Nonlinear Solver Research Articles

Accelerated nonlinear domain decomposition solver for multi-phase flow and transport in porous media

Development of robust and efficient nonlinear solution strategies for multi-phase flow and transport within natural porous media is challenging. Fully Implicit Method (FIM) is widely used in reservoir simulation for solving nonlinear algebraic systems. Previous studies showed that nonlinearities of flow and transport problems could exhibit large degrees of locality across timesteps and nonlinear iterations. Nonlinear domain decomposition (NDD) is an attractive class of methods for achieving localization and resolving spatially unbalanced nonlinearities. However, NDD is not robust as iterative solvers, because the outer iterations are essentially fixed-point type.We design a solution framework to accelerate the outer-loop convergence of NDD. We first apply a nonlinear acceleration (NA) technique to the fixed-point iterations, for correcting the boundary conditions of subdomains. Because flow problems are parabolic and exhibit global behaviors, long-range information is essential to capture main features of the flow field. We incorporate a preconditioning step based on CPR-AMG before each timestep, to provide better initial solutions. Additionally, smoother iterations can be taken to improve the boundary conditions during the NDD process.We evaluate the NDD methods using heterogeneous multi-phase models. The nonlinearities associated with the flow, transport and the coupling are locally resolved through the NDD iterations. Results demonstrate that the basic NDD solver is afflicted with severe nonlinear difficulties. By comparison, the accelerated NDD solver greatly improves the outer convergence performance. Moreover, the new solver can effectively exploit locality on the Newton level, for both the flow and transport problems.

Design, analysis and test of conventional and winglet propellers under highly loaded conditions

In this paper, some conventional and winglet inboard propellers are designed, analyzed and tested under highly loaded conditions. The design objective is to achieve the highest efficiency at a given propeller RPM and target power. A non-linear optimization solver is coupled with the panel method to design the conventional propellers. A robust parameterized geometry definition is given for the winglet propellers, with the flexibility to change the shapes of the winglet, including its location and direction, and its sections’ pitch and camber. The winglet propellers are designed using the SHERPA (Simultaneous Hybrid Exploration that is Robust, Progressive, and Adaptive) algorithm and the performance of each geometry in the optimization process is predicted by the steady RANS (Reynolds-Averaged Navier–Stokes) method with periodic boundary conditions. Both the conventional propellers and winglet propellers are designed under the open water condition and multiple designs are further analyzed by the unsteady RANS method, together with the hull, the rudder, the free surface and an inclined shaft angle. It is found that (i) in the conventional propeller designs, the unsteadiness in propeller-induced forces on the hull reduces with increased distance from the propeller blade tip to the bottom of the hull; (ii) the efficiency of optimized conventional propellers generally decreases with the decrease of propeller diameter; (iii) under the specified design conditions, the blade tips of the optimized winglet propellers exhibit a tendency to bend towards the pressure side of the blade; (iv) when appropriately designed, winglet propellers can attain enhanced efficiency and reduced unsteadiness in propeller-induced forces acting on the hull’s underside, in comparison to conventional propellers of the same overall diameter. Consequently, winglet propellers are good design choices for highly loaded conditions with strict constraints on diameter. Finally, one conventional propeller design and one winglet propeller design are selected to make full-scale prototypes and tested on the water, where the fuel economy of these prototypes is measured and compared with the numerical predictions. The results demonstrate that the winglet design surpasses the optimal conventional propeller design, yielding a 13.2% improvement in fuel economy under light conditions and a 9.3% improvement under heavy conditions. These findings substantiate the advantages of utilizing winglet designs for propellers operating under highly loaded conditions.

