Robust Solver Research Articles

Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity

Abstract The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments.

A robust solver for incompressible high-Reynolds-number two-fluid flows with high density contrast

A new numerical method is developed for the robust simulation of high-Reynolds number two-fluid flows with high density contrast. The proposed method is a combination of the consistent mass–momentum transport framework of Nangia et al. (2019) [5] and the coupled level-set and volume-of-fluid method of Sussman and Puckett (2000) [23]. The conservative form of the momentum equation is solved, and the convection equation of density is solved together with the momentum equation at all cell faces. Moreover, consistent spatial and temporal discretization schemes are used to advance the governing equations of density and momentum to ensure a stable simulation. In the proposed method, the diffusion of the density or viscosity around the interface is not needed. In the test of a canonical case, the convection of a high density droplet, the proposed method makes an accurate prediction of the interface dynamics with a relatively low grid resolution. This feature is useful for reducing the computational cost for two-fluid flows with complex interface geometries. Furthermore, the test results show that the simulation is also stable for high Reynolds number two-fluid flows. In the case of wave breaking, the proposed method is stable for a high Reynolds number of Re=108. In the test case of a collapsing and impinging water column, the present method makes good predictions on the water height and pressure that agree well with experimental results.

Fast, lean-and-agile, multi-parameter multi-trending robust quality screening in a 3D-printed product

Additive manufacturing (AM) has revolutionized the local production realization of highly customizable items. However, the high process complexity - inherent to AM operations - renders uncertain the quality performance of the final products. Consequently, there is often a need to assess the unique fabrication capabilities of AM against the reoccurring issues of process instability and end-product inconsistency. Improvement opportunities may be identified by empirically exploring the complex phenomena that regulate the quality performance of the final products. Thus, focused quality-screening and process optimization studies should additionally take into account the special need for speedy, practical and economical experimentation. Robust multi-factorial solvers should predict effect strength by relying on small samples while possibly dealing with non-linear and non-normal trends. We propose a nonparametric modification to the classical Taguchi method in order to enable the generation of rapid and robust screening/optimization predictions for an arbitrary 3D-printing process. The new methodology is elucidated in a recently published dataset that involves the difficult Taguchi screening/optimization application of a fused deposition process. We compare differences in the predicted effect-strength magnitudes between the two approaches. We comment on the practical advantages that the new technique might offer over the traditional Taguchi-based improvement analysis. The emphasis is placed on the ‘assumption-free’ aspect, which is embodied in the new solver. It is shown that the proposed tool is agile. It could also reliably support a customized 3D-printing process by offering robust and faster quality improvement predictions.

Fundamental solutions to the stochastic perturbed nonlinear Schrödinger’s equation via gamma distribution

We introduce a new stochastic robust solver to solve several classes of nonlinear stochastic partial differential equations (NSPDEs). This solver presents the closed formula for the stochastic solutions. In this regard, this solver is developed to accurately present the complete wave structure of the NSPDEs. To verify this solver, an application is given. The acquired stochastic solutions may be applicable for some new observations in superfluid, new physics, engineering and biology. The theoretical study shows that the proposed solver is sturdy and efficacious. The obtained stochastic optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, the constraint conditions are utilized to verify the existence solutions. In the view of the characterization for the behavior of random solutions utilizing their statistical features, the expectation and variance values of these solutions are considered.

A new efficient two-stage method for damage localization and quantification in shell structures

The article proposes a new efficient two-stage approach for damage localization and quantification in shell structures using a modal flexibility sensitivity-based damage indicator abbreviated as MFBDI and a recently developed parameter-free optimization algorithm named golden ratio optimization method (GROM). In the first stage, the damage indicator MFBDI is employed to localize possible damage elements in the monitored shell structure. These possible damage elements also help define the search space of optimization problem in the next step. In the second stage, the GROM as a robust optimization solver is implemented to update the finite element (FE) model of the shell structure for refined localization of damage and quantification of its severity. The accuracy and efficiency of the proposed two-stage approach are demonstrated by two numerical simulation examples including a hypar shell and a spherical shell. The simultaneous influences of spatially-incomplete and inaccurate vibration data on damage prediction results are also taken into consideration. The obtained results reveal that the proposed approach can provide an efficient and accurate damage localization and quantification procedure for the studied shell structures.

