Nonlinear Iteration Research Articles

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.

A consistent linearization scheme for KGD problems using fracture tip asymptotic solutions

AbstractRecently, a fluid volume enrichment strategy based on the asymptotic solutions near the crack tip was proposed for fluid‐driven fracture propagation problems. Despite its successes in various benchmark and field‐scale problems of hydraulic fracturing simulations, the aforementioned enrichment strategy has the following limitations. First, the tightly coupled solid‐fluid nonlinear system cannot be consistently linearized due to the fast marching method applied to solve the Eikonal equation for fracture tracking. As a result, an approximated Jacobian had to be deployed for the Newton‐Raphson iterations. This is particularly troublesome when the fracture front propagates into newly fractured cells, since a large number of nonlinear iterations are required for convergence because of the inconsistent linearization. Second, the existing method only focused on the viscosity‐dominated fracture propagation regime. Even though the extension of the method to the toughness‐dominated regime could be relatively straightforward, it is not immediately clear how to apply the enrichment technique to the transition regime. This work is dedicated to address the above two limitations. Specifically, a unified fracture propagation criterion is proposed, which not only works for the viscosity‐dominated regime, but also for the toughness‐dominated regime and the transition regime in between. The techniques to consistently linearize the coupled solid‐fluid system and properly initialize the primary unknowns are demonstrated, which result in the significant reduction of required number of nonlinear iterations for convergence. The proposed technique is demonstrated in the context of the Khristianovic‐Geertsma‐de Klerk (KGD) problems due to its relative simplicity.

A novel robust iterated CKF for GNSS/SINS integrated navigation applications

In challenging circumstances, the estimation performance of integrated navigation parameters for tightly coupled GNSS/SINS is impacted by outlier measurements. An effective solution that employs a novel iterative sigma-point structure with a modified robustness optimization approach for enhancing the error compensation effectiveness and robustness of filters utilized in GNSS challenge conditions is proposed in this paper. The proposed method modifies the CKF scheme by incorporating nonlinear regression and numerous iteration processes for ameliorating error compensation. Subsequently, a loss function and penalty mechanism are implemented to enhance the filter's robustness to outlier measurements. Furthermore, to fully incorporate valid information of the innovation and speed up the operation of the proposed method, the outlier measurement detection criteria are established to bypass the penalty mechanism against measurement weights in the absence of outliers in GNSS measurements. Field experiments demonstrate that the proposed method outperforms traditional methods in mitigating navigation errors, particularly when multipath errors and non-line-of-sight (NLOS) reception are increased.

Toward the use of LES for industrial complex geometries. Part II: Reduce the time-to-solution by using a linearised implicit time advancement

The strong increase in computational power observed during the last few years has allowed to use Large Eddy Simulation (LES) for industrial configurations. Nevertheless, the time-to-solution is still too large for a daily use in the design phases. The objective of this work is to develop a new time integration method to reduce the time-to-solution of LES of incompressible flows by allowing the use of larger time step. The projection method, probably the most commonly used method in the context of LES of incompressible flow, is generally applied using explicit time advancement which constrains the time-step value for stability reasons (CFL and Fourier constraints). The time step can then be small with respect to the physical characteristic times of the studied flow. In this case, an implicit time advancement method, which is unconditionally stable, can be used. However, this leads to non-linear resolution of momentum equation which can strongly increase time-to-solution because of non-linear iterations inside a physical iteration. To relax the stability constraints while minimising the computational cost of an iteration, a linearised implicit time advancement based on Backward Differentiation Formula (BDF) scheme is proposed in this work. The linearisation is performed using an extrapolated velocity field based on the previous fields. This time integration is first evaluated on a turbulent pipe test case. It is observed a time-to-solution up to five times lower than the explicit time integration while keeping the same accuracy in terms of mean and fluctuating velocity fields. To incorporate this new time advancement method in the automatic mesh convergence developed in Part I, a time-step control method based on the local truncation error is used. The resulting automatic time-step and mesh procedure is evaluated on a turbulent round jet case and on PRECCINSTA configuration, a swirl burner which is a representative case of an industrial aeronautical injection system. This new procedure leads to a time-to-solution up to three times lower than the previous procedure, presented in Part I.

