Large Nonlinear Systems Research Articles

Greedy Randomized Kaczmarz with momentum method for nonlinear equation

Based on the nonlinear greedy randomized Kaczmarz (NGRK) method and the heavy ball method, a nonlinear greedy randomized Kaczmarz with momentum (NGRKm) method is presented for solving large nonlinear systems. Furthermore, the convergence of the NGRKm method is proved and the corresponding numerical experiments show the effectiveness of the NGRKm method.

A new numerical path to retrieve isolated branches on large scale nonlinear mechanical systems

A new numerical path to retrieve isolated branches on large scale nonlinear mechanical systems

High-order energy stable discrete variational derivative schemes for gradient flows

Abstract The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.

Two-Level method for blind image deblurring problems

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h (h≤H). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.

Unconditionally Stable Central Difference Dissipative Algorithm for Multi-Directional Real-Time Hybrid Simulations of Large Nonlinear Structural Systems

ABSTRACT The central difference is a popular algorithm used to integrate the equations of motion, yet suffers from two drawbacks: (1) it is only conditionally stable and requires a small-time step to maintain numerical stability; (2) it is non-dissipative, and high-frequency spurious oscillations may appear and compromise the accuracy of the solution. These drawbacks are detrimental to applying the algorithm to the real-time hybrid simulation of large, complex nonlinear structural systems. In this paper, the conventional central difference algorithm is modified to overcome these drawbacks, and the modified algorithm is applied to the real-time hybrid simulation of complex structural systems.

Automatic initial value generation procedure for nonlinear process models by interval arithmetic based cutting and splitting

To solve large nonlinear equation systems, we present a hybrid method based on interval arithmetic and real-valued root-finding. The proposed approach includes two new interval arithmetic based methods, the so-called cutting and a special kind of bisection of consistent variable spaces. Applied to three examples from chemical engineering, it is shown that the approach is now able to find all system solutions within a variable space as well as only one process-relevant solution in much less time thanks to the built-in root-finding step. For the latter, a conventional Newton method as well as IPOPT have been examined. The hybrid approach no longer needs a well-estimated initial point to converge to a solution, only rough initial variable bounds are required.

3D concrete fracture simulations using an explicit phase field model

Phase field models can effectively capture complicated crack evolution characteristics such as propagation, bifurcating, intersecting and merging. However, the simulation of three-dimensional (3D) quasi-brittle fracture remains a challenge due to large nonlinear equation systems and significant computational costs, which are often intractable by iteration-based implicit approaches, especially in complex mixed-mode fracture. This work presents an efficient explicit phase field model based on the unified phase field theory in order to overcome the above issues. In this model, the displacement field is second-order time dependent while the damage-phase field follows a first-order time dependence with a viscosity term. Efficient explicit central- and forward-difference algorithms for each field are developed by combining VUEL and VUMAT subroutines in the software ABAQUS/Explicit; hence, the convergence issue in the implicit phase field modelling is avoided. Several typical 3D fracture benchmarks with different failure modes are analysed for verification purposes and compared with the available experimental data. The results indicate that the developed model and computational implementation method can simulate complex 3D fracture of brittle/quasi-brittle materials with salient accuracy and efficiency, and are promising to meet the requirements in structural-level engineering practices.

Agents of social change in the institutional field of medicine

The pragmatic goal of this article is to study agent activity aimed at changes in the medical institutional field, under conditions of high social instability. The implementation of the empirical part of аrticle required an adequate methodological and conceptual approach, on the basis of which it could be designed and implemented. As a general methodological platform, the author chose the paradigm of complexity, which builds its methodological proposals on the assumption that large complex nonlinear systems change according to laws that are fundamentally different from those by which simple linear system formations function. Society refers precisely to such complex system objects, and therefore the appeal to such a methodological basis is adequate and fruitful. The work offers the author’s view on the processes of social changes, which are conceptualized as the process of changing the social order. The latter, in turn, is considered as a set of all the practices of social interactions that are currently present in society. The key point in this concept is the statement about the presence of two fundamentally different mechanisms of social change — organizational and self-organizational. It is emphasized that high social instability increases the weight of the self-organizing component of changes, which can fade into the background in relatively stable periods. Applying this theoretical framework to the processes in the institutional field of medicine, we get a model of relevant changes, where both organizational (formal rules of the game, laws, government regulations) and self-organization (informal, spontaneously formed rules of everyday interactions in this institutional field) are equally important. Both of these mechanisms have their conductors-agents, through whom they are implemented. The issue of consistency and balance between these different rules of the game is key for institutional management. At the empirical level, the work examines in detail the agentic actions of U. Suprun in the process of her attempts to reform domestic medicine during 2016-2019 (organizational mechanisms), as well as the agentic activity of volunteers from the "Svoi" Foundation, which is headed by Lesia Lytvynova (self-organizational mechanisms). All the features of these two processes of change, their fundamental difference and the complexities that arise in the process of their coordination are shown. The need to build managerial strategies that take these circumstances into account is emphasized.

