different from the two‐dimensional (2‐D) case, for which the singular part of the boundary is only a point, it is a curve for the 3‐D case in this paper. This difference brings new difficulties to the mathematical analysis of the stability problem. A modified partial hodograph transformation is introduced such that the extension technique developed for the 2‐D case can be employed to establish the well‐posed theory for the initial‐boundary value problem of the linearized hyperbolic equation of second order in a dihedral‐space domain. Moreover, the extension technique is improved in this paper such that loss of regularity in the a priori estimates on the shock front does not occur. Thus, the classical nonlinear iteration scheme can be constructed to prove the existence of the solution to the stability problem, which shows the dynamic stability of the steady planar normal shock without applying the Nash–Moser iteration method.

Deep Unfolding Contrast Source Inversion for Strong Scatterers via Generative Adversarial Mechanism

To alleviate the extremely intrinsical ill-posedness and nonlinearity of electromagnetic inverse scattering under high contrast and low signal to noise ratio (SNR), we propose a deep unfolding network based on generative adversarial network (GAN) under contrast source inversion (CSI) framework, termed UCSI-GAN. The method solves inverse scattering problems (ISPs) using end-to-end generating confrontation way by incorporating a physical model together with its iterative updating formulation into the internal architecture of GAN. First, the nonlinear iterative scheme is extended to a deep unfolding generator network, and the contrast source and contrast updates are mapped to each module of the generator network. Second, to stabilize the imaging process, we add a refinement network to each variable update. Finally, the discriminator network is employed to ensure the authenticity of reconstructed images. The generator network and discriminator network are alternately trained with a generative adversarial learning strategy to reconstruct the properties of the medium object. Numerical experiments demonstrated that the performance of UCSI-GAN is better than traditional CSI and state-of-the-art learning approaches under high contrast and low SNR condition.

On Admissible Positions of Transonic Shocks for Steady Isothermal Euler Flows in a Horizontal Flat Nozzle under Vertical Gravity

In this paper we are concerned with the existence of transonic shocks for 2-D steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. In particular, we focus on the contribution of the vertical gravity in determining the position of the shock front. For steady horizontal flows, the existence of normal shocks with the position of the shock front being arbitrary in the nozzle can be easily established. This paper will try to determine the position of the shock front as the state of the flow at the entrance of the nozzle and the pressure at the exit are slightly perturbed. To this end, this paper proposes a free boundary problem of the linearized Euler system with vertical gravity, whose solution could be an initial approximation for the shock solution with the free boundary being the approximation for the shock front. Due to the existence of the vertical gravity, difficulties arise in solving the boundary value problem in the approximate subsonic domain behind the shock front. The linearized Euler system is elliptic-hyperbolic composite for subsonic flows, and the elliptic part and the hyperbolic part are coupled in the $0$-order terms depending on the acceleration of gravity $g$. Moreover, the coefficients are not constants since the unperturbed shock solution depends on the vertical variable. New ideas and techniques are developed to deal with these difficulties and, under certain sufficient conditions on the perturbation, the existence of the solution to the proposed free boundary problem is established as the acceleration of gravity $g>0$ and the perturbation are sufficiently small. Then, with the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed which leads to a transonic shock solution with the position of the shock front being close to the initial approximating position.

Stability of stationary subsonic compressible Euler flows with mass-additions in two-dimensional straight ducts

We show existence, uniqueness and stability for a family of stationary subsonic compressible Euler flows with mass-additions in two-dimensional rectilinear ducts, subjected to suitable time-independent multi-dimensional boundary conditions at the entrances and exits. The stationary subsonic Euler equations consist a quasi-linear system of elliptic-hyperbolic composite-mixed type, while addition-of-mass destructs the usual methods based upon conservation of mass and Lagrangian coordinates to separate the elliptical and hyperbolic modes of the system. We establish a new decomposition and nonlinear iteration scheme to overcome this major difficulty. It reveals that mass-additions introduce very strong interactions in the elliptic and hyperbolic modes, and lead to a class of second-order elliptic equations with multiple integral nonlocal terms. The linearized problem is solved by studying algebraic and analytical properties of infinite weakly coupled boundary-value problems of ordinary differential equations, each with multiple nonlocal terms, after applications of Fourier analysis methods.