A Morley–Wang–Xu Element Method for a Fourth Order Elliptic Singular Perturbation Problem

A Morley–Wang–Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods. The sharp error analysis is given for this MWX element method. And the Nitsche’s technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers. An important feature of the MWX element method is solver-friendly. Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming \(P_1\)–\(P_0\) element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method. Some numerical examples are provided to verify the theoretical results.

A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

AbstractIn this work, we consider equal‐order discontinuous Galerkin (DG) solver for incompressible Navier–Stokes equations based on high‐order dual splitting scheme. In order to stay stable, the time step size of this method has been reported that is strictly limited. The upper bound of time step size is restricted by Courant–Friedrichs–Lewy (CFL) condition (Hesthaven and Warburton, 2007) and lower bound is required to be larger than the critical value which depends on Reynolds number and spatial resolution (Ferrer et al., 2014). For high‐Reynolds‐number flow problems, if the spatial resolution is low, the critical value may be larger than CFL condition, then instability will occur for any time step size. Therefore, sufficiently high spatial resolution is indispensable in order to maintain stability, which increases the computational cost. To overcome these difficulties and develop a robust solver for high‐Reynolds‐number flow problem, it is necessary to further study the instability problem at small time steps. We numerically investigate the effect of the pressure gradient term in projection step and the velocity divergence term in pressure Poisson equation on the stability for small time step size, respectively, and conclude that the DG formulation of the pressure gradient term has a more significant effect on the stability of the scheme than that of the velocity divergence term. Integration by parts of these terms is essential in order to improve the stability of the scheme. Based on this discretization format, an appropriate penalty parameter for pressure Poisson equation is utilized so as to provide the scheme with an inf‐sup stabilization. Moreover, the lid‐driven cavity flow is considered to verify that this numerical algorithm enhances the stability without additional stabilization term at small time step size and high‐Reynolds number for equal‐order polynomial approximations.

A robust inverse design solver for controlling the potential aggressiveness of cavitating flow on hydrofoil cascades

AbstractThis article presents the development of a new inverse design algorithm capable of generating blade geometries for cavitating cascade flows. With this methodology, we demonstrate the controllability of the pressure distribution in and around the cavity and thereby provide a means to regulate the aggressiveness of blade cavitation phenomena. The solver proposed here uses the Tohoku–Ebara equation of state to model phase change, combined with bespoke preconditioning and multigrid methods designed to handle the system's ill conditioning and cope with the hypersonic flow regime of the mixture, respectively. Blade geometries and the cavitating flow field are calculated simultaneously in a robust and efficient manner, with a blade loading that matches the target distribution. In this article, the accuracy of the cavitating flow solver is first demonstrated for the NACA0015 hydrofoil case and associated experimental data. The inverse design procedure is then applied to a typical axial flow pump cascade: a new blade profile is generated with a topology that successfully reduces the gradient of the pressure jump at cavity closure.

Correction to: Development of efficient and robust Eikonal solver variants for first-arrival seismic modeling

Correction to: Development of efficient and robust Eikonal solver variants for first-arrival seismic modeling

MPS‐Based Model to Solve One‐Dimensional Shallow Water Equations

AbstractHere, a moving particle simulation method is presented to spatially integrate the cross‐sectional average shallow water equations using a prediction‐correction procedure for the time discretization. A density‐ratio equation is derived for the water depth computation according to the particle number density concept. The newly derived equation does not miscalculate the water depth in case an incorrect searching radius parameter is adopted, unlike the typical volume‐summation formula in meshless shallow water flows. A new one‐dimensional Spiky kernel function is developed to satisfy the unity condition employed in the Newton‐Raphson iteration to calculate the water depth. Dynamic stabilization is adopted to capture shockwave problems, a case‐independent technique based on the inelastic collision with unequal masses. The convective flux term is eliminated under the Lagrangian framework, and so the momentum is adequately conserved without the need for any special treatment. The proposed scheme maintains the exact C‐property, meaning that the water depth gradient and bed slope are hydrostatically well balanced within a discretized solution domain. Compared to analytical solutions and experimental data, the results of this study reveal that the present model is a robust numerical solver without unphysical oscillations. It can capture various shock problems, including steep gradient shock‐front, discontinuous wet bed, transcritical flow regimes, and irregular bed topography. The present model can also simulate the dry‐wet flow transition influenced by friction without experiencing any divisions by zero, negative values, or unphysical perturbations. This advantage is basically due to the water particles' absence and presence for the dry and wet regions, respectively.

