Nonlinear Solver Research Articles

Minimizing the Formation of Thermal Residual Stresses of the Elastic Zone in Functionally Graded Materials

Abstract: Functionally graded materials (FGM’s) are advanced composites consisting of two or more materials whose composition and structure changes gradually over the volume, leading to gradual change in its property. Typically one component is a ceramic, while the other is metal or metal alloy. In common embodiments, the composition can change gradually from all ceramic on one side to all metal on the other side, the ceramic contributing high resistance to temperature with the metal contributing high ductility. It’s a way to incorporate favorable properties of two materials into different locations of a single structure. In ideal cases the composition changes gradually but for manufacturing reasons, the changes occurs in homogeneous layers. The boundaries between the layers cause thermal residual stresses to arise, especially during material processing and in cases of cyclic loading. These thermal residual stresses are a result of differences in the thermal expansion coefficient of the varying compositions of materials. Thermal residual stresses may lead to cracking and ultimate failure of FGM’s. This study investigates the formation of thermal residual stresses for a Nickel-Alumina Ni-Al₂O₃ FGM. The Ni-Al₂O₃ system is chosen because it is one of the most common systems used in practice. This work explores the formation, impact and minimization of thermal residual stresses for a number of practical conditions that may arise during processing. All relevant properties are calculated using the rule of mixture equations. Abaqus, a non-linear FEA solver, is used for all the simulation work. In addition, the study highlights the need for examining elasto-plastic analysis to illuminate the process of crack development which is our future research goal.

Finite Element Modeling of the Plastic Zone Thermal Residual Stresses in Functionally Graded Materials

Abstract: Functionally graded materials (FGM’s) are advanced composites consisting of two or more materials whose composition and structure changes gradually over the volume, leading to gradual change in its property. Typically one component is a ceramic, while the other is metal or metal alloy. In common embodiments, the composition can change gradually from all ceramic on one side to all metal on the other side, the ceramic contributing high resistance to temperature with the metal contributing high ductility. It’s a way to incorporate favorable properties of two materials into different locations of a single structure. In ideal cases the composition changes gradually but for manufacturing reasons, the changes occurs in homogeneous layers. The boundaries between the layers cause thermal residual stresses to arise, especially during material processing and in cases of cyclic loading. These thermal residual stresses are a result of differences in the thermal expansion coefficient of the varying compositions of materials. Thermal residual stresses may lead to cracking and ultimate failure of FGM’s. This study investigates the formation of thermal residual stresses for a Nickel-Alumina Ni-Al₂O₃ FGM. The Ni-Al₂O₃ system is chosen because it is one of the most common systems used in practice. This work explores the formation, impact and minimization of thermal residual stresses for a number of practical conditions that may arise during processing. All relevant properties are calculated using the rule of mixture equations. Abaqus, a non-linear FEA solver, is used for all the simulation work. In addition, the study includes an elasto-plastic analysis to illuminate the process of crack development.

Monitoring of spatio-temporal properties with nonlinear SAT solvers

The automotive industry is increasingly dependent on computing systems with different critical requirements. The verification and validation methods for these systems are now leveraging complex AI methods, for which the decision algorithms introduce non-determinism, especially in autonomous driving. This paper presents a runtime verification technique agnostic to the target system, which focuses on monitoring spatio-temporal properties that abstract the evolution of objects’ behavior in their spatial and temporal flow. First, a formalization of three known traffic rules (from the Vienna convention on road traffic) is presented, where a spatio-temporal logic fragment is used. Then, these logical expressions are translated to a monitoring model written in first-order logic, where they are processed by a non-linear satisfiability solver. Finally, the translation allows the solver to check the validity of the encoded properties according to an instance of a specific traffic scenario (a trace). The results obtained from our tool, which automatically generates a monitor from a formula, show that our approach is feasible for online monitoring in a real-world environment.

