Nonlinear Solver Research Articles

Applications of Catalan Neural Network (CaNN) model

In this paper, we construct Catalan neural network (CaNN) model and examine the nonlinear solvers of the ordinary nonlinear differential equation using the CaNN model with linear and nonlinear activation functions. Using a single-layer functional linking neural network (FLANN) architecture, we extend input models using the first five Catalan polynomials, update network parameters that we initially randomly select, and use error backpropagation algorithms. The CaNN trial solution for solving nonlinear differential equations is in excellent agreement with the analytical solution. Also, the validity of the CaNN method is investigated by treating second-order differential equations of Lane–Emden type as boundary and initial value problems.

On thermal conduction in the solar atmosphere: An analytical solution for nonlinear diffusivity without compact support

Context. The scientific community employs complicated multiphysics simulations to understand the physics in solar, stellar, and interstellar media. These must be tested against known solutions to ensure their validity. Several well-known tests exist, such as the Sod shock tube test. However, a test for nonlinear diffusivity is missing. This problem is highly relevant in the solar atmosphere, where various events release energy that subsequently diffuses by Spitzer thermal conductivity. Aims. The aim is to derive an analytical solution for nonlinear diffusivity in 1D, 2D, and 3D, which allows for a nonzero background value. The solution is used to design a test for numerical solvers and study Spitzer conductivity in the solar atmosphere. Methods. There exists an ideal solution assuming zero background value. We performed an analytical first-order perturbation of this solution. The first-order solution was first tested against a dedicated nonlinear diffusion solver, whereupon it was used to benchmark the single- and multifluid radiative magnetohydrodynamics code Ebysus, used to study the Sun. The theory and numerical modeling were used to investigate the role of Spitzer conductivity in the transport of energy released in a nanoflare. Results. The derived analytical solution models nonlinear diffusivity accurately within its region of validity and approximately beyond. Various numerical schemes implemented in the Ebysus code is found to model Spitzer conductivity correctly. The energy from a representative nanoflare is found to diffuse 9 Mm within the first second of its lifetime due to Spitzer conductivity alone, strongly dependent on the electron density. Conclusions. The analytical first-order solution is a step forward in ensuring the physical validity of intricate simulations of the Sun. Additionally, since the derivation and argumentation are general, they can easily be followed to treat other nonlinear diffusion problems.

Nonlinear Optimization and Adaptive Heuristics for Solving Irregular Object Packing Problems

We review and present several challenging model classes arising in the context of finding optimized object packings (OP). Except for the smallest and/or simplest general OP model instances, it is not possible to find their exact (closed-form) solution. Most OP problem instances become increasingly difficult to handle even numerically, as the number of packed objects increases. Specifically, here we consider classes of general OP problems that can be formulated in the framework of nonlinear optimization. Research experience demonstrates that—in addition to utilizing general-purpose nonlinear optimization solver engines—the insightful exploitation of problem-specific heuristics can improve the quality of numerical solutions. We discuss scalable OP problem classes aimed at packing general circles, spheres, ellipses, and ovals, with numerical (conjectured) solutions of non-trivial model instances. In addition to their practical relevance, these models and their various extensions can also serve as constrained global optimization test challenges.

Nonlinear dynamic characteristics of smart FG-GPLRC sandwich varying thickness truncated conical shell with internal resonance for first three order modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.

SAM: A Modern System Code for Advanced Non-LWR Safety Analysis

The System Analysis Module (SAM), developed at Argonne National Laboratory and by collaborators at other organizations, is for advanced non–light water reactor safety analysis. SAM aims to provide fast-running, modest-fidelity, whole-plant transient analysis capabilities that are essential for fast-turnaround design scoping and engineering analyses of advanced reactor concepts. To facilitate code development, SAM utilizes the MOOSE object-oriented application framework, its underlying finite element library, and linear and nonlinear solvers to leverage modern advanced software environments and numerical methods. SAM aims to solve tightly coupled physical phenomena, including fission reaction, heat transfer, fluid dynamics, and thermal-mechanical responses in advanced reactor structures, systems, and components with high accuracy and efficiency. This paper gives an overview of the SAM code development, including goals and functional requirements, physical models, current capabilities, verification and validation, software quality assurance, and examples of simulations for advanced nuclear reactor applications.