Receding Galerkin Optimal Control with High-Order Sliding Mode Disturbance Observer for a Boiler-Turbine Unit

The control of the boiler-turbine unit is important for its sustainable and robust operation in power plants, which faces great challenges due to the control unit’s serious nonlinearity, unmeasurable states, variable constraints, and unknown time-varying lumped disturbances. To address the above issues, this paper proposes a receding Galerkin optimal controller with a high-order sliding mode disturbance observer in a composite scheme, in which a high-order sliding mode disturbance observer is first employed to estimate the lumped disturbances based on a deviation form of the mathematical model of the boiler-turbine unit. Subsequently, under the hypothesis of state constraint, a receding Galerkin optimal controller is designed to compensate the lumped disturbances by embedding their estimates into the mathematically based predictive model at each sampling time instant. With the help of an interpolation polynomial, Gauss integration, and nonlinear solvers, an optimal control law is then obtained based on a Galerkin optimization algorithm. Consequently, disturbance rejection, target tracking, and constraint handling performance of a controlled closed-loop system are improved. Some simulation cases are conducted on a mathematical boiler-turbine unit model to demonstrate the effectiveness of the proposed method, which is supported by the quantitative result analysis, such as tracking and disturbance rejection performance indexes.

Long-term analysis of wave-induced loads using High Order Spectral Method and direct sampling of extreme wave events

An approach for running long-term analysis of the wave environment and associated wave-induced structural loads, which includes effects of nonlinearity, irregularity and short crestedness of ocean waves is presented. The suggested approach combines a novel event-sampling technique with the efficient nonlinear wave solver High Order Spectral Method (HOSM) to represent the long-term wave environment by a large, yet computationally feasible, number of short-crested wave events which are simulated in HOSM. For the present study, HOSM is enhanced with a wave-breaking model, which improves the description of steep sea states in which wave breaking plays a role.Extensive validation of HOSM and the implemented wave-breaking model is carried out through comparison with model-test results for wave-crest statistics in long- and short-crested sea states and through direct comparison with CFD simulations.The proposed methodology is demonstrated on a case study considering a 106 years long-term analysis of wave crests and wave-in-deck loads, using a simplified load model to estimate drag loads on a deck structure.

ARKODE: A Flexible IVP Solver Infrastructure for One-step Methods

We describe the ARKODE library of one-step time integration methods for ordinary differential equation (ODE) initial-value problems (IVPs). In addition to providing standard explicit and diagonally implicit Runge–Kutta methods, ARKODE supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge–Kutta methods and multirate infinitesimal (MRI) methods. We present the role of ARKODE within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKODE infrastructure for utilities common to large classes of one-step methods, as well as its use of “time stepper” modules enabling easy incorporation of novel algorithms into the library. Numerical results show example problems of increasing complexity, highlighting the algorithmic flexibility afforded through this infrastructure, and include a larger multiphysics application leveraging multiple algorithmic features from ARKODE and SUNDIALS.

Nonlinear dynamics of turbine bladed disk with friction dampers: Experiment and simulation

Accurately predicting the nonlinear dynamic response of aero-engine components is critical, as excessive vibration amplitudes can considerably reduce the operational lifespan. This paper compares experimental and numerical nonlinear dynamic responses of an industrial aero-engine, specifically focusing on the first stage turbine bladed disk with under-platform dampers (UPDs). The friction forces between UPDs and blades result in a strongly nonlinear dynamic response, influenced by stick, slip and separation contact states at the interfaces. These contact states, and the resulting global dynamic responses, are predicted with an advanced industrial modelling approach for nonlinear dynamics. The predictions are compared, updated and validated against measurement data from an operational engine test. Results highlight the importance to validate models against industrial data and show that realistic contact pressure distributions are required for increased prediction reliability. The novelty of this work includes (1) the use of unique industrial experimental data from a fully operational aero-engine, (2) the observation, at the end of engine testing, of real contact conditions in blade/UPD interfaces, (3) detailed modelling of these contact conditions with high-fidelity finite element representations in nonlinear dynamic solvers. Based on this unique industrial validation work, guidelines are proposed to improve the state-of-the-art modelling of nonlinear dynamics in structures with friction contacts.

A unified steady and unsteady formulation for hydrodynamic potential flow simulations with fully nonlinear free surface boundary conditions

This work discusses the correct modeling of the fully nonlinear free surface boundary conditions to be prescribed in water waves flow simulations based on potential flow theory. The main goal of such a discussion is that of identifying a mathematical formulation and a numerical treatment that can be used both to carry out transient simulations, and to compute steady solutions — for any flow admitting them. In the literature on numerical towing tank in fact, steady and unsteady fully nonlinear potential flow solvers are characterized by different mathematical formulations. The kinematic and dynamic fully nonlinear free surface boundary conditions are discussed, and in particular it is proven that the kinematic free surface boundary condition, written in semi-Lagrangian form, can be manipulated to derive an alternative non penetration boundary condition by all means identical to the one used on the surface of floating bodies or on the basin bottom. The simplified mathematical problem obtained is discretized over space and time via Boundary Element Method (BEM) and Implicit Backward Difference Formula (BDF) scheme, respectively. The results confirm that the solver implemented is able to solve steady potential flow problems just by eliminating null time derivatives in the unsteady formulation. Numerical results obtained confirm that the solver implemented is able to accurately reproduce results of classical steady flow solvers available in the literature.