The semi-implicit DLN algorithm for the Navier-Stokes equations

AbstractDahlquist, Liniger, and Nevanlinna design a family of one-leg, two-step methods (the DLN method) that is second order, $$\varvec{A}-$$ A - and $$\varvec{G}-$$ G - stable for arbitrary, non-uniform time steps. Recently, the implementation of the DLN method can be simplified by the refactorization process (adding time filters on the backward Euler scheme). Due to these fine properties, the DLN method has strong potential for the numerical simulation of time-dependent fluid models. In the report, we propose a semi-implicit DLN algorithm for the Navier-Stokes equations (avoiding non-linear solver at each time step) and prove the unconditional, long-term stability, and second-order convergence with the moderate time step restriction. Moreover, the adaptive DLN algorithms by the required error or numerical dissipation criterion are presented to balance the accuracy and computational cost. Numerical tests will be given to support the main conclusions.

High-Order Spectral Irregular Wave Generation Procedure in Experimental and Computational Fluid Dynamics Numerical Wave Tanks, with Application in a Physical Wave Tank and in Open-Source Field Operation and Manipulation

The accurate generation of a target sea state in numerical or experimental wave tanks is a fundamental line of research for the ocean engineering community. It guarantees the quality and relevance of wave–structure interaction tests. This study presents a reproducible irregular wave generation and qualification procedure, accounting for the nonlinear aspects of wave propagation. It can be used for both numerical simulation and experiments. The presented numerical and experimental results are obtained from the OpenFOAM solver and the Ecole Centrale Nantes wave tank facilities, respectively. The procedure comprises two steps: First, the wavemaker motion is calibrated numerically to generate the target wave spectrum at the position of interest. This is achieved with a wavemaker-equipped nonlinear potential flow solver. The open-source HOS-NWT solver, based on the high-order spectral method, was employed in this study. Then, the corrected wavemaker motion is used directly in the experimental wave tank. OpenFOAM simulations were performed to generate waves with the relaxation method, using wave elevation and velocity field data from HOS-NWT. The procedure was finally tested for mild and extreme breaking sea states. The waves generated by the HOS-NWT solver, the experiment, and the OpenFOAM simulation were compared from both stochastic and deterministic perspectives.

ML‐Augmented Bayesian Optimization of Pain Induced by Microneedles

AbstractMicroneedles (MNs) have emerged as a promising solution for drug delivery and extraction of body fluids. Pain is an important physiological attribute to be examined when designing MNs. There is no known representation of pain with geometric features of a MN despite the focus on experimental work. This study focuses on optimizing MN designs with the aim of minimizing pain through means of machine learning, finite element analysis, and optimization tools. Three distinct approaches are proposed. The first approach involves training multiple regression models on data obtained through finite element analysis in COMSOL. The second approach uses COMSOL's built‐in nonlinear optimization solver. Finally, the third approach utilizes the LiveLink interface between COMSOL and MATLAB, combined with Bayesian optimization. Each approach presents unique strengths and challenges, with the third approach demonstrating significant promise due to its efficiency, practicality, and time‐saving.

Advancing Convergence Rates in Nonlinear Equation Solvers

It is crucial to achieve high-order convergence for nonlinear equations in numerical analysis and scientific computing. Newton-Raphson and other conventional techniques frequently converge slowly. High-order algorithms offer quicker convergence and less expensive processing.This abstract outlines important ideas and innovations, such as improved Newtonian methods for better convergence. To hasten convergence, these techniques make use of enhanced Jacobian matrix approximations and higher-order derivatives. Additionally, it covers useful topics like dealing with stability problems and computing higher-order derivatives, which could lead to efficiency and accuracy gains in scientific and engineering applications. Keywords: Non- linear Equations, High Order Convergence, Numerical Methods, Jacobian Matrix, High Order Derivatives

Distributed Nonlinear Least-Squares Solver for Practical Network Determination

Distributed Nonlinear Least-Squares Solver for Practical Network Determination

A Dual-Arm Robot Cooperation Framework Based on a Nonlinear Model Predictive Cooperative Control