Accelerated nonlinear domain decomposition solver for multi-phase flow and transport in porous media

Development of robust and efficient nonlinear solution strategies for multi-phase flow and transport within natural porous media is challenging. Fully Implicit Method (FIM) is widely used in reservoir simulation for solving nonlinear algebraic systems. Previous studies showed that nonlinearities of flow and transport problems could exhibit large degrees of locality across timesteps and nonlinear iterations. Nonlinear domain decomposition (NDD) is an attractive class of methods for achieving localization and resolving spatially unbalanced nonlinearities. However, NDD is not robust as iterative solvers, because the outer iterations are essentially fixed-point type.We design a solution framework to accelerate the outer-loop convergence of NDD. We first apply a nonlinear acceleration (NA) technique to the fixed-point iterations, for correcting the boundary conditions of subdomains. Because flow problems are parabolic and exhibit global behaviors, long-range information is essential to capture main features of the flow field. We incorporate a preconditioning step based on CPR-AMG before each timestep, to provide better initial solutions. Additionally, smoother iterations can be taken to improve the boundary conditions during the NDD process.We evaluate the NDD methods using heterogeneous multi-phase models. The nonlinearities associated with the flow, transport and the coupling are locally resolved through the NDD iterations. Results demonstrate that the basic NDD solver is afflicted with severe nonlinear difficulties. By comparison, the accelerated NDD solver greatly improves the outer convergence performance. Moreover, the new solver can effectively exploit locality on the Newton level, for both the flow and transport problems.

A monolithic, finite element-based strategy for solving fluid structure interaction problems coupled with electrostatics

In this work, we present a monolithic finite-element-based strategy for problems involving the deformation of a hyperelastic solid, an incompressible fluid and electrostatics. We use a two-field hybrid Lagrangian formulation for the structure, and a velocity-based ALE mixed formulation for the fluid with appropriately chosen interpolations for the various field variables to ensure stability of the resulting numerical procedure. The equations of electrostatics are solved on the reference configuration over both the solid and fluid domains, with voltage and electric displacement continuity imposed at the interface. Keeping in view that the thickness of typical MEMS structures is small, a stress-based hybrid formulation is used to prevent locking. The use of a monolithic strategy provides a robust and stable algorithm, and allows the user to take large time steps. The consistent linearization ensures a quadratic rate of convergence of the non-linear iterations at each time step. Detailed expressions for the tangent stiffness and associated matrices for the three-way coupled problem are provided. The robustness and accuracy of the proposed method are demonstrated by solving several benchmark problems from the literature.

A finite volume scheme preserving the invariant region property for a class of semilinear parabolic equations on distorted meshes

AbstractIn this article, we present a finite volume scheme preserving invariant‐region‐property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a specially designed iteration is proposed to preserve the IRP. The IRPs are proved for both, the finite volume scheme and the nonlinear iteration. Numerical examples are presented to verify the accuracy and IRP of our scheme.

Radial Basis Function Finite Difference Method Based on Oseen Iteration for Solving Two-Dimensional Navier-Stokes Equations.

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier-Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier-Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration.

Application of linear iterative methods for the proximity effect correction in electron beam lithography

The Jacobi linear iterative method and weight Jacobi method (WJM) are introduced for solving the large-scale linear problem in the proximity effect correction (PEC) of electron beam lithography. Based on the discussion of PEC physics, a symmetrical and positive defined proximity interaction matrix is constructed to ensure the convergence of the methods. It shows that zeroing the center exposure fraction in the point spread function matrix is equal to the operation of splitting the proximity interaction matrix. Then, the Jacobi method is ready for the PEC calculation. The iterative can be performed in the Fourier domain due to the inherent parallelization of the Jacobi method. The convergent property of the Jacobi method is discussed and then testified by the PEC simulation. Compared with the classical Jacobi method, an improvement of 100% in convergence efficiency can be achieved by introducing the optimized relaxation parameter quasi-ωopt in the WJM. By combining the WJM and Gold nonlinear iteration method, a new method that shows an order of magnitude superior in accuracy to the WJM is proposed. Results indicate the methods introduced here could be used to calculate the PEC problem efficiently.