Flow dynamics over cylinder in two‐dimensional benchmark problem: A finite element approximation

This research investigates 2D benchmark flow around a circular cylinder, utilizing the incompressible Navier–Stokes equations alongside the continuity and energy equations. The numerical solution is achieved through finite element discretization for space variable, combined with a second‐order Crank–Nicolson scheme for time integration. The computational results are derived using the FEATFLOW finite element based library package. Our study focuses on the dimensionless form of the flow equations and examines three key dimensionless parameters: drag, lift, and pressure drop. Upon applying finite element method (FEM) discretization, the system is converted into a set of linear or nonlinear ordinary differential equations or algebraic equations for steady‐state scenarios. We then apply the Newton–Raphson method as the outer nonlinear solver, and the multigrid method for efficiently resolving the linear subproblems. To ensure numerical accuracy, we evaluated the and errors, confirming that the experimental order of convergence matches the theoretical predictions. Flow profiles were both graphically represented and tabulated, offering a detailed understanding of the simulation results.

Optimizing the design and operation of water networks: Two decomposition approaches

We consider the design and operation of water networks simultaneously. Water network problems can be divided into two categories: the design problem and the operation problem. The design problem involves determining the appropriate pipe sizing and placements of pump stations, while the operation problem involves scheduling pump stations over multiple time periods to account for changes in supply and demand. Our focus is on networks that involve water co-produced with oil and gas. While solving the optimization formulation for such networks, we found that obtaining a primal (feasible) solution is more challenging than obtaining dual bounds using off-the-shelf mixed-integer nonlinear programming solvers. Therefore, we propose two methods to obtain good primal solutions. One method involves a decomposition framework that utilizes a convex reformulation, while the other is based on time decomposition. To test our proposed methods, we conduct computational experiments on a network derived from the PARETO case study.

Collaborative optimization for energy hub and load aggregator considering the carbon intensity-driven and uncertainty-aware

To achieve optimization operation between the energy hub operator and the load aggregators, this paper proposes a collaborative model for the energy hub and load aggregator considering the carbon intensity-driven and uncertainty. Firstly, a time-coupled carbon emission flow model for energy storage is proposed to cope with the change in carbon intensity at the injection node. Meanwhile, a carbon intensity-based incentive price is developed for interruptible loads with diverse carbon intensity. Secondly, the Wasserstein metric-based distributionally robust optimization method is leveraged to height against the energy hub uncertainty, in which the infinite-dimensional distributionally robust optimization model is reformulated into a tractable linear programming. Thirdly, considering the multi-entity characteristics of energy hub and load aggregators, a collaborative optimization method based on the alternating direction method of multipliers algorithm is constructed. Finally, to meet the convergence requirements of the alternating direction method of multipliers algorithm, the iterative convexification algorithm is applied to transform the nonconvex bi-level energy hub-users model into an iterative convex model by avoiding binary variables and big-M parameters. Consequently, the energy hub and load aggregator can operate cooperatively and protect privacy in uncertain conditions. Simulation results verify the effectiveness of the proposed model and method. Simulation results demonstrate the effectiveness of the proposed model and method. Specifically, the iterative convexification algorithm achieved a profit of ¥1653.82 for the load aggregator, which is 24.4 % higher than the profit obtained by CONOPT (¥1329.73) and 20.3 % higher than that obtained by IPOPT (¥1374.69), thus providing substantial advantages over traditional nonlinear solvers.

Numerical simulation of carbon dioxide flooding in fractured reservoirs using generic projection-based embedded discrete fracture model

This paper, for the first time, integrates the generic projection-based embedded discrete fracture model (pEDFM) with the commercial reservoir simulator ECLIPSE for carbon dioxide (CO2) flooding in fractured reservoirs. The integrated method first obtains inter-grid connections and corresponding transmissibilities within the reservoir model based on the generic pEDFM. It then constructs an equivalent CO2 flooding ECLIPSE model to the original pEDFM reservoir model, thereby calculating the global equations of the compositional flow model to obtain distributions of pressure, saturation, component concentrations, and well performance data. We implemented three numerical examples to verify that the proposed method can achieve significantly higher computational accuracy compared to the widely used embedded discrete fracture model in both high and low permeability fracture scenarios, while also avoiding the difficulties associated with generating matching grids for complex fracture networks. Furthermore, the proposed integrated method uses the solver within ECLIPSE to solve the global equations, thus avoiding the high cost of developing a robust nonlinear solver for complex compositional model of CO2 flooding. This demonstrates the practicality of the method and its significant potential for subsequent application to various reservoir models.

