Large Systems Of Nonlinear Equations Research Articles

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.

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.

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.

General Power Flow Calculation for Multi-Terminal HVDC System Based on Sensitivity Analysis and Extended AC Grid

The power flow calculation of the multiterminal high voltage direct current (MTDC) system is essential for planning sustainable energy sources and power flow analysis for the MTDC system. However, the traditional unified methods require a large system of non-linear equations leading to low calculational efficiency. Also, for a large DC grid, there is a concern about the convergence of sequential methods. This paper proposes a general power flow calculation method for voltage source converter (VSC) based MTDC systems. Based on an extended topology of an AC grid, a generalized calculation model of the MTDC power flow is proposed. Then, a novel sensitivity analysis-based power flow (SAPF) method is proposed, in which the state variables of the extended AC grid are calculated via sensitivity analysis. With a smaller system of equations, the proposed SAPF method has less computational burden than the traditional unified methods and causes no convergence problem compared to sequential methods. To further improve the calculational efficiency, the sensitivity-based variable updating is adopted to accelerate the iterative process. By comparing with the existing methods in calculating power flow for different MTDC systems, including large systems with multiple AC/DC grids, the effectiveness and scalability of the proposed methods are verified.

Robust Modeling of Volumes for Dynamic Simulations of Thermo-Fluid Stream Networks

Modeling of complex thermo-fluid systems often leads to either many states or large non-linear systems of equations, which are both undesirable especially for real-time applications. Both of these issues can be solved by our algebraic stream-dominated approach to build hard real-time capable simulations. In this approach volume elements play a central role as boundaries and loop-breakers for fluid streams.In this paper, we first derive a lumped state standard volume model from first principle and then several specialized volumes for different applications. We regularize the models to perform robustly, even in degenerated operating conditions that might appear in dynamic simulations. While doing so, we show all approximations and assumptions used to arrive at the final models. Finally, we present some application examples showcasing the use of the different volumes for dynamic simulations. Alongside the models developed in this paper these examples are part of our open-source Modelica library and their implementation is therefore available.

CMMSE: A novel scheme having seventh-order convergence for nonlinear systems

In this paper, we present a new three-point iterative scheme for obtaining the solution of nonlinear system having seventh-order convergence. The beauty of our scheme is that we obtained the seventh-order convergence with minimal computational cost as compared to the existing ones. In addition, we also analyze the theoretical convergence properties of the proposed scheme. Moreover, we show its applicability on a total six numbers of nonlinear models: first three of them are boundary value, Hammerstein integral and 2D Bratu’s problems; the last three are standard academic large systems of nonlinear equations of order 50 × 50, 100 × 100 and 120 × 120, respectively. Finally, we concluded on the basis of obtained numerical experiments that our iterative method performs better in terms of residual error, computational efficiency, error between the two consecutive iterations, CPU-time, asymptotic error constant term and approximated root.

Composite Backward Differentiation Formula for the Bidomain Equations.

The bidomain equations have been widely used to model the electrical activity of cardiac tissue. While it is well-known that implicit methods have much better stability than explicit methods, implicit methods usually require the solution of a very large nonlinear system of equations at each timestep which is computationally prohibitive. In this work, we present two fully implicit time integration methods for the bidomain equations: the backward Euler method and a second-order one-step two-stage composite backward differentiation formula (CBDF2) which is an L-stable time integration method. Using the backward Euler method as fundamental building blocks, the CBDF2 scheme is easily implementable. After solving the nonlinear system resulting from application of the above two fully implicit schemes by a nonlinear elimination method, the obtained nonlinear global system has a much smaller size, whose Jacobian is symmetric and possibly positive definite. Thus, the residual equation of the approximate Newton approach for the global system can be efficiently solved by standard optimal solvers. As an alternative, we point out that the above two implicit methods combined with operator splittings can also efficiently solve the bidomain equations. Numerical results show that the CBDF2 scheme is an efficient time integration method while achieving high stability and accuracy.

Numerical optimization of a multiphysics calculation scheme based on partial convergence

This paper deals with the numerical optimization of a multiphysics calculation scheme. The purpose of this tool is the fine-scale neutronic and thermal-hydraulic modelling of Pressurized Water Reactors (PWRs), under steady-state nominal conditions and fission products equilibrium concentrations. The neutronic model follows a two-steps approach with pin-cell homogenization. The considered coupling scheme includes the neutronic core model, the subchannel thermal-hydraulic and heat conduction one and the isotopic depletion one. From the numerical standpoint, this problem is a large system of non-linear equations. For its resolution, two standard numerical methods are considered, the damped fixed-point method and the Anderson acceleration. In particular, this work focuses on the application of the partial-convergences within these methods. The partial-convergence technique deals with the research of a progressive internal convergence of the considered neutronic and thermal-hydraulic solvers. Within this approach, the degree of convergence is controlled either by limiting the number of single-solver iterations or by adjusting the required precisions on the respective key variables. To realise a large number of simulations in an affordable time, the chosen case study is a mini-core (5 × 5 PWR fuel assemblies plus reflector). The application of the partial-convergences to the fixed-point algorithm is rather common in the industry, but few systematic studies have been published. The impact of limiting both the neutronic and the thermal-hydraulic iterations is deeply analysed. In this context, the partial-convergences allow to strongly increase the robustness and the efficiency of the fixed-point method. For what concerns the application of this technique to the Anderson algorithm, a new strategy is proposed and preliminary tests show promising results.