Simultaneous recovery of Robin boundary and coefficient for the Laplace equation by shape derivative

We study the simultaneous recovery of the boundary and coefficient of the Robin boundary condition for the Laplace equation from a pair of solution measurements on another part of the boundary. We derive the variational derivatives of the data-fitting objective functional with respect to the Robin boundary and coefficient, which are then used to device a nonlinear conjugate gradient iterative scheme for the numerical recovery of both the Robin boundary and coefficient together. Numerical examples are presented to illustrate the effectiveness of the recovery algorithms.

Analysis on the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes

This paper concerns the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes. One of the key points is to quantitatively determine the position of the shock front as the gravity is small. To this end, a free boundary problem for the linearized Euler system with small gravity will be proposed. Then one can obtain the information of the approximating position of the shock front. Once the initial approximation of the shock solution is obtained, a nonlinear iteration scheme can be constructed to show that the position of the shock front is a small perturbation of the initial approximation, which establishes a quantitative expression for the shock solution. Such an expression shows the asymptotic behavior as the gravity goes to zero.

Application of Nonlinear Krylov Solvers for Conjugate Heat Transfer Simulations of Electrical Battery Packs

Abstract Krylov-based methods are an attractive alternative to traditional fixed-point iterative schemes, being much more robust and accurate when solving elliptic equations (e.g., the energy equation in the solid domain). This study assesses the performance of a Krylov-based accelerator, when used for conjugate heat transfer (CHT) simulations of an electrical battery pack. The nonlinear nature of CHT simulations (due to spatial and temporal changes in boundary conditions) necessitates the use of the non-inear form of the Krylov-based accelerator (termed NKA), which utilizes the generalized minimized residual (GMRES) method, and works by accelerating an existing fixed-point iteration scheme. NKA is used while performing steady-state CHT simulations of an air-cooled lithium-ion battery pack, specifically to help accelerate the solution of the solid domain energy equation. The effect of using either isotropic or anisotropic thermal conductivity within the cylindrical lithium-ion battery cells is also evaluated. Results obtained using the NKA accelerator are compared, in terms of accuracy and speed, with those obtained from a traditional nonlinear fixed-point iterative scheme based on successive over-relaxation (SOR). The NKA accelerator is found to perform quite well for the problem at hand, providing results with the specified accuracy, while also being between 5 and 20 times faster than SOR (while solving the solid energy equation). The robust nature of NKA also leads to better global heat balance within the battery pack at all times during the simulation. These observations hold for both the isotropic and anisotropic thermal conductivity conditions. Overall, computational cost reductions of 30–40% are observed when using NKA for the battery pack simulations. Although the performance of NKA is demonstrated for a battery cooling application, NKA performs quite well in other applications also.

Superconvergence analysis of BDF-Galerkin FEM for nonlinear Schrödinger equation

A nonlinear iteration scheme for nonlinear Schrodinger equation with 2-step backward differential formula (BDF) finite element method (FEM) is proposed. Energy stability is testified for the constructed scheme, which leads to the boundedness of $\|{U_{h}^{n}}\|_{0}$ and $\sqrt {\tau }\|\nabla {U_{h}^{n}}\|_{0}$ . Auxiliary equation known as a time-discrete system is constructed to get rid of the restriction of τ. The solutions of the time-discrete equation in H1-norm is deduced. On the one hand, $\sqrt {\tau }\|\nabla U^{n}\|_{0}$ reduces to the temporal error in H2-norm. On the other hand, with the help of the boundedness about $\sqrt {\tau }\|\nabla {U_{h}^{n}}\|_{0}$ , the unconditional optimal estimate for spatial error is derived. Without any restriction of the time step, $\|{U_{h}^{n}}\|_{0,\infty }$ is bounded through surmounting the difficulty of nonlinear term. By taking difference between two time levels n and n − 1 of the error equation, an unconditional superconvergence estimate is derived. At last, global superconvergence result is achieved through the known interpolated postprocessing technique. Here, τ is the time step, and Un and ${U_{h}^{n}}$ denote the solutions of the time-discrete system and the finite element approximation equation, respectively. Furthermore, by introducing modified mass and energy functions, the numerical scheme is proved to preserve the total mass and energy in the discrete senses. Finally, numerical results are given to support the theoretical analysis.