Closed-form solutions to the new coupled Konno–Oono equation and the Kaup–Newell model equation in magnetic field with novel statistic application

This paper employs the new robust solver to retrieve exact solutions to a Kaup–Newell model equation and the nonlinear coupled Konno–Oono equation. This solver yields the closed formula for the solutions. Different types of travelling wave solutions, i.e., rational function, hyperbolic function and trigonometric function solutions with many capricious parameters are revealed. Subsequently, by utilizing Matlab 18 we plot 2D and 3D surfaces of obtained analytical solutions for suitable values of the free parameters. The depiction of the solver is direct, vital, sturdy and can be applied to other nonlinear partial differential equations. We also show that some proposed rational solutions can be rapprochement with some known probability distributions.

Constraint programing for solving four complex flexible shop scheduling problems

In recent years, with the advent of robust solvers such as Cplex and Gurobi, constraint programing (CP) has been widely applied to a variety of scheduling problems. This paper presents CP models for formulating four scheduling problems with minimal makespan and complex constraints: the no-wait hybrid flow shop scheduling problem, the hybrid flow shop scheduling problem with sequence-dependent setup times, the flexible job shop scheduling problem with worker flexibility and the semiconductor final testing problem. The advantages of CP method in solving these four complex scheduling problems are explored. Finally, a set of benchmark instances are adopted to demonstrate the effectiveness and efficiency of the CP method. Experiment results show that the proposed CP models outperform existing algorithms; in particular, several best-known solutions of benchmark instances are improved by our CP method.

3D actual microstructure-based modeling of non-isothermal infiltration behavior and void formation in liquid composite molding

In the liquid composite molding (LCM) non-isothermal infiltration process, the effect of the complex interaction between viscous pressure and capillary pressure on the fluid invasion morphology and void formation mechanism is not well understood. Based on high-resolution, three-dimensional (3D) computed microtomography (µCT) images of porous silicon carbide particulate-reinforced composite (SiCp) preforms, a pore-scale, non-isothermal two-phase flow numerical model was implemented to describe the flow displacement pattern that evolves with the decrease in displacement velocity during the infiltration process. The simulation approach was performed by coupling Cahn–Hilliard phase field and heat equations using a robust finite element solver. The numerical results indicate that (i) the displacement pattern experiences a transition from stable displacement to capillary fingering, which mainly depends on the transition of complex yet intriguing pore-scale events responsible for local meniscus dynamics, from viscous self-correcting smoothing transition to the noncooperative Haines jump; (ii) the void trapping stability is higher in the capillary fingering pattern, and it is difficult for the macrovoids generated by the bypass trapping mechanism to migrate; (iii) there is a critical capillary number (Ca) value, and as long as the Ca of the displacement front during infiltration process is not lower than this value, the local macroscopic void concentration can be avoided; (iv) the non-isothermal effect in the LCM simulation cannot be ignored, where the change in two-phase physical properties caused by the evolution of the temperature field affects the local pore-scale flow behavior, i.e., the disappearance of the partial interface pinning phenomenon.

Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDE

The aim of this paper is to propose a robust numerical solver, which is capable of efficiently solving a three-dimensional elliptic problem in a data-sparse quantized tensor format. In particular, ...

Robust Preconditioners for Perturbed Saddle-Point Problems and Conservative Discretizations of Biot's Equations Utilizing Total Pressure

We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-fie...

Power flow solution of Ill-conditioned systems using current injection formulation: Analysis and a novel method

Along the conventional approach in power mismatches, the current injection form of the Power Flow (PF) equations is also quite popular and extensively considered for PF analysis, however, this approach has not been profusely studied for solving ill-conditioned systems yet. This paper aims at filling this gap by studying various conventional and modern robust solvers in current injection formulation. In addition, a novel robust PF technique using this formulation is developed. Several snapshots of the Polish Transmission systems are used to validate the studied PF solvers. It is concluded that the PF equations tend to be notably worse conditioned when they are handled in the current injection form compared with its power mismatch form, which complicates the solution of ill-conditioned systems. In this regard, the results obtained with the proposed technique are promising, showing a good trade-off between robustness and efficiency.