Robust object-oriented formulation of directed thermofluid stream networks

ABSTRACT Object-oriented formulation of thermal fluid streams often yields large non-linear equation systems whose numerical solution is difficult to achieve. This paper revisits the fundamental equations for thermal fluid streams and introduces a new term: the steady mass flow pressure . Using this term, the equations can be brought into a form where all non-linear computations are explicit. This enables a robust and object-oriented formulation of even complex architectures. The modelling of aircraft environmental control systems is presented as one possible application example.

A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend the earlier scalar scheme [16] to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one more additional step, we obtain minimum 5th-order convergence for every value of a free parameter, θ∈ℝ, and for θ=−1, the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Furthermore, numerical results demonstrate the validity and reliability of the suggested methods.

A manifold-based approach to sparse global constraint satisfaction problems

We consider square, sparse nonlinear systems of equations whose Jacobian is structurally nonsingular, with reasonable bound constraints on all variables. We propose an algorithm for finding good approximations to all well-separated solutions of such systems. We assume that the input system is ordered such that its Jacobian is in bordered block lower triangular form with small diagonal blocks and with small border width; this can be performed fully automatically with off-the-shelf decomposition methods. Five decades of numerical experience show that models of technical systems tend to decompose favorably in practice. Once the block decomposition is available, we reduce the task of solving the large nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The proposed method resolves this issue with the novel backsolve step. We study the effect of the decomposition on a sequence of challenging problems. Beyond a certain problem size, the computational effort of multistart (no decomposition) grows exponentially. In contrast, thanks to the decomposition, for the proposed method the computational effort grows only linearly with the problem size. It depends on the problem size and on the hyperparameter settings whether the decomposition and the more sophisticated algorithm pay off. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size hyperparameter improves the robustness of the proposed method.

A New Broyden rank reduction method to solve large systems of nonlinear equations

We propose a modification of limited memory Broyden methods, called dynamical Broyden rank reduction method, to solve high dimensional systems of nonlinear equations. Based on a thresholding process of singular values, the proposed method determines a priori the rank of the reduced update matrix. It significantly reduces the number of singular values decomposition calls of the update matrix during the iterations. Local superlinear convergence of the method is proved and some numerical examples are displayed.

Extending the applicability of the inexact Newton-HSS method for solving large systems of nonlinear equations

We revisit the study of the semi-local convergence of the inexact Newton-HSS method (INHSS) introduced by Amiri et al. (2018), for solving large systems of nonlinear equations. In particular, first we present the correct convergence criterion, since the one in the preceding reference is incorrect. Secondly, we present an even weaker convergence criterion using our idea of recurrent functions. Moreover, the bound functions are compared. Finally, numerical examples are provided to show that the earlier convergence criteria are not satisfied but the new ones are satisfied. Hence, the applicability of the INHSS method is extended and under the same information as in the earlier studies.

Determination of Gas Pressure Distribution in a Pipeline Network using the Broyden Method

A potential problem in natural gas pipeline networks is bottlenecks occurring in the flow system due to unexpected high pressure at the pipeline network junctions resulting in inaccurate quantity and quality (pressure) at the end user outlets. The gas operator should be able to measure the pressure distribution in its network so the consumers can expect adequate gas quality and quantity obtained at their outlets. In this paper, a new approach to determine the gas pressure distribution in a pipeline network is proposed. A practical and user-friendly software application was developed. The network was modeled as a collection of node pressures and edge flows. The steady state gas flow equations Panhandle A, Panhandle B and Weymouth to represent flow in pipes of different sizes and a valve and regulator equation were considered. The obtained system consists of a set of nonlinear equations of node pressures and edge flowrates. Application in a network in the field involving a large number of outlets will result in a large system of nonlinear equations to be solved. In this study, the Broyden method was used for solving the system of equations. It showed satisfactory performance when implemented with field data.

A simple iterative method for water distribution network analysis

In this paper, a new method categorized as a modification to the Q–h equations is proposed for the analysis of pipe networks. The method is inspired by the Kani method which is used for analysis of structural frames. The new method can be considered as an iterative procedure which does not require a large system of nonlinear equations to be solved simultaneously. The main advantages of the proposed method are relative simplicity in formulation and programming, and rapid and smooth convergence of initially guessed values. To show the robustness and convergence of the present method, several pipe networks are analyzed, and the results are compared with those of the conventional methods.

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equation. Motivated by the PMHSS method, we develop a new method of solving a class of linear equations with block two-by-two complex coefficient matrix by introducing two coefficients, noted as DPMHSS. By making use of the DPMHH iteration as the inner solver to approximately solve the Newton equations, we establish modified Newton-DPMHSS (MN-DPMHSS) method for solving large systems of nonlinear equations. We analyze the local convergence properties under the Holder continuous conditions, which is weaker than Lipschitz assumptions. Numerical results are given to confirm the effectiveness of our method.

A parallel Navier–Stokes solver using spectral discretisation in time

ABSTRACTWe investigate the performance of a multi-harmonic space–time approach to solve time-periodic flow problems. It employs a Fourier spectral discretisation of the time domain. The resulting large system of nonlinear equations is solved iteratively and in parallel. For illustration, a three-dimensional channel flow disturbed by an oscillating force is simulated.