Nonlinear Solution Method Research Articles

Comparison of Frequency Domain and Time-Domain Methods for Aeromechanical Analysis

Unsteady flow around an oscillating plate cascade and that through a single compressor rotor subject to vibration have been computationally studied, aimed at examining the predictive ability of two low fidelity frequency methods compared with a high fidelity time-domain solution method for aeroelasticity. The computational solutions demonstrate the capabilities of the frequency domain methods compared with the nonlinear time-domain solution method in capturing small perturbations in the unsteady flow. They also show the great advantage of significant CPU time saving by the frequency methods over the nonlinear time method. Comparisons of two different frequency methods, nonlinear harmonic and phase solution method, show that these methods can produce different results due to the differences in numeric and physical conditioning. The results obtained using phase solutions method are in better agreement with the nonlinear time-domain solution. This is because the same numeric and physical conditioning are used in both the nonlinear time-domain method and phase solution frequency domain method.

Neighborhood influences on stress and strain partitioning in dual-phase microstructures

The mechanical response of multiphase metallic materials is governed by the strain and stress partitioning behavior among their phases, crystals, and subgrains. Despite knowledge about the existence of these complex and long-ranging interactions, the experimental characterization of such materials is often limited to surface observations of microstructure evolution and strain partitioning, i.e. ignoring the influence of the underlying features. Hence, for the interpretation of the observed surface behavior it is imperative to understand how it might be influenced by the subsurface microstructure. In the present study, we therefore systematically change the subsurface microstructure of synthetic dual-phase polycrystals and investigate the altered response of a 2D region of interest. The series of high-resolution crystal plasticity simulations are conducted with a fast and efficient spectral-based iterative scheme for calculating the mechanical response of complex crystalline materials. To overcome the slow convergence of the conventional spectral-based solver when dealing with heterogeneous materials of large contrast in stiffness (or strength), direct and mixed variational conditions for mechanical equilibrium and strain compatibility have been formulated such that they can be combined with a general class of non-linear solution methods. The different solution techniques have been implemented into DAMASK, the Dusseldorf Advanced Material Simulation Kit, and the ones showing the best performance are used in this study. The results show that the subsurface microstructure has a dominant influence on the observed stress and strain partitioning. Additionally, it can be seen that the zone of influence increases with increasing heterogeneity of the microstructure.

A Stepped Nonlinear Solver for Nonlinear Magnetic Materials With Hysteresis

A stepped nonlinear solution method based on differential susceptibility is presented for the quasi-magnetostatic analysis of nonlinear magnetic materials. The stepped solver is applicable to hysteretic materials and supports a permanent magnetization. A locally corrected Nyström discretization of the magnetostatic volume integral equation is used to implement the stepped solver. Characterization of the stepped solver is performed using various magnetic material models which support hysteresis. Solver accuracy is validated using Team Workshop Problem 13, analytic results of a magnetic sphere with hysteresis, and against measured data for a hollow steel pipe.

Impact assessment of a hybrid energy-generation system on a residential distribution system in Taiwan

In this paper, a Monte Carlo-based three-phase power-flow method is proposed to evaluate the impact of a variety of hybrid energy-generation systems on residential distribution systems. The proposed deterministic power-flow method was based on graph theory, circuit theory and nonlinear numerical solution methods, and was extended to develop the proposed Monte Carlo based power flow method using the commercial software package OPTIMUS. Various technologies for distributed generation were combined in the proposed method. The hybrid energy system at the University of Yuan Ze was used in this research. It consists of a 5kW photovoltaic system, a 3kW wind power system and a 2kW hydrogen-fuel cell system. Then, a comprehensive study of the hybrid-energy micro-grid system was performed to evaluate its impact on the residential distribution power grids. The factors studied included steady-state voltage deviations, power flows, power losses, reverse power flows, short-circuit fault currents, maximum allowable capacity for distributed generation, and voltage unbalance factor. The results of this research should be of value for pre-determining the impacts of hybrid energy systems on the residential distribution systems.