Distributed Nonconvex Optimization for Control of Water Networks with Time-coupling Constraints

AbstractIn this paper, we present a new control model for optimizing pressure and water quality operations in water distribution networks. Our formulation imposes a set of time-coupling constraints to manage temporal pressure variations, which are exacerbated by the transition between pressure and water quality controls. The resulting optimization problem is a nonconvex, nonlinear program with nonseparable structure across time steps. This problem proves challenging for state-of-the-art nonlinear solvers, often precluding their direct use for near real-time control in large-scale networks. To overcome this computational burden, we investigate a distributed optimization approach based on the alternating direction method of multipliers (ADMM). In particular, we implement and evaluate two algorithms: a standard ADMM scheme and a two-level variant that provides theoretical convergence guarantees for our nonconvex problem. We use a benchmarking water network and a large-scale operational network in the UK for our numerical experiments. The results demonstrate good convergence behavior across all problem instances for the two-level algorithm, whereas the standard ADMM approach struggles to converge in some instances. With an appropriately tuned penalty parameter, however, both distributed algorithms yield good quality solutions and computational times compatible with near real-time (e.g. hourly) control requirements for large-scale water networks.

Conjunctive optimal design of water and power networks

Water distribution systems (WDS) and power grids (PG) are critical infrastructure systems that are vital to all human activity. As such, their quality of service is of great importance for economic, environmental, and human welfare reasons. Although traditionally being analyzed separately, the two systems are interconnected and can mutually affect one another. In order to utilize the potential benefits that the two systems can produce for each other, their design and operation should be analyzed conjunctively. In this paper, a conjunctive optimal design approach for water and power networks is presented, with the objective of finding the dimensions of the systems’ facilities that will result in minimal overall costs, for both design and operation. The model is formulated and implemented on two example applications using an off-the-shelf nonlinear solver by MATLAB and compared to the optimal design of the independent WDS. A sensitivity analysis is performed to provide validity to the obtained results. The conjunctive design is compared to the design of an independent WDS to emphasize the effect of including the PG in the optimization problem. Results show a clear link between the availability of renewable energy and sizing of WDS components. The design of the independent WDS leads to the violation of PG constraints, which are satisfied when including both systems under a single optimization model, demonstrating the importance of a holistic design approach.

Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem

This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the lsqnonlin non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below 0.006% when the discretization elements are M=N=80. The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems.

Significance of Multi-Variable Model Calibration in Hydrological Simulations within Data-Scarce River Basins: A Case Study in the Dry-Zone of Sri Lanka

Traditional hydrological model calibration using limitedly available streamflow data often becomes inadequate, particularly in dry climates, as the flow regimes may abruptly vary from arid conditions to devastating floods. Newly available remote-sensing-based datasets can be supplemented to overcome such inadequacies in hydrological simulations. To address this shortcoming, we use multi-variable-based calibration by setting up and calibrating a lumped-hydrological model using observed streamflow and remote-sensing-based soil moisture data from Soil Moisture Active Passive Level 4. The proposed method was piloted at the Maduru Oya River Basin, Sri Lanka, as a proof of concept. The relative contributions from streamflow and soil moisture were assessed and optimised via the Kling–Gupta Efficiency (KGE). The Generalized Reduced Gradient non-linear solver function was used to optimise the Tank Model parameters. The findings revealed satisfactory performance in streamflow simulations under single-variable model validation (KGE of 0.85). Model performances were enhanced by incorporating soil moisture data (KGE of 0.89), highlighting the capability of the proposed multi-variable calibration technique for improving the overall model performance. Further, the findings of this study highlighted the instrumental role of remote sensing data in representing the soil moisture dynamics of the study area and the importance of using multi-variable calibration to ensure robust hydrological simulations of river basins in dry climates.

Generalized flow calculation of the gas flow in a network of capillaries used in gas chromatography.