Dual-arm robots show better task adaptability but face more constraints than single-arm robots. Inspired by human cooperation options, a nonlinear model predictive cooperative control (NMPCC) coupled with a cooperative index is proposed in this article. By adjusting the cooperative index, dual-arm cooperation can be classified into four modes: tracking, motion synchronization, impedance, and force priority. Thus, the cooperation operation problem is converted to a multiobjective optimization problem. Then, a nonlinear model predictive solver coupled with the ACADO toolkit is designed to solve the multiobjective optimization problem, where the robotic torque control input can be calculated in real time (less than 1 ms). The motion synchronization, cooperative transportation, and human dual-arm robot interaction experiments were conducted on dual Franka panda robots. Experiments reveal that the NMPCC control coupled with the cooperative index is easy to apply in dual-arm robots and can adapt to different complex manipulation tasks.

A compact, equality-based weighted residual formulation for periodic solutions of systems undergoing frictional occurrences.

A very compact weighted residual formulation is proposed for the construction of periodic solutions of oscillators subject to frictional occurrences. Coulomb's friction is commonly expressed as a differential inclusion which can be cast into the complementarity formalism. When targeting periodic solutions, existing algorithms rely on a procedure alternating between the frequency domain, where the dynamics is solved, and the time domain, where friction is dealt with. In contrast, the key idea of the present work is to express all governing equations including friction as equalities, which are then satisfied in a weak integral sense through a weighted residual formulation. The resulting algebraic nonlinear equations are solved numerically using an adapted trust region nonlinear solver and basic integral quadrature schemes. To increase efficiency, the Jacobian of the friction forces is calculated analytically in a piecewise linear fashion. The shape functions considered in this work are the classical Fourier functions. It is shown that periodic solutions with clear multiple sticking and sliding phases can be found with a high degree of accuracy. The equality-based formulation is shown to be effective and efficient, convergence being achieved in all cases considered with low computational cost, including for large numbers of harmonics. Importantly, this new friction formulation does not suffer from the typical limitations or hypotheses of existing frequency-time domain methods for non-smooth systems, such as regularization, penalization, or massless frictional interfaces.

A Novel Projection-based Embedded Discrete Fracture Model (pEDFM) for Anisotropic Two-phase Flow Simulation Using Hybrid of Two-point Flux Approximation and Mimetic Finite Difference (TPFA-MFD) Methods

This paper presents a novel projection-based embedded discrete fracture model (pEDFM) applicable to general two-phase flow simulation in anisotropic reservoirs for the first time. The novel pEDFM is developed based on hybrid two-point flux approximation and mimetic finite difference (TPFA-MFD) methods, which retains the basic features of two-point flux approximation (TPFA) based generic pEDFM workflow and incorporates new features of mimetic finite difference (MFD) to handle full-tensor permeabilities. This is achieved by introducing the concept of effective connection, formulating additional pressure degrees of freedom for effective connections, assigning effective matrix-matrix (m-m) and matrix-fracture (m-f) connections to the corresponding sides of matrix-cell interfaces, presenting methods corresponding to different types of connections to handle low-conductivity fractures, and further deriving formulas of numerical fluxes for effective connections. The resulting global equation system is solved by a Newton-Raphson nonlinear solver with the temporary discretization of the implicit backward Euler scheme. The 2D and 2.5D numerical examples show the novel pEDFM could achieve the same level of computation accuracy and convergence as the reference discrete fracture model (DFM), while avoiding the gridding difficulty of complex fracture networks. It also significantly outperforms classical EDFM based on MFD and generic pEDFM workflow in the computational accuracy for cases with anisotropic full tensor permeabilities. The novel pEDFM, to our best knowledge, is for the first time proposed and developed for general anisotropic two-phase flow simulation and maybe the numerical simulation framework for multi-phase flow in fractured reservoirs with the best comprehensive performance so far.