Hybrid time integration and coupled solution methods for nonlinear finite element analysis of partially saturated deformable porous media at small strain

SummaryThe goal of the paper is to determine the most efficient, yet accurate and stable, finite element nonlinear solution method for analysis of partially saturated deformable porous media at small strain. This involves a comparison between fully implicit, semi‐implicit, and explicit time integration schemes, with monolithically coupled and staggered‐coupled nonlinear solution methods and the hybrid combination thereof. The pore air pressure pa is assumed atmospheric, that is, pa=0 at reference pressure. The solid skeleton is assumed to be pressure‐sensitive nonlinear isotropic elastic. Coupled partially saturated ‘consolidation’ in the presence of surface infiltration and traction is simulated for a simple one‐dimensional uniaxial strain example and a more complicated plane strain slope example with gravity loading. Three mixed plane strain quadrilateral elements are considered: (i) Q4P4; (ii) stabilized Q4P4S; and (iii) Q9P4; “Q” refers to the number of solid skeleton displacement nodes, and “P” refers to the number of pore fluid pressure nodes. The verification of the implementation against an analytical solution for partially saturated pore water flow (no solid skeleton deformation) and comparison between the three time integration schemes (fully implicit, semi‐implicit, and explicit) are presented. It is observed that one of the staggered‐coupled semi‐implicit schemes (SIS(b)), combined with the fully implicit monolithically coupled scheme to resolve sharp transients, is the most efficient computationally. Copyright © 2015 John Wiley & Sons, Ltd.

Position Accuracy Analysis on Disc Storing Mechanism for Benthic Drill

The positional accuracy of disc storing mechanism for benthic drill is the guarantee of long hole coring in deep sea. Aiming the lack of positional accuracy analysis on disc storing mechanism, the mathematic model of the positional accuracy for disc storing mechanism is presented by using complex vector and matrix analyzing method. The analytical formula of crank rotation positional accuracy is acquired through rotation positional analysis of crank in disc storing mechanism driven by hydraulic cylinder. Adopting numerical nonlinear iteration solution method of Newton-Simpson, the variation rule of rotation positional error for disc storing mechanism to cylinder length is acquired, which supports an important theory, leading to tolerance design for dimensional parameters of disc storing mechanism.

Safety of Nuclear Power Plants against the Aircraft Attack

This paper presents the nonlinear dynamic analysis of the reinforced concrete building of nuclear power plant to the aircraft impact. The dynamic load is defined in time on the base of the simulation of the airplane impact considering real stiffness, masses, direction and velocity of the flight. The dynamic response is calculated in the system ANSYS using the transient nonlinear analysis solution method. The damage of the concrete wall is evaluated in accordance with the standard NDRC considering the spalling, scabbing and perforation effects. The simple and detailed calculations of the wall damage are compared.

Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials

Efficient spectral methods are developed to predict the micromechanical behaviour of plastically deforming heterogeneous materials. The direct and mixed variational conditions for mechanical equilibrium and strain compatibility are formulated in a framework that couples them to a general class of non-linear solution methods. Locally evolving micromechanical fields in a sheared polycrystalline material governed by a phenomenological crystal plasticity constitutive law are used to validate the methods, and their performance at varying material heterogeneities is benchmarked. The results indicate that the solution method has a dominant influence on performance and stability at large material heterogeneities, and significant improvements over the conventional fixed-point approach are obtained when higher-order solution methods are employed. Optimal solution strategies are devised based on this and applied to an idealised dual-phase polycrystalline aggregate.

Derivative-Free Augmented Lagrangian for Global Optimization: Cost Minimization in a Simplified Activated Sludge System Model

A methodology for finding the optimal values of the decision variables from an efficient simplified mathematical model of an activated sludge system is addressed in this paper. The work herein presented arises in a wastewater treatment plant design context, where investment and operational costs are to be minimized and computational effort is to be reduced. To achieve the best design, a non-linear optimization solution method based on an augmented Lagrangian approach is proposed. At each iteration, a subproblem is globally solved by a derivative-free recursive branching technique, known as the multilevel coordinate search algorithm of Huyer and Neumaier [20]. The presented technique has been shown to work quite well when solving the herein proposed non-convex and non-smooth constrained optimization model. The numerical results show the reliability of the obtained solutions at a reduced computational cost.