On the construction of an efficient finite-element solver for phase-field simulations of many-particle solid-state-sintering processes

We present an efficient solver for the simulation of many-particle solid-state-sintering processes. The microstructure evolution is described by a system of equations consisting of one Cahn–Hilliard equation and a set of Allen-Cahn equations to distinguish neighboring particles. The particle packing is discretized in space via multicomponent linear adaptive finite elements and implicitly in time with variable time-step sizes, resulting in a large nonlinear system of equations with strong coupling between all components to be solved. Since on average 10k degrees of freedom per particle are necessary to accurately capture the interface dynamics in 3D, we propose strategies to solve the resulting large and challenging systems. This includes the efficient evaluation of the Jacobian matrix as well as the implementation of Jacobian-free methods by applying state-of-the-art matrix-free algorithms for high and dynamic numbers of components, advances regarding preconditioning, and a fully distributed grain-tracking algorithm. We validate the obtained results, examine in detail the node-level performance and demonstrate the scalability up to 10k particles on modern supercomputers. Such numbers of particles are sufficient to simulate the sintering process in (statistically meaningful) representative volume elements. Our framework thus forms a valuable tool for the virtual design of solid-state-sintering processes for pure metals and their alloys.

An aneurysm-specific preconditioning technique for the acceleration of Newton-Krylov method with application in the simulation of blood flows.

In this paper, we develop an algorithm to simulate blood flows in aneurysmal arteries and focus on the construction of robust and efficient multilevel preconditioners to speed up the convergence of both linear and nonlinear solvers. The work is motivated by the observation that in the local aneurysmal region, the flow is often quite complicated with one or more vortices, but in the healthy section of the artery, the principal component of blood flows along the centerline of the artery. Based on this observation, we introduce a novel two-level additive Schwarz method with a mixed-dimensional coarse preconditioner. The key components of the preconditioner include (1) a three-dimensional coarse preconditioner covering the aneurysm; (2) a one-dimensional coarse preconditioner covering the central line of the healthy section of the artery; (3) a collection of three-dimensional overlapping subdomain preconditioners covering the fine meshes of the entire artery; (4) extension/restriction operators constructed by radial basis functions. The blood flow is modeled by the unsteady incompressible Navier-Stokes equations with resistance outflow boundary conditions discretized by a stabilized finite element method on fully unstructured meshes and the second-order backward differentiation formula in time. The resulting large nonlinear algebraic systems are solved by a Newton-Krylov algorithm accelerated by the new preconditioner in two ways: (1) the initial guess of Newton is obtained by solving a linear system defined by the coarse preconditioner; (2) the Krylov solver of the Jacobian system is preconditioned by the new preconditioner. Numerical experiments indicate that the proposed preconditioner is highly effective and robust for complex flows in a patient-specific artery with aneurysm, and it significantly reduces the numbers of linear and nonlinear iterations.

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

Continuous conditional generative adversarial networks for data-driven modelling of geologic CO[formula omitted] storage and plume evolution

Storing CO2 in geological formations requires accurate monitoring of the plume evolution for safe long-term operations, and traditionally, carbon storage practices rely on the numerical simulation of the multiphase flow and plume movement. Such simulations may be impractical for rapid, real-time monitoring, in which the risks involved are dynamically anticipated and/or mitigated, as they require the numerical solution of large non-linear systems of equations that also change over time. In this work, we investigate the application of continuous conditional generative adversarial networks (CCGAN) for predicting the CO2 plume evolution in cases in which limited data is available. We adapt the CCGAN for mapping the sparsely available input data from three wells to the spatially distributed CO2 saturation over the whole domain. Our main focus in this work is the accuracy and performance of the CCGAN when only sparse data is available. As such, we limit our investigation to the 2-D conceptual model of an aquifer and employ a simple immiscible and isothermal two-phase flow model. Our results show that the CCGAN can predict the CO2 plume from sparse data with at least 90% of accuracy while allowing a substantial reduction in the computational cost when compared to traditional methods based on numerical simulations, with a speed-up of no less than 700-folds. Future works may include the extension to 3-D reservoir models with irregular geometry.

