Nonlinear Solver Research Articles

A multi-fidelity approach for the evaluation of extreme wave loads using nonlinear response-conditioned waves

In the present study, a multi-fidelity methodology based on response-conditioned waves (RCW) is demonstrated for the probabilistic assessment of wave-induced loads and responses of offshore structures. The methodology consists of two steps: (i) the RCW are determined using a surrogate response model and (ii) they are reproduced in an experimental or CFD-based numerical wave tank (NWT) to obtain the fully nonlinear response, as a high-fidelity evaluation. The main novelty of the proposed response-conditioning technique is that it uses a fully nonlinear wave solver that calculates the wave propagation inside an NWT. In this work, the extreme Vertical Bending Moment (VBM) of a zero-speed containership is investigated. It is found that the proposed approach provides nonlinear wave sequences that can be explicitly and exactly reproduced experimentally, up to very extreme sea states (Hs=17 m, Tp=15.5 s). Moreover, through comparison of the obtained results with an experimental Monte Carlo approach, it is shown that the overall framework can predict the short-term distribution of the VBM accurately and efficiently. Finally, given that model tests are costly and not widely accessible, the implementation of the high-fidelity response evaluation within a CFD-based NWT is also demonstrated and validated against the available experimental results.

Accurate and Efficient Numerical Simulation of Land Models Using SUMMA With SUNDIALS.

Numerical simulation of land models without error control can be highly inaccurate. We present the incorporation of the Suite of Nonlinear and Differential-Algebraic Equation Solvers (SUNDIALS) package to solve the equations that simulate thermodynamics and hydrologic processes in the Structure for Unifying Multiple Modeling Alternatives (SUMMA) land model. The algorithmic features of SUNDIALS, such as error estimation and adaptive order and step-size control, result in a SUMMA-SUNDIALS model that delivers substantially improved accuracy and relative computational efficiency compared to integration with the previous SUMMA model, which uses the low-order backward Euler method with no rigorous error control. The results are demonstrated through simulations over the North American continent with more than 500,000 spatial elements. Compared to the previous SUMMA model, we find that the simulations produced by the SUMMA-SUNDIALS model are orders of magnitude closer to converged solutions for the same computational cost. Being able to efficiently perform more reliable simulations makes the SUMMA-SUNDIALS model a powerful tool for improving our understanding of the terrestrial component of the Earth System.

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.

Comparison of the Biomechanical Effect of Distal İmplants Placed at Different Angles in the All-On-Four Technique: A Nonlinear Finite Element Analysis: (Effect of Distal Implant Angles in All-on-Four)

IntroductionThe "All-on-four" technique addresses this by placing two vertical implants in the anterior region and two posterior implants angled up to 45 degrees. This study evaluates stress distribution on bone, implants, abutments, gingiva, and prostheses based on posterior implant angulation, using both standard and angled neck implants. Additionally, angled neck implants were used alongside standard implants. MethodsTo evaluate the biomechanical stress and displacement behavior of implants placed at 15°, 30°, 45°, and 30° neck-angled in the interforaminal region of using the All-on-four technique, under a 200N masticatory force on bone, implant, abutment, and prosthesis, using the FEA method with a nonlinear solver. ResultsThe stress values measured in the neck region of the angled implant and abutment placed in the distal region were highest in model IV (305 and 420 MPa), followed model III (282 and 359 MPa), model II (192 and 337 MPa) and lowest in model I (177 and 225 MPa), respectively. The mean elemental von Mises stress in the mandibular bone region where the implant was placed was highest in model I (56 MPa) and lowest in model III (40.4 MPa). ConclusionAs the angle of insertion of the implant into the bone increases, the biomechanical stress value on the implant and abutment also increases. Among the models, the lowest von Mises stress values on the implant and abutment were measured in model I and model II. Models with low stress values can be recommended for patients with clinical conditions and biomechanical risk factors.

Compact High Order Finite Volume Method and Its Application

This paper presents a compact high-order finite volume method based on recording the weighted integral value of a function on a grid cell. Extending the general finite volume method, it employs a single unit template and a limiter method to suppress oscillations, addressing the convection-diffusion equation. By utilizing implicit methods for nonlinear and linear solvers of the discrete equation, the scheme achieves enhanced compactness, accuracy, and computational efficiency.

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.

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.

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.

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.

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.

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.

Evaluation of beam convergence driven by spherical gradient refractive index lenses based on nonparaxial beam propagation.

We utilize a theoretical method based on nonlinear beam propagation and finite difference eigenmode solver methods to precisely simulate Gaussian beam propagation in electrical fields through spherical gradient refractive index lenses. The theoretical computation uses second-order partial differentiation of propagation coordinates to generate microwave field propagation. Consequently, it offers accurate simulation results for any complex refractive index profile. The reliability of the proposed method is verified by comparing it with existing experimental and theoretical results. We employ the theoretical method to assess Gaussian beam convergence in terms of four key parameters: beam waist, maximum intensity, focal position, and Rayleigh range. The results indicate that gradient index spherical lenses have better convergence than convex thin lenses, as evidenced by a significant reduction in beam waist size. However, these lenses prompt an extremely short back focal length. Consequently, we propose a slight shift in the boundary and index distribution of spherical lenses to expand their back focal lengths.

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.