Advanced two-phase subchannel method via non-linear iteration

A fast-running, robust two-phase flow, sub-channel model is presented based on non-linear solution of the steady-state subchannel fluid flow equations. The drift-flux model solves for conservation of liquid and vapor mass, mixture energy, and axial and transverse mixture momentum as part of an efficient planar marching scheme and nonlinear, nested outer and inner iteration. Models based on mechanistic subcooled boiling, two-phase turbulent void mixing, and drift are included. Solution verification and mesh convergence studies were performed for modern GE 10 × 10 fuel geometry and are shown to have excellent convergence behavior. Run time performance for a 50 axial mesh model showed 2.2 seconds on a single CPU core to tightly converge all 3D distributions (flow, void, pressure) for the GE 10 × 10 fuel geometry, supporting its efficient use within the Virtual Environment for Reactor Applications boiling water reactor framework.

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Nonlinear fully coupled thermoelastic transient analysis of axial functionally graded composite panel

In this article, the two-way coupled thermoelastic behavior of axially graded composite plate and doubly-curved (cylindrical, spherical, hyperbolic, and elliptical) panels is examined under conductive-convective boundary and uniformly distributed loading conditions. Here, the material properties of the axial functionally graded panel are computed by employing the power-law-based Voigt’s scheme. The material and geometric nonlinearities are incorporated using the cubic-polynomial-based temperature-dependent constitutive model and higher-order kinematics-based Green-Lagrange strain, respectively. The temperature profile is obtained through thermally nonlinear theory associated with high temperature, which is used to extract the energy equations obtained from the first law of thermodynamics and deformation-dependent entropy relation. The weak forms of motion and heat-transfer equations are derived using Galerkin’s method and further combined into the two-way coupled field equation through 2D-finite element approximation via Lagrangian elements. The transient responses are computed via the Newmark and the Crank-Nicolson schemes, whereas the nonlinear iterations are executed through Picard’s iteration technique. The performance of the fully coupled model for an axially graded (metal/ceramic) structure is verified with analytical, numerical, and experimental results and tested through a variety of numerical illustrations under various sets of conditions.

Particle swarm optimisation strategies for IOL formula constant optimisation.

To investigate particle swarm optimisation (PSO) as a modern purely data driven non-linear iterative strategy for lens formula constant optimisation in intraocular lens power calculation. A PSO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as achieved refraction minus predicted refraction) for the Castrop formula in a dataset of N= 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. The hyperparameters were set to inertia: 0.8, accelerations c1=c2=0.1. The algorithm was initialised with NP = 100 particles having random positions and velocities within the box constraints of the constant triplet parameter space C=0.25 to 0.45, H=-0.25 to 0.25 and R=-0.25 to 0.25. The performance of the algorithm was compared to classical gradient-based Trust-Region-Reflective and Interior-Point algorithms. The PSO algorithm showed fast and stable convergence after 37 iterations. The rmsPE reduced systematically to 0.3440 diopters (D). With further iterations the scatter of the particle positions in the swarm decreased but without further reduction of rmsPE. The final constant triplet was C/H/R=0.2982/0.2497/0.1435. The Trust-Region-Reflective/Interior-Point algorithms showed convergence after 27/17 iterations, respectively, resulting in formula constant triplets C/H/R=0.2982/0.2496/0.1436 and 0.2982/0.2495/0.1436, both with the same rmsPE as the PSO algorithm (rmsPE=0.3440 D). The PSO appears to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation even in formulae with more than 1 constant. It acts independently of an analytical or numerical gradient and is in general able to search for the best solution even with multiple local minima of the target function.