Joint Stocking and Product Offer Decisions Under the Multinomial Logit Model

This article studies a joint stocking and product offer problem. We have access to a number of products to satisfy the demand over a finite selling horizon. Given that customers choose among the set of offered products according to the multinomial logit model, we need to decide which sets of products to offer over the selling horizon and how many units of each product to stock so as to maximize the expected profit. We formulate the problem as a nonlinear program, where the decision variables correspond to the stocking quantity for each product and the duration of time that each set of products is offered. This nonlinear program is intractable due to its large number of decision variables and its nonseparable and nonconcave objective function. We use the structure of the multinomial logit model to formulate an equivalent nonlinear program, where the number of decision variables is manageable and the objective function is separable. Exploiting separability, we solve the equivalent nonlinear program through a dynamic program with a two dimensional and continuous state variable. As the solution of the dynamic program requires discretizing the state variable, we study other approximate solution methods. Our equivalent nonlinear program and approximate solution methods yield insights for good offer sets.

Compact Implementation Strategy for a Harmonic Balance Method Within Implicit Flow Solvers

A two-step approximate factorization technique for implementing a computationally stable nonlinear unsteady frequency-domain harmonic balance solution method within existing implicit computational fluid dynamic flow solver codes is presented. The approach uses an explicit discretization of the harmonic balance source term, and no new implicit code development is required. Both of these features enable the harmonic balance method to be implemented within existing implicit flow solver codes with minimal modification necessary to the underlying flow solver code. The resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows. The methodology is applied to the NASA OVERFLOW flow solver code, and results are presented for transonic viscous flow past an unsteady pitching airfoil section.

Hydrothermal unit commitment with AC constraints by a new solution method based on benders decomposition

This paper presents a new approach based on Benders decomposition (BD) to solve hydrothermal unit commitment problem with AC power flow and security constraints. The proposed method decomposes the problem into a master problem and two sets of sub-problems. The master problem applies integer programming method to solve unit commitment (UC) while the sub-problems apply nonlinear programming solution method to determine economic dispatch for each time period. If one sub-problem of the first set becomes infeasible, the corresponding sub-problem of the second set is called. Moreover, strong Benders cuts are proposed that reduce the number of iterations and CPU time of the Benders decomposition method. All constraints of the hydrothermal unit commitment problem can be completely satisfied with zero penalty terms by the proposed solution method. The methodology is tested on the 9-bus and IEEE 118-bus test systems. The obtained results confirm the validity of the developed approach.

Numerical simulation of water resources problems: Models, methods, and trends

Mechanistic modeling of water resources systems is a broad field with abundant challenges. We consider classes of model formulations that are considered routine, the focus of current work, and the foundation of foreseeable work over the coming decade. These model formulations are used to assess the current and evolving state of solution algorithms, discretization methods, nonlinear and linear algebraic solution methods, computational environments, and hardware trends and implications. The goal of this work is to provide guidance to enable modelers of water resources systems to make sensible choices when developing solution methods based upon the current state of knowledge and to focus future collaborative work among water resources scientists, applied mathematicians, and computational scientists on productive areas.

Nonlinear Buckling Analysis of Wind Turbine Towers

Wind turbine towers are belonging to towering cylinder shell structures, which are easy to appear buckling instability under wind or other complicated loads, and on which integral elastic-plastic buckling analyses have great theoretical and practical significances. This paper used large deflection nonlinear pre-buckling and Koiter initial post-buckling theories, and adopted the finite element scheme of updated integration algorithm and LDC nonlinear solution method, then analyzed the linear buckling, elastic-plastic static buckling, and post-buckling response of the towers with initial imperfections in different location and size. It has obtained that: 1) the critical load of towers with elastic-plastic buckling is much smaller than it with elastic buckling; 2) gravity has certain influence on the critical buckling load; 3) the critical buckling load is insensitive to initial imperfections, meanwhile the imperfections which located on the top or the bottom of the tower are inferior for the stability of tower.

A Possible Solution for the Boltzmann Equation

Recently there has been an interest in utilizing recent advances in nonlinear partial differential equations solution methods, the equivalent description by stochastic calculus methods, and the state function level of information entropy to the solution of several nonlinear PDEs that are well known in application to problems of gases and fluids and regular to turbulent flow in physics. A recent reported solution has been to the Boltzmann equation. Previously advances were made in solutions to the Navier-Stokes equations of gaseous flow. In this article we derive a possible solution to the 3D Boltzmann equation utilizing transformation methods at the macroscopic functional level, the PDF distribution and stochastic differential equation level. The collision integrals are evaluated utilizing the hypothesis of semiclassical collisions and are obtained from a maximum entropy approach. The collision integral evaluated beyond the molecular chaos approximation, it is 'added' to the phase space variational principle and the Boltzmann equation is transformed to a Fokker-Planck equation which is solved by several known methods both for computational and mathematical analytical applications.