A general method for the calculation of the flow and pressure of a gas in a network of cylindrical capillaries is presented. This method is used specifically for gas chromatographic systems in this work. With this approach, it is possible to easily calculate flow and pressures in complex gas chromatographic systems, like flow-modulated or thermal-modulated multidimensional gas chromatographic systems, or systems with multiple outlets at different pressures. A mathematic abstraction using graph theory is used to represent the system of capillaries. With this graph, the flow balance equations at the connections of the capillaries can easily be set up. Using a computer algebra system, the system of flow balance equations can be solved for the pressures at the connection points. For simple systems, this approach is presented, and calculated flows, pressures, and hold-up times are compared with measured values. In addition, two complex systems (4-Way-Splitter, Deans Switch system) of capillaries are presented with calculations only. For these systems, certain conditions were formulated, that is, a certain difference in hold-up times and a defined split ratio between different paths of these systems. Using a numeric non-linear solver, configurations of these systems were found, that fulfill these conditions.

Dynamic security constrained AC optimal power flow for microgrids

To ensure the continuous operation of a microgrid, proactive planning is essential, especially when contemplating possible dynamic events that alter the scheduled scenarios for the islanded operation or during its transition from connected to islanded mode. This paper introduces an innovative mathematical programming model for the AC optimal power flow (AC-OPF) with dynamic security constraints (DSCs), considering two scenarios: the islanded operation and the transition. This model uses positive-sequence equations to represent the inverter-based resources (IBRs) for grid-forming and grid-following roles, along with a fourth-order synchronous generator model equipped with excitation and frequency control systems. They assess the dynamic response to specific events such as short-circuits, decreases in PV generation, increases in load demand during the islanded operation, and the transition itself. The DSCs are applied to dispatchable distributed energy resources (DERs), which react to variations in the microgrid, considering generation and opportunity costs to minimize the discrepancy between planned and secure operating points. The mathematical programming model is implemented using AMPL, and solutions are obtained through the nonlinear optimization solver IPOPT. The tests are conducted in an adapted version of the microgrid being developed at the University of Campinas that includes a synchronous generator, photovoltaic (PV) generation, and a battery energy storage system (BESS). Results demonstrate the model’s effectiveness in adjusting generation dispatch to withstand defined events and optimizing generation resources, even when limits are not reached.

Effects of non-equilibrium phase behavior in nanopores on multi-component transport during CO2 injection into shale oil reservoir

Injecting CO2 into shale oil reservoirs is one of the most effective methods for enhancing oil recovery (EOR) while simultaneously facilitating CO2 capture, utilization, and storage (CCUS). However, it is not always that gas dissolution or condensate evaporation reaches equilibrium instantaneously during the CO2 injection process. In cases where equilibrium is not attained instantaneously, conventional vapor-liquid equilibrium models will have certain limitations in characterizing phase transition hysteresis. These limitations will affect the accuracy of simulating multi-component transport in porous media. In this paper, a robust and generic vapor-liquid non-equilibrium thermodynamic model considering nano-confinement effects (NC) was established by introducing component mass transfer rates proportional to the chemical potential difference. The established model was then used to modify the fully-compositional, projection-based embedded discrete fracture model (pEDFM). Furthermore, C++ hybrid programming and the Graphics Processing Unit (GPU) parallel method were both used to accelerate both the nonlinear and linear solvers. By comparing with results from MATLAB Reservoir Simulation Tools (MRST), Eclipse, and tNavigator, the accuracy and advantages of our solver were demonstrated. The results show that if the underground fluid undergoes a phase transition from a vapor-liquid two-phase state to a single-phase state, non-equilibrium thermodynamic effects need to be considered, and the simulation results show a larger two-phase zone, higher bottom hole pressure, and lower cumulative liquid production. But the nano-confinement effects will weaken the non-equilibrium thermodynamic effects.

Rheological modelling of carbonyl-iron particles (CIP) paraffin oil-based magneto-rheological fluids

New types of magneto-rheological fluids are increasingly being developed lately, but there is a dearth of information on the performance of commonly used rheological models for emerging MRFs such as carbonyl-iron particles (CIP) paraffin-oil-based MRF. This work aims to investigate the performance of some rheological models for application in predicting shear stress and yield strength in an emerging MRF suitable for flow-mode applications. CIP, low viscosity paraffin oil, and lithium grease were used as magnetic particles, carrier fluid, and additives, respectively, to prepare the MRF. Based on different mixing proportions determined with the Taguchi method of experimental design, sixteen samples were prepared following a standard procedure. For each sample, the values of viscosity and shear stress were determined using a viscometer and rheometer, respectively, with an incorporated self-developed magnetic device. By fitting the data and using the multi-objective nonlinear programming solver in Micro-soft Excel to determine optimum parameters for each model, the Bingham Model, Herschel–Buckley Model, Casson Model, Cross models, and Power-law were used to model the experimental data. Predicted shear stress values and yield strength were then analyzed using ANOVA at a 5% confidence level. The relative errors were determined using RMSE, Mean Square Error, and Mean Absolute Error. There was a significant variation in the predicted outcomes of all the models. Overall, all the models gave relatively acceptable results. However, the Herschel-Buckley model gave the best results, while the Casson model gave the worst results, judging by their values of errors. It is shown that the Herschel-Buckley model should be best used for predicting the rheological characteristics of CIP and paraffin oil-based MRF.