On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with the 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observations and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

Optimal control of convection-cooling and numerical implementation

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion–convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.

Nonlinear iterative perturbation scheme with simplified spherical harmonics (SP3 ) light propagation model for quantitative photoacoustic tomography.

When using quantitative photoacoustic tomography (q-PAT) reconstruction to recover the optical absorption coefficients of tissue, the commonly used diffusion equation has several limitations in the case of the objects that have small geometries and high-absorption or low-scattering areas. Furthermore, the conventional perturbation reconstruction strategy is unsatisfactory when the target tissue containing large heterogeneous features. We herein present a modified q-PAT implementation that employs the higher-order photon migration model achieving the tradeoff between mathematical rigidity and computational efficiency. Besides, a nonlinear iterative method is proposed to obtain the perturbations of optical absorption considering the updating of the sensitivity matrix in calculating the fluence perturbations. Consequently, the distribution of tissue optical properties can be recovered in a robust way even if the targets with high absorption are included. The proposed approach has been validated by simulation, phantom and in vivo experiments, exhibiting promising performances in image fidelity and quantitative feasibility for practical applications.

ONE- AND TWO-NODE HYBRID LEAKAGE-CORRECTED 3-D CMFD ALGORITHM FOR FAST AND ACCURATE PIN-WISE PWR ANALYSIS

The HCMFD (Hybrid Coarse-Mesh Finite Difference) with GPS (GET Plus SPH) method and its 3-D application are investigated in this paper. In the HCMFD algorithm, a parallel computing for a pin-by-pin core calculation can be performed very effectively with a non-linear local-global iterative scheme. For an effective parallel computing, the one-node CMFD is used to solve a global eigenvalue problem. The conventional two-node CMFD is used for solving local fixed source problems with incoming current boundary conditions. The GPS method is a leakage correction method to correct the pin-wise XSs of the conventional GET-based two-step procedure. In the GPS method, the XS-dependent SPH factors are parameterized as a function of the pin-wise albedo information, current-to-flux ratio (CFR). With updated XS-wise SPH factors, the pin-wise XSs are corrected in order to improve the accuracy of the conventional two-step core analysis. The GPS method is implemented to an in-house pin-by-pin diffusion solver with the HCMFD algorithm to keep their strong points together. In this paper, several 3-D variant cores of KAIST-1A benchmark problem were chosen to demonstrate the combination of HCMFD and GPS (HCMFD-GPS) method.

Global existence and decay in multi-component reaction–diffusion–advection systems with different velocities: oscillations in time and frequency

It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e.~nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs $L^1$-$L^p$ estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.

A variational flexible multibody formulation for partitioned fluid–structure interaction: Application to bat-inspired drones and unmanned air-vehicles

We present a three-dimensional (3D) partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow. While the incompressible Navier–Stokes system is discretized using a stabilized Petrov–Galerkin procedure, the multibody structural system consists of a generic interaction of multiple components such as rigid body, beams and flexible thin shells along with various types of joints and connections among them. A co-rotational framework is utilized for the category of small strain problems where the displacement of the body is decomposed into a rigid body rotation and a small strain component. This assumption simplifies the structural equations and allows for the incorporation of multiple bodies (rigid as well as flexible) in the system. The displacement and rotation constraints at the joints are imposed by a Lagrange multiplier method. The equilibrium conditions at the fluid–structure interface are satisfied by the transfer of tractions and structural displacements via the radial basis function approach, a scattered data interpolation technique, which is globally conservative. For the coupled stability in low structure-to-fluid mass ratio regimes, a nonlinear iterative force correction scheme is employed in the partitioned staggered predictor–corrector scheme. The convergence and generality of the radial basis function mapping are analyzed by carrying out systematic error analysis of the transfer of fluid traction across the non-matching fluid–structure interface where a third-order of convergence is observed. The proposed aeroelastic framework is then validated by considering a flow across a flexible pitching plate configuration with serration at the trailing edge. Finally, we demonstrate the flow across a flexible flapping wing of a bat modeling the bone fingers as beams and the flexible membrane as thin shells in the multibody system along with the joints.