Reconstructing Meteoroid Trajectories Using Forward Scatter Radio Observations From the BRAMS Network

AbstractIn this paper, we aim to reconstruct meteoroid trajectories using a forward scatter radio system transmitting a continuous wave (CW) with no modulation. To do so, we use the meteor echoes recorded at the receivers of the BRAMS (Belgian RAdio Meteor Stations) network. This system consists, at the time of writing, of a dedicated transmitter and 44 receiving stations located in and nearby Belgium, all synchronized using GPS clocks. Our approach processes the meteor echoes at the BRAMS receivers and uses the time delays as inputs to a nonlinear optimization solver. We compare the quality of our reconstructions with and without interferometric data to the trajectories given by the optical CAMS (Cameras for Allsky Meteor Surveillance) network in Benelux. We show that the general CW forward scatter trajectory reconstruction problem can be solved, but we highlight its strong ill‐conditioning. With interferometry, this high sensitivity to the inputs is alleviated and the reconstructed trajectories are in good agreement with optical ones, displaying an uncertainty smaller than 10% on the velocity and 2° on the inclination for most cases. To increase accuracy, the trajectory reconstruction with time delays only should be complemented by information about the signal phase. The use of at least one interferometer makes the problem much easier to solve and greatly improves the accuracy of the retrieved velocities and inclinations. Increasing the number of receiving stations also enhances the quality of the reconstructions.

Collision-Free Model Predictive Trajectory Tracking Control for UAVs in Obstacle Environment

In this paper, we propose a collision-free model predictive trajectory tracking control algorithm for unmanned aerial vehicles (UAVs) in environments with both static obstacles and dynamic obstacles. Collision avoidance is ensured by obtaining outer polyhedral approximations of each interval of the dynamic obstacles trajectories based on MINVO basis, and then optimizing a plane to separate the polyhedra and the trajectory of the UAV. By incorporating the resulting computationally efficient collisionfree constraints and divers physical constraints, a model predictive control (MPC) optimization problem is formulated with a tailored terminal constraint set, which can be solved by a standard nonlinear programming solver. Moreover, the control theoretic properties are established, including recursive feasibility, the guarantee of collision avoidance, as well as closed-loop stability. Finally, the efficacy of the proposed algorithm is successfully evaluated by a simulation in a multi-obstacle environment.

Stein-based preconditioners for weak-constraint 4D-var

Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. Variational approaches, such as the weak-constraint four-dimensional variational data assimilation formulation considered in this problem, can ultimately be interpreted as a minimization problem. One of the main challenges of such a formulation is the solution of large linear systems of equations which arise within the inner linear step of the adopted nonlinear solver. Depending on the adopted approach, these linear algebraic problems amount to either a saddle point linear system or a symmetric positive definite (SPD) one. Both formulations can be solved by means of a Krylov method, like GMRES or CG, that needs to be preconditioned to ensure fast convergence in terms of the number of iterations. In this paper we illustrate novel, efficient preconditioning operators which involve the solution of certain Stein matrix equations. In addition to achieving better computational performance, the latter machinery allows us to derive tighter bounds for the eigenvalue distribution of the preconditioned linear system for certain problem settings. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.