Nodal discontinuous Galerkin methods for Maxwell's equations in Lorentz-Kerr-Raman medium without nonlinear algebraic solver

The propagation of electromagnetic waves is modeled by time-dependent Maxwell's equations coupled with constitutive laws that describe the responses of the media. In this work, we consider a nonlinear model that describes the electromagnetic wave in an optical medium with the linear Lorentz effect and the cubic nonlinear instantaneous Kerr and delayed Raman effects. Mathematically this model obeys an energy conservative/dissipative law. Though there have been active efforts in designing numerical methods (e.g. of finite difference / finite element /discontinuous Galerkin type) to simulate this model, the methods proposed here are distinctive in that they are free of any nonlinear algebraic solvers. Moreover, in the absence of the Raman effect, our methods also enjoy a provable discrete energy law, and optimal a priori error estimates are further established when the exact solutions are sufficiently smooth. The key ingredients of the new methods include some novel treatment in time discretizations and nodal discontinuous Galerkin spatial discretization for the specific nonlinearities, and they also render a local nature of the methods and hence their suitability for parallel implementation with great efficiency. Numerical experiments are performed to illustrate the accuracy, stability, computational efficiency and parallel scalability of the proposed methods. We further apply the methods to simulate some physically relevant problems in one, two, and three dimensions.

Performance analysis of linearization schemes for modelling multi-phase flow in porous media

Reservoir simulation is crucial for understanding the flow response in underground reservoirs, and it significantly helps reduce uncertainties in geological characterization and optimize methodologies for field development strategies. However, providing efficient and accurate solutions for the strong heterogeneity remains challenging, as most of the discretization methods cannot handle this complexity. In this work, we perform a comprehensive assessment of various numerical linearization techniques employed in reservoir simulation, particularly focusing on the performance of the nonlinear solver for problem dealing with fluid flow in porous media. The primary linearization methods examined are finite difference central (FDC), finite forward difference (FDF), and operator-based linearization (OBL). These methods are rigorously analyzed and compared in terms of their accuracy, computational efficiency, and adaptability to changing reservoir conditions. The results demonstrate that each method has distinct strengths and limitations. The FDC method is more accurate particularly in complex simulations where strong heterogeneity are introduced but is generally slower in convergence. The OBL on the other hand, is more efficient and converges quickly, which makes it suitable for scenarios with limited computational resources and simple physics, while the FDF method provides a balanced combination of precision and computational speed, contingent upon careful step size management of the derivative estimations. This paper aims to guide the selection of appropriate linearization techniques for enhancing nonlinear solvers’ accuracy and efficiency in reservoir simulation .

Harmonic-modal hybrid frequency approach for parameterized non-linear dynamics

Structural dynamics systems are represented by discretized partial differential equations, whose solutions depend on various parameters. Developing high-fidelity numerical models for multi-dimensional systems or those with multiple parameters can be computationally expensive, particularly if the systems are non-linear. Consequently, the concept of a precalculated library of the system's response to a wide range of parameters is appealing. Thus, a global non-linear space-frequency solver is proposed that produces a low-rank representation of the solution using Modal Basis analysis known as the Harmonic Modal Hybrid Method. The DEIM is also used to accelerate its convergence by creating a reduced basis of the non-linear function(s) based on either calculated or experimental values. The optimized solver is then employed for rapid offline computations to construct surrogate models that can give real-time predictions of the parametrized dynamic response using the sPGD technique. These models serve as building blocks for virtual twins that necessitate near-instantaneous calculations when the system parameters and/or conditions are changed. A proof of concept is illustrated by using this technique to analyze a well-known non-linear system, the cantilevered beam with a non-linear cubic spring attached to its end. This method can be easily extended to solve other dynamical systems quickly and effectively.