BilevelJuMP.jl: Modeling and Solving Bilevel Optimization Problems in Julia

In this paper, we present BilevelJuMP.jl, a new Julia package to support bilevel optimization within the JuMP framework. The package is a Julia library that enables the user to describe both upper and lower-level optimization problems using the JuMP algebraic syntax. Because of the generality and flexibility that our library inherits from JuMP’s syntax, our package allows users to model bilevel optimization problems with conic constraints in the lower level and all constraints supported by JuMP in the upper level including conic, quadratic, and nonlinear constraints. Moreover, the models defined with the syntax from BilevelJuMP.jl can be solved by multiple techniques that are based on reformulations as mathematical programs with equilibrium constraints (MPEC). Manipulations on the original problem data are possible due to MathOptInterface.jl’s structures and Dualization.jl features. Hence, the proposed package allows quick modeling, deployment, and thereby experimenting with bilevel models based on off-the-shelf mixed-integer linear programming and nonlinear solvers. History: Accepted by Ted Ralphs, Area Editor for Software Tools. Funding: The authors were partially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The work of A. Street was also partially supported by Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). The work was partially funded by the project P&D ANEEL PD-00403-0050/2020 sponsored by ENGIE BRASIL ENERGIA S.A. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0135 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0135 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Performance of a Raft-Type Wave Energy Converter with Diverse Mooring Configurations

The development and utilization of wave energy, heralded as a potential leading source of clean energy worldwide, have garnered considerable attention from the global research community. Among the diverse array of wave energy converters (WECs), the raft-type WEC stands out for its potential to efficiently harness and utilize wave energy, offering high energy conversion rates and a broad frequency response range. This paper delves into the evaluation of a raft-type WEC’s performance in various mooring configurations under different wave conditions. Our analysis primarily focuses on the dynamics of the two-body WEC using a weakly nonlinear three-dimensional potential flow solver. The considered device comprises two interconnected floating barges, incorporating a power take-off system at the hinged connection point. This investigation involves the use of equivalent linear damping to model the power take-off (PTO) system. To validate the numerical simulations, we conduct physical model experiments with WECs. Additionally, the coupling of the raft-type WEC’s dynamics and its mooring dynamics was examined, highlighting the performance differences between various mooring systems through a comparative analysis.

Analysis of a simplified approach for the determination of modified Kostiakov model parameters.

Accurate estimation of infiltration losses is very important in many hydrologic projects. However, the lack of available data often forces hydrologists to resort to the application of empirical and semi-empirical models for estimating the infiltration rates. Estimation of infiltration rate from these empirical and semi-empirical models is highly dependent on the method adopted to determine the parameters of models. One of the most common infiltration models known as the Kostiakov model was modified to rectify zero infiltration at the start of the infiltration process. Recently, a simplified approach for determining the parameters of the modified Kostiakov model was proposed. The present article focuses on analyzing the reliability of the simplified approach for determining the parameters of the modified Kostiakov model. In this regard, six observed datasets were analyzed from different parts of the world to arrive at some generalized results on the reliability of the simplified approach. Cumulative infiltration obtained from simplified approaches was compared with that obtained through the application of an excel spreadsheet-based nonlinear optimization solver. It was found that on average Sum of Square of Error reduced by 72% on application of nonlinear optimization as compared to the simplified approach.

A comparison of Rosenbrock–Wanner and Crank–Nicolson time integrators for atmospheric modelling

AbstractNonhydrostatic atmospheric models often use semi‐implicit temporal discretisations in order to negate the time‐step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the nonlinear solver is typically truncated to a fixed number of iterations, often using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third‐order, four stage Rosenbrock–Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank–Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock–Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly‐stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow‐water equations and the 3D compressible Euler equations at both planetary and nonhydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank–Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock–Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at nonhydrostatic scales.

Transformed primal–dual methods for nonlinear saddle point systems

Abstract A transformed primal–dual (TPD) flow is developed for a class of nonlinear smooth saddle point systemThe flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit–explicit, and Gauss–Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex–concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed.