Trust–Region Nonlinear Optimization Algorithm for Orientation Estimator and Visual Measurement of Inertial–Magnetic Sensor

This paper proposes a novel robust orientation estimator to enhance the accuracy and robustness of orientation estimation for inertial–magnetic sensors of the small consumer–grade drones. The proposed estimator utilizes a trust–region strategy within a nonlinear optimization framework, transforming the orientation fusion problem into a nonlinear optimization problem based on the maximum likelihood principle. The proposed estimator employs a trust–region Dogleg gradient descent strategy to optimize orientation precision and incorporates a Huber robust kernel to minimize interference caused by acceleration during the maneuvering process of the drone. In addition, a novel method for evaluating the performance of orientation estimators is also presented based on visuals. The proposed method consists of two parts: offline calibration of the basic cube using Augmented Reality University of Cordoba (ArUco) markers and online orientation measurement of the sensor carrier using a nonlinear optimization solver. The proposed measurement method’s accuracy and the proposed estimator’s performance are evaluated under low–dynamic (rotation) and high–dynamic (shake) conditions in the experiment. The experimental findings indicate that the proposed measurement method obtains an average re–projection error of less than 0.1 pixels. The proposed estimator has the lowest average orientation error compared to conventional orientation estimation algorithms. Despite the time–consuming nature of the proposed estimator, it exhibits greater robustness and precision, particularly in highly dynamic environments.

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

Mixed and Nitsche’s discretizations of Coulomb frictional contact-mechanics for mixed dimensional poromechanical models

This work deals with the discretization of single-phase Darcy flows in fractured and deformable porous media, including frictional contact at the matrix–fracture interfaces. Fractures are described as a network of planar surfaces leading to so-called mixed-dimensional models. Small displacements and a linear poro-elastic behavior are considered in the matrix. One key difficulty to simulate such coupled poro-mechanical models is related to the formulation and discretization of the contact mechanical sub-problem. Our starting point is based on the mixed formulation using facewise constant Lagrange multipliers along the fractures representing normal and tangential stresses. This is a natural choice for the discretization of the contact dual cone in order to account for complex fracture networks with corners and intersections. It leads to local expressions of the contact conditions and to efficient semi-smooth nonlinear solvers. On the other hand, such a mixed formulation requires to satisfy a compatibility condition between the discrete spaces restricting the choice of the displacement space and potentially leading to sub-optimal accuracy. This motivates the investigation of two alternative formulations based either on a stabilized mixed formulation or on the Nitsche’s method. These three types of formulations are first investigated theoretically in order to enhance their connections. Then, they are compared numerically in terms of accuracy and nonlinear convergence. The sensitivity to the choice of the formulation parameters is also investigated. Several 2D test cases are considered with various fracture networks using both P1 and P2 conforming Finite Element discretizations of the displacement field and an Hybrid Finite Volume discretization of the mixed-dimensional Darcy flow model.

Phase field modeling and computation of vesicle growth or shrinkage.

We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen-Cahn equation describing the evolution of the phase field parameter that describes the shape of the vesicle and a Cahn-Hilliard-type equation describing the evolution of the ionic fluid. We establish conditions for vesicle growth or shrinkage via a common tangent construction using free energy curves. During the membrane deformation, the model ensures total mass conservation of the ionic fluid, and we weakly enforce a surface area constraint of the vesicle. We develop a stable numerical scheme and an efficient nonlinear multigrid solver to evolve the phase and concentration fields, and we use this to evolve the fields to near equilibrium for 2D vesicles. Convergence tests confirm an [Formula: see text] accuracy for our scheme and near-optimal convergence for our multigrid solver. Numerical results reveal that the diffuse interface model captures the main features of cell shape dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies.

An experimental and numerical study on the behaviors of adhesively-bonded steel double-hat section components under axial impact loading

A significant consideration in the deployment of adhesively-bonded joints in applications such as automotive body structures is their performance under impact loading. Due to the distinct behaviors of an epoxy-based structural adhesive under normal and shear loads combined with its propensity to undergo brittle fracture in cohesion and adhesion, special care needs to be taken in the constitutive modeling of such an adhesive as traditional Von Mises type yielding and elastoplastic material behavior do not seem to be applicable. In the current study, to start with, experimental load-displacement behaviors till failure of single lap shear and T-peel joints with steel substrates are predicted with the aid of a cohesive zone material model using LS-DYNA, an explicit nonlinear finite element analysis solver. With confidence established on cohesive zone modeling of the joints mentioned, the procedure is extended to robust prediction of the load-displacement responses and deformed shapes of steel double-hat section components subjected to axial impact loading. Three variants of steel hat section components have been considered from the viewpoint of joining of flanges with conventional discrete spot-welds, only continuous adhesive bonding, and a hybrid configuration with a combination of adhesive bonding and sparse spot-welds. The study reported here provides insights into mechanical behaviors of adhesively-bonded steel double-hat sections under impact loading, and demonstrates an efficient finite element modeling approach for adhesively-bonded steel hat sections using cohesive zone elements with fracture mechanics-based criteria for failure.