On maximum residual nonlinear Kaczmarz-type algorithms for large nonlinear systems of equations

Recently, a class of nonlinear Kaczmarz (NK) algorithms has been proposed to solve large-scale nonlinear systems of equations. The NK algorithm is a generalization of the Newton–Raphson (NR) method and does not need to compute the entire Jacobian matrix. In this paper, we present a maximum residual nonlinear Kaczmarz (MRNK) algorithm for solving large-scale nonlinear systems of equations, which employs a maximum violation row selection and acts only on single rows of the entire Jacobian matrix at a time. Furthermore, we also establish the convergence theory of MRNK. In addition, inspired by the effectiveness of block Kaczmarz algorithms for solving linear systems, we further present a block MRNK (MRBNK) algorithm based on an approximate maximum residual criterion. Based on sketch-and-project technique and sketched Newton–Raphson method, we propose the deterministic sketched Newton–Raphson (DSNR) method which is equivalent to MRNBK, and then the global convergence theory of DSNR is established based on some assumptions and μ-strongly quasi-convex condition. Furthermore, the convergence theory of DSNR is provided under star-convex assumption. Finally, some numerical examples are tested to show the effectiveness of our new technique.

Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper. Convergence analysis demonstrates that the local order of convergence of the numerical method is five. The computer algebra system CAS-Maple, Mathematica, or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects. Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms. A dynamic evaluation of the presented methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this methodology provides useful information on iterations, such as the number of diverging-converging points and the average number of iterations as a function of initial points. Solving numerous highly nonlinear boundary value problems and large nonlinear systems of equations of higher dimensions demonstrate the performance, efficiency, precision, and applicability of a newly presented technique.

The DLR ThermoFluid Stream Library

This paper introduces the DLR Thermofluid Stream Library: a free open-source library for the robust modeling of complex thermofluid architectures. Designed to be easy to use, easy to adapt, and enriched by a number of examples, this library contains the fundamental components for many different applications, such as thermal management of electric cars, power plants, or building physics. Different from many previous software implementations, the library exploits a new computational concept on the basis of the fluid inertance that derives the mass flow balance in such a manner that large non-linear equations systems in implicit form can be avoided. Hence, there is a reliable and efficient computational scheme for simulation and initialization. Although this scheme is explained in detail in previous publications, this paper focuses on the implementational aspects of the library, explaining its structure, the underlying equations of key components and providing basic examples. In addition to the practical value of the library, we also aim to display the underlying thought driving the design of the software.

Linear active disturbance rejection control of heavy-haul train operation based on an interval type-2 fuzzy logic system model

Abstract The heavy-haul train (HHT) has large capacity and high efficiency, which represents the level of freight and makes the amelioration of control performance a trend in various countries. Improving the model reliability and increasing the anti-disturbance ability of the operation controller are two main ways to improve the operation control accuracy of HHTs. Herein, to describe the large nonlinear system more precisely, an interval type-2 fuzzy logic system (IT2FLS) is introduced to obtain a dynamic model. Then, a linear active disturbance rejection controller (LADRC) is designed to achieve precise operational control. In addition, the ‘bandwidth method’ is combined with the sparrow search algorithm (SSA) to solve the difficulty of controller parameters adjustment. Afterwards, the stability analysis of the closed-loop control system is given. The simulation experiments are conducted based on data collected from HXD1 locomotives driven by excellent drivers. Results show that the speed tracking error is no more than 0.5 km/h, and demonstrate that the proposed method significantly improves the operational performance of HHTs.