Power Plastics: A Hybrid Lagrangian/Eulerian Solver for Mesoscale Inelastic Flows

We present a novel hybrid Lagrangian/Eulerian method for simulating inelastic flows that generates high-quality particle distributions with adaptive volumes. At its core, our approach integrates an updated Lagrangian time discretization of continuum mechanics with the Power Particle-In-Cell geometric representation of deformable materials. As a result, we obtain material points described by optimized density kernels that precisely track the varying particle volumes both spatially and temporally. For efficient CFL-rate simulations, we also propose an implicit time integration for our system using a non-linear Gauss-Seidel solver inspired by X-PBD, viewing Eulerian nodal velocities as primal variables. We demonstrate the versatility of our method with simulations of mesoscale bubbles, sands, liquid, and foams.

Robust conservative scheme and nonlinear solver for phase transitions in heterogeneous permafrost

In this paper we study computational schemes to simulate freezing and thawing in permafrost modeled by a nonlinear heat equation with constitutive properties resembling those in the well known Stefan problem but featuring distinct challenges. The models are discretized with low order conservative discretization and fully implicit time stepping. We explore the challenges due to the nonsmooth nature of the temperature–enthalpy relationship and the domain heterogeneity, with focus on the solver and the use of enthalpy as a primary variable in contrast to the temperature variable used commonly in applications literature. We prove and demonstrate the convergence of our algorithms in realistic physical scenarios.

A Time-Domain Volume Integral Equation Solver to Analyze Electromagnetic Scattering From Nonlinear Dielectric Objects

A time domain electric field volume integral equation (TD-EFVIE) solver is proposed for analyzing electromagnetic scattering from dielectric objects with Kerr nonlinearity. The nonlinear constitutive relation that relates electric flux and electric field induced in the scatterer is used as an auxiliary equation that complements TD-EFVIE. The ordinary differential equation system that arises from TD-EFVIE’s Schaubert-Wilton-Glisson (SWG)-based discretization is integrated in time using a predictor-corrector method for the unknown expansion coefficients of the electric field. Matrix systems that arise from the SWG-based discretization of the nonlinear constitutive relation and its inverse obtained using the Padé approximant are used to carry out explicit updates of the electric field and the electric flux expansion coefficients at the predictor and the corrector stages of the time integration method. The resulting explicit marching-on-in-time (MOT) scheme does not call for any Newton-like nonlinear solver and only requires solution of sparse and well-conditioned Gram matrix systems at every step. Numerical results show that the proposed explicit MOT-based TD-EFVIE solver is more accurate than the finite-difference time-domain method that is traditionally used for analyzing transient electromagnetic scattering from nonlinear objects.

Innovations in Computational Approaches for Nonlinear Problems and Complex System Simulations

Nonlinear problem solving and complex system simulation have become critical issues in many fields of science. The development of novel computational methods is crucial to understanding these complex systems. In this abstract, we explore the dynamic landscape of computational techniques, focusing on their uses in simulating complex systems and tackling nonlinear challenges. Creating complex algorithms that can deal with nonlinearity, chaos, and emergent behaviours is where it's at. Tools for modelling and comprehending such complex systems are few, but machine learning, artificial intelligence, and evolutionary computation are at the forefront. The way problems are solved has been completely rethought because of their nonlinearity-tolerance and ability to operate in high-dimensional domains. In addition, novel opportunities have arisen due to the combination of classical mathematical models with computer methods. The behaviour and emergent features of complex systems are best understood by hybrid approaches that combine differential equations, agent-based modelling, and cellular automata. These techniques provide a fine-grained comprehension of component interactions, illuminating emergent events. Moreover, the advent of high-performance computing has substantially expanded the breadth and resolution of simulations. Scientists are now able to probe increasingly complex systems, shedding light on their dynamics and behaviours. Computational capacities have been vastly improved by parallel computing, distributed systems, and cloud computing infrastructures, allowing for the study of systems that were once thought to be intractable. Nonlinear issues and complex system simulations can benefit greatly from the combination of cutting-edge computational approaches with domain-specific expertise. This abstract is a testament to the expanding significance and potential of these computational approaches in understanding complex systems and opening up new frontiers for research and solving problems.