Development of efficient and robust Eikonal solver variants for first-arrival seismic modeling

We have developed and tested a new Eikonal first-arrival forward model scheme by combining a fast marching method (FMM) algorithm, an upwind Eikonal solver scheme first described by Sethian and Popovici in 1996, with a more accurate but less robust Eikonal scheme described by Vidale in 1990 as a finite difference method (VFD) in order to produce a robust forward model based in FMM, with application of VFD schemes for refinement of computed model times (FMM-VFD). The developed model was tested through a uniform velocity base case, and the Southern California Earthquake Center (SCEC) Harvard Velocity Model (CVM-H) of upper crust including the sedimentary basin under Los Angeles, CA. Its performance was evaluated by measuring error at different grid sizes against a set of reference times generated by an independent model. In comparison against first-order FMM (FMM-O1), second-order FMM (FMM-O2), and an improved FMM scheme known as factored FMM (FMM-F) against a set of generated reference times, FMM-VFD error was found to be significantly lower than first- and second-order FMM, and competitive with FMM-F, outperforming it in the presented scenarios.

Coordinate parameterisation and spectral method optimisation for Beltrami field solver in stellarator geometry

The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus, but ill-conditioning of the discretised linear equations occurs in the (solid) toroid due to the artificially singular coordinate parameterisation near the axis. In this work, we propose an improved choice for the reference axis to prevent coordinates surfaces from overlapping. Then, we examine the parity and asymptotics of the magnetic vector potential near the axis and suggest the use of recombined and rescaled Zernike radial basis functions. The maximum relative error in the magnetic field of the Wendelstein 7-X geometry is shown to reach 10−9 at high resolution in a series of convergence tests and benchmarks against the boundary integral equation solver for Taylor states. The new method is also reported to significantly improve the accuracy of multi-volume SPEC calculations. A comparison between free-boundary SPEC and the analytical Dommaschk potential is presented with higher-than-usual Fourier resolution. It is illustrated that we are able to resolve low amplitude current sheets when an interface is placed where there is no flux surface in the analytic solution. This was previously concealed because of insufficient numerical resolution.

Transported PDF modeling of compressible turbulent reactive flows by using the Eulerian Monte Carlo fields method

Although the transported probability density function (PDF) method has been developed for decades, its application has been mainly focused on the low-Mach number flow problems. This work extends the transported PDF method to compressible flow problems. The Eulerian Monte Carlo fields (EMCF) solution method is employed to solve the transported PDF equation for compressible flow problems. A pseudo stagnation enthalpy is introduced and its stochastic partial differential equation is derived to ensure total energy conservation numerically. A new mixing model called interaction by partial exchange with mean (IPEM) is introduced to expand the available choices of mixing models for the EMCF method. The consistency of the EMCF method is examined for solving the transported PDF equation. Numerical implementation details are discussed, such as the density coupling between the compressible flow solver and the EMCF solver, discretization schemes for the mixing terms and the stochastic terms. The implemented compressible flow solver coupled with the EMCF solver is verified and validated in a series of test cases with increasing level of complexity, ranging from a statistically one-dimensional turbulent mixing layer to a self-excited resonance model rocket combustor. It is observed that in general with the increase of compressibility, there is an increase in the sensitivity of the modeling results to the different models and algorithms. This makes it necessary to develop a thorough understanding of the model sensitivity in order to develop a robust and accurate simulation solver for highly compressible turbulent reactive flows. The thermo-acoustic instability inside the model rocket combustor case is captured reasonably, which demonstrates the overall capability of the developed compressible turbulent combustion solver based on the transported PDF method.

A staggered‐grid multilevel incomplete LU for steady incompressible flows

AbstractAlgorithms for studying transitions and instabilities in incompressible flows typically require the solution of linear systems with the full Jacobian matrix. Other popular approaches, like gradient‐based design optimization and fully implicit time integration, also require very robust solvers for this type of linear system. We present a parallel fully coupled multilevel incomplete factorization preconditioner for the 3D stationary incompressible Navier‐Stokes equations on a structured grid. The algorithm and software are based on the robust two‐level method developed by Wubs and Thies. In this article, we identify some of the weak spots of the two‐level scheme and propose remedies such as a different domain partitioning and recursive application of the method. We apply the method to the well‐known 3D lid‐driven cavity benchmark problem, and demonstrate its superior robustness by comparing with a segregated SIMPLE‐type preconditioner.