Optimal Regularity for Elliptic Equations With Measurable Nonlinearities Under Nonstandard Growth

Abstract We are concerned with weak solutions of elliptic equations involving measurable nonlinearities with Orlicz growth to address what would be the weakest regularity condition on the associated nonlinearity for the Calderón–Zygmund theory. We prove that the gradient of weak solution is as integrable as the nonhomogeneous term under the assumption that the nonlinearity is only measurable in one of the variables while it has a small BMO assumption in the other variables. To this end, we develop a nonlinear Moser-type iteration argument for such a homogeneous reference problem with one variable–dependent nonlinearity under Orlicz growth to establish $W^{1,q}$–regularity for every $q>1$. Our results open a new path into the comprehensive understanding of the problem with nonstandard growth in the literature of optimal regularity theory in highly nonlinear elliptic and parabolic equations.

A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System

AbstractModified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well as conservativity of a production–destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i. e. the iterates are generated by a nonlinear map g even when the PDS is linear. Moreover, due to the conservativity of the method, the map g possesses non‐hyperbolic fixed points.Recently, a new theorem for the investigation of stability properties of non‐hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.

Multiscale solver for multi-component reaction–diffusion systems in heterogeneous media

Coupled nonlinear system of reaction–diffusion equations describing multi-component (species) interactions with heterogeneous coefficients is considered. Finite volume method based approximation for the space is used to construct semi-discrete form for the computation of numerical solutions. Two techniques for time approximations, namely, a fully implicit (FI) and a semi-implicit (SI) schemes are examined. The fully implicit scheme is constructed using Newton’s method and leads to the coupled system of equations on each nonlinear and time iterations which is computationally rather expensive. In order to minimize the latter hurdle, an efficient and fast multiscale solver is proposed for reaction–diffusion systems in heterogeneous media. To construct fast solver, we apply a semi-implicit scheme that leads to an uncoupled system for each individual component. Problems in heterogeneous domains require a very fine grid for accurate solutions of large systems of equations at each time step iteration. Here we present a multiscale model reduction technique to reduce the size of the discrete system. Multiscale solver is based on the uncoupled operator of the problem and constructed by the use of Generalized Multiscale Finite Element Method (GMsFEM). In GMsFEM we use a diffusion part of the operator and construct multiscale basis functions by solving spectral problems in each local domain associated with the coarse grid nodes. We collect multiscale basis functions to construct a projection/prolongation matrix and generate reduced order model on the coarse grid for fast solution. Moreover, the prolongation operator is used to reconstruct a fine-scale solution and accurate approximation of the reaction part of the problem which then leads to a very accurate and computationally effective multiscale solver. We provide numerical results for two species competition test problems in two-dimensional domain with heterogeneous inclusions. Our computed solutions predict strong competition between the two species and demonstrate that diffusion can dictate the dominance of one species over the other in heterogeneous environments. We investigate the influence of number of the multiscale basis functions to the method accuracy and ability to work with different values of the diffusion coefficients.

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.

Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against C_{textrm{ub}}^m-Perturbations

The real Ginzburg–Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure L^infty -estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show the wider relevance of our method, we also apply it to the amplitude equation as it appears for pattern-forming systems with an additional conservation law.

SUPG formulation augmented with YZβ shock‐capturing for computing shallow‐water equations

AbstractWe demonstrate that the streamline‐upwind/Petrov–Galerkin (SUPG) formulation enhanced with YZβ discontinuity‐capturing, that is, the SUPG‐YZβ formulation, is an efficient and robust method for computing 2D shallow‐water equations (SWEs). The SUPG‐stabilized semi‐discrete formulation is discretized in time by employing the backward Euler time‐integration scheme. The nonlinear equation systems arising from the space and time discretizations are handled using the Newton–Raphson (N–R) method at each time step. The resulting linear equation systems are solved directly at each nonlinear iteration. Two challenging test problems are provided to examine the performance of the proposed formulation and techniques. To that end, we consider a full dam‐break and a partial dam‐break problem. We develop the solvers in the FEniCS environment. Test computations reveal that the SUPG‐YZβ formulation successfully eliminates spurious oscillations that cannot be captured with the SUPG‐stabilized formulation alone in narrow regions where steep gradients occur.

Exponential Euler and Backward Euler Methods for Nonlinear Heat Conduction Problems

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).