Multilevel field-split preconditioners with domain decomposition for steady and unsteady flow problems

With the incremental applications of Newton–Krylov methods for solving large sparse nonlinear systems of equations, the design of robust and scalable linear preconditioners plays an essential role for the whole solver. In this paper, we investigate the family of field-split (FS) preconditioners with different combinations of physics-based and domain decomposition methods, applied to the two typical fluid problems, i.e., the unsteady flow through fractured porous media and the steady buoyancy driven flow. In the implementation, several new versions of FS preconditioners are considered under the framework of the domain decomposition technique: additive FS, multiplicative FS, Schur-complement FS, and the constrained pressure residual (CPR) method, where the inverse of corresponding matrices is approximated by using the restricted additive Schwarz (RAS) algorithm. Rigorous eigenvalue analysis for various FS preconditioners is also provided for facilitating the design of algorithms. In particular, our approach further enhances the numerical performance by presenting a family of multilevel field-split methods for efficiently preconditioning. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the proposed preconditioning strategies for both standard benchmarks as well as realistic flow problems on a supercomputer.

High-resolution cerebral blood flow simulation with a domain decomposition method and verified by the TCD measurement

Background: An efficient and accurate blood flow simulation can be useful for understanding many vascular diseases. Accurately resolving the blood flow velocity based on patient-specific geometries and model parameters is still a major challenge because of complex geomerty and turbulence issues. In addition, obtaining results in a short amount of computing time is important so that the simulation can be used in the clinical environment. In this work, we present a parallel scalable method for the patient-specific blood flow simulation with focuses on its parallel performance study and clinical verification.Methods: We adopt a fully implicit unstructured finite element method for a patient-specific simulation of blood flow in a full precerebral artery. The 3D artery is constructed from MRI images, and a parallel Newton–Krylov method preconditioned with a two-level domain decomposition method is adopted to solve the large nonlinear system discretized from the time-dependent 3D Navier-Stokes equations in the artery with an integral outlet boundary condition. The simulated results are verified using the clinical data measured by transcranial Doppler ultrasound, and the parallel performance of the algorithm is studied on a supercomputer.Results: The simulated velocity matches the clinical measured data well. Other simulated blood flow parameters, such as pressure and wall shear stress, are within reasonable ranges. The results show that the parallel algorithm scales up to 2160 processors with a 49% parallel efficiency for solving a problem with over 20 million unstructured elements on a supercomputer. For a standard cerebral blood flow simulation case with approximately 4 million finite elements, the calculation of one cardiac cycle can be finished within one hour with 1000 processors.Conclusion: The proposed method is able to perform high-resolution 3D blood flow simulations in a patient-specific full precerebral artery within an acceptable time, and the simulated results are comparable with the clinical measured data, which demonstrates its high potential for clinical applications.

Simulation of Network Systems Based on Loop Flows Algorithms

This paper presents a simulation scheme for water distribution systems based on loop flows algorithms. Water networks are large scale and non-linear systems. The operational control of such system has posed difficulties in the past to the human operator that had to take the right decisions, such as pumping more water or closing a valve, within a short period of time and quite frequently in the absence of reliable measurement information such as pressure and flow values. Computer simulations of such systems have alleviated these difficulties. They allowed 'what-if' scenarios to be run by the human operator, giving him the possibility to know in advance the operational problems that can arise in the real-life networks due to malfunctions of valves or burst in pipes. However, these computer simulations are generally relying on solving a system of equations and the nodal heads variables have been the most frequently used variables in these computer simulations. More recently, the numerical algorithms based on loo

Complexity of periodic sequences of large-scale rotating machinery system coupled by multi-fault

Taking the steam turbine cracked rotor system coupling multi-fault as an example, this paper reports systematic numerical experiments exploring two-parameter periodic complexification cascades. Specifically, we report high-resolution phase diagrams predicting and describing how the periodic sequences unfold over a quite extended range of parameter planes. Surprisingly, such diagrams reveal that the distribution of periodic sequences either agrees well with the characteristics of “eye” of chaos, or is characterized by the derived Farey sum tree and the bidirectional symmetrical Stern–Brocot sum tree. These cascades show that the frequent occurrence of transient multi-period motion is a generic feature for the large rotating machinery system with multi-fault. And the paper obtains a cluster of sequences of the Farey sum tree and the Stern–Brocot sum tree specially suitable for the large nonlinear and nonautonomous systems, which can provide a powerful reference for failure prediction and parameter matching.