Symmetry preserving neural network models for spur gear static transmission error curves

This paper proposes the use of neural networks to predict static transmission error (STE) curves for spur gears. Initially, a dataset spanning a parametric space of 17 parameters and comprising 20000 STE curves is created with a physics based solver, utilizing a dimensionless formulation. This data is used to train and evaluate different neural network architectures, which incorporate the symmetries of periodicity and input–output interchangeability. Results show that a small fully connected network with 3 hidden layers of 60 neurons can capture the highly non-linear STE response accurately, achieving a mean absolute percentage error of 0.075% on previously unseen data, while the incorporation of symmetries noticeably improves performance. The highest errors are below 1% and occur in border regions of the dataset, where training data is sparse. These results indicate that neural network models can faithfully reproduce the predictions of traditional, non-linear solvers and are thus a promising approach for modeling the static response of spur gears over extended parametric spaces. Finally, indicative dynamic simulations investigate the extension of these results to the dynamic regime.

Jacobian weighted element differential method for solid mechanics

In this paper, a novel collocation method is proposed to solve the problems in solid mechanics. In the proposed method, the gradients across the element boundaries are evaluated by a weight-averaging algorithm to achieve higher order continuity. Then, the collocation method is employed to discretize the governing equations for all domain nodes, including the nodes at the interfaces of elements for which the earlier collocation methods have to use the traction-equilibrium equations. As a result of this difference, linear and serendipity elements can be used in the proposed method to reduce the total number of freedoms and easily used for very complex geometries. Moreover, the proposed method significantly improves the accuracy compared with the earlier collocation methods. The provided numerical test cases show that the method can also avoid the degeneration of accuracy when odd-order elements are employed, which further demonstrates the advantages of the proposed method. The proposed method are also leads to better convergency in nonlinear cases and iterative matrix solver. By analyzing the coefficient matrix, we found that the proposed method is better conditioned. The proposed method also yields promising advantages in transient problems for explicit time marching because of the advantage of the collocation methods.

Unilateral frictional contact between a rigid wheel traversing on a flexible beam: An analytical investigation

This paper proposes a non-linear dynamic solver to simulate the dynamics of a rigid wheel traversing over a flexible beam involving a unilateral contact at the interface. Euler’s discretization scheme is extended incorporating Linear Complementary approach to perform explicit integration over time domain. The proposed solver is analytically designed to incorporate frictional force acting at the wheel-beam interface along with the possibility of generation of normal gap between the two bodies. Moreover, rotation of the wheel is also appended in the proposed algorithm and change in contact point between the wheel and beam is evaluated at every time step. After validation with existing literature, the solver is applied to study various dynamic responses of wheel and beam. Further, the technique is applied to moving oscillator model traversing over a flexible beam. The effect of initial wheel velocity, coefficient of friction, frequency of the oscillator on overall dynamic response of beam and wheel have been thoroughly investigated. The proposed algorithm delivers an exact solution based on the system’s fundamental mechanics, thus reducing the mesh reliance which amounts to high computational cost in finite element solver. The suggested method provides a broad future potential for solving rigid-flexible contact multibody dynamic problems.

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Anderson acceleration for a regularized Bingham model

AbstractThis article studies a finite element discretization of the regularized Bingham equations that describe viscoplastic flow. An efficient nonlinear solver for the discrete model is then proposed and analyzed. The solver is based on Anderson acceleration (AA) applied to a Picard iteration, and we show accelerated convergence of the method by applying AA theory (recently developed by the authors) to the iteration, after showing sufficient smoothness properties of the associated fixed point operator. Numerical tests of spatial convergence are provided, as are results of the model for 2D and 3D driven cavity simulations. For each numerical test, the proposed nonlinear solver is also tested and shown to be very effective and robust with respect to the regularization parameter as it goes to zero.