Concurrent Blade Aerodynamic-Aero-elastic Design Optimization Using Adjoint Method

Increasing aerothermal and aero-elastic performance requirements and constraints are closely linked in modern blading designs. There is thus a need for more concurrent interaction between the disciplines at earlier stages of a design process. Presented in this paper are the development, validation, and demonstration of the adjoint approach to concurrent blading aerodynamic and aero-elastic design optimizations. A nonlinear harmonic phase solution method is adopted to solve the unsteady Reynolds-averaged Navier–Stokes equations. The flow field response in terms of both the mean aerothermal performance and aero-elastic stability to a geometrical perturbation can be obtained by three “steadylike” flow solutions at three distinctive temporal phases. This unsteady flow solution method is computationally very efficient and provides a convenient and consistent base for formulating the corresponding adjoint equations. The adjoint system for the unsteady flow solver is solved effectively by a relatively simple extension of the method and techniques previously developed for a steady flow adjoint solver. As a result, the sensitivities of both the steady (time-mean) flow loss and the aerodynamic damping/forcing to detailed blade geometry changes can be very efficiently obtained by solving equivalently three steadylike adjoint equations. Several case studies are presented to illustrate the validity and effectiveness of this new concurrent approach.

The Output Effect of a Transition to Price Stability When Velocity Is Time Varying

This paper explores the effect of time‐varying velocity on output responses to policies for reducing/stopping inflation. We study a dynamic general equilibrium model with sticky prices in which we introduce time‐varying velocity. Specifically, we endogenize time‐varying velocity into the model developed by Ireland (1997) for analyzing optimal disinflation. The nonlinear solution method reveals that, depending on velocity, the “disinflationary boom” found by Ball (1994) may disappear even under perfect credibility and that early output losses may be much larger than previously thought. Indeed, we find that a gradual disinflation from a low inflation may even be undesirable.

Performance of a parallel algebraic multilevel preconditioner for stabilized finite element semiconductor device modeling

In this study results are presented for the large-scale parallel performance of an algebraic multilevel preconditioner for solution of the drift-diffusion model for semiconductor devices. The preconditioner is the key numerical procedure determining the robustness, efficiency and scalability of the fully-coupled Newton–Krylov based, nonlinear solution method that is employed for this system of equations. The coupled system is comprised of a source term dominated Poisson equation for the electric potential, and two convection–diffusion-reaction type equations for the electron and hole concentration. The governing PDEs are discretized in space by a stabilized finite element method. Solution of the discrete system is obtained through a fully-implicit time integrator, a fully-coupled Newton-based nonlinear solver, and a restarted GMRES Krylov linear system solver. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the nonzero block structure of the Jacobian matrix. Representative performance results are presented for various choices of multigrid V-cycles and W-cycles and parameter variations for smoothers based on incomplete factorizations. Parallel scalability results are presented for solution of up to 10 8 unknowns on 4096 processors of a Cray XT3/4 and an IBM POWER eServer system.

Time-Centered Split Method for Implicit Discretization of Unsteady Advection Problems

A new method for implicit solution of unsteady nonlinear advection equations is presented. This time-centered split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically nonlinear coupled equations for any number of variables. The TCS algorithm is compared to other nonlinear solution methods (local linearization, Picard iteration, and Newton iteration) and applied to the Crank-Nicolson discretization of the one-dimensional Burgers’ equation. The temporal accuracy and practical stability of the method is confirmed using an unsteady flow test case with an analytical solution. The method is shown to require computational effort similar to local linearization, but does not require discrete computation of a functional Jacobian for solution.

Nonlinear solution methods for infinitesimal perfect plasticity

AbstractWe review the classical return algorithm for incremental plasticity in the context of nonlinear programming, and we discuss the algorithmic realization of the SQP method for infinitesimal perfect plasticity. We show that the radial return corresponds to an orthogonal projection onto the convex set of admissible stresses. Inserting this projection into the equilibrium equation results in a semismooth equation which can be solved by a generalized Newton method. Alternatively, an appropriate linearization of the projection is equivalent to the SQP method, which is shown to be more robust as the classical radial return. This is illustrated by a numerical comparison of both methods for a benchmark problem.