Robust Solution Techniques Research Articles

A novel approach for solving weakly singular fractional integro-differential equations

In this essay, we introduce a novel idea to tackle the challenges of fractional integro-differential equations (FIDEs) featuring weakly singular kernels (WSKs). New idea leverages B-splines for solving such equations, offering a robust numerical solution technique. We delve into the operational matrices integral to this method, providing comprehensive insights into their functionality. Furthermore, we establish the convergence of our approach and substantiate its effectiveness through various numerical examples.

Acceleration of nonlinear solvers for natural convection problems

Abstract This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.

An improved version of the Continuous Newton’s method for efficiently solving the Power-Flow in Ill-conditioned systems

This paper tackles the efficient Power-Flow solution of ill-conditioned cases. In that sense, those methods based on the Continuous Newton’s philosophy look very promising, however, these methodologies still present some issues mainly related with the computational efficiency or the robustness properties. In order to overcome these drawbacks, we suggest several modifications about the standard structure of the Continuous Newton’s method. Thus, the standard Continuous Newton’s paradigm is firstly modified with a frozen Jacobian scheme for reducing its computational burden; secondly, it is extended for being used with High-order Newton-like method for achieving higher convergence rate and, finally, a regularization scheme is introduced for improving its robustness features. On the basis of the suggested improvements, a Power-Flow solution paradigm is developed. As example, a novel Power-Flow solver based on the introduced solution framework and the 4th order Runge-Kutta formula is developed. The novel technique is validated in several realistic large-scale ill-conditioned systems. Results show that the suggested modifications allow to overcome the drawbacks presented by those methodologies based on the Continuous Newton’s method. On the light of the results obtained it can be also claimed, that the developed solution paradigm constitutes a promising framework for developing robust and efficient Power-Flow solution techniques.

On the micromechanics of deep material networks

We investigate deep material networks (DMNs), recently introduced by Liu et al. [Comput. Method Appl. M., vol. 345, pp. 1138–1168, 2019], from the viewpoint of classical micromechanics at small strains. We aim to establish the basic micromechanical principles of deep material networks, shed light on the characteristics of the building blocks and introduce a simple, robust and fast solution technique for inelastic deep material networks.In their original formulation, DMNs are solely trained by linear elastic data, but applied to nonlinear and inelastic problems with astonishing accuracy. We clarify this phenomenon theoretically by showing that, to first order in the strain rate, the effective inelastic behavior of composite materials is determined by linear elastic localization. Our argumentation applies to arbitrary microstructures comprising nonlinear generalized standard materials at small strains. The main technical tool is a Volterra series approximation of the stress of a generalized standard material, which we adapt from nonlinear dynamical systems theory.Next, we establish that deep material networks inherit thermodynamic consistency and stress-strain monotonicity from their phases. These properties root in the definition of the DMN as a tree of hierarchical laminates and contrast with other applications of neural networks to the approximation of material laws, where consistency and monotonicity typically cannot be guaranteed far away from the training set.Last but not least, we introduce rotation-free DMNs with arbitrary directions of lamination and exploit a novel formulation, uniting the implementation of DMNs of arbitrary tree topology and multi-phase laminates, and apply our insights to microstructures of industrial complexity.

A Three-Stage Algorithm Based on a Semi-Implicit Approach for Solving the Power-Flow in Realistic Large-Scale ill-Conditioned Systems

Solving the Power-Flow in realistic large-scale ill-conditioned systems supposes a challenging task for most of available solution methodologies. This paper tackles this issue by developing a novel efficient and robust Power-Flow method. It is mainly based on a Semi-Implicit approach but incorporates other numerical arrangements for enhancing its features. The resulting three-stage algorithm is validated using several realistic ill-conditioned systems ranging from 3012 to 70000-buses. Results show that the developed methodology constitutes an efficient and robust Power-Flow solution technique, outperforming the results obtained with other available approaches.

High detail stationary optimization models for gas networks: validation and results

Due to strict regulatory rules in combination with complex nonlinear physics, major gas network operators in Germany and Europe face hard planning problems that call for optimization. In part 1 of this paper we have developed a suitable model hierarchy for that purpose. Here we consider the more practical aspects of modeling. We validate individual model components against a trusted simulation tool, give a structural overview of the model hierarchy, and use its large variety of approximations to devise robust and efficient solution techniques. An extensive computational study demonstrates the suitability of our models and techniques for previously unsolvable problems in gas network planning.

The mixed finite element multigrid method for stokes equations.

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.

A linear solver based on algebraic multigrid and defect correction for the solution of adjoint RANS equations

SUMMARYA new solution approach for discrete adjoint Reynolds‐averaged Navier–Stokes equations is suggested. The approach is based on a combination of algebraic multigrid and defect correction. The efficient interplay of these two methods results in a fast and robust solution technique. Algebraic multigrid deals very well with unstructured grids, and defect correction completes it in such a way that a required second‐order accuracy of the solution is achieved. The performance of the suggested solution approach is demonstrated on a number of representative benchmarks.Copyright © 2014 John Wiley & Sons, Ltd.

A supercoarsening multigrid method for poroelasticity in 3D coupled flow and geomechanics modeling

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO2 injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.

Modeling of Energy Processes in Wheel-Rail Contacts Operating under Influence of Periodic Discontinuous Forces

In this paper we present new numerical simulation approaches for determining the energy processes under periodic conditions caused by time-discontinuous forces in the wheel-rail contacts. The main advantage of the presented method is the total elimination of frequency analysis, which in effect introduces important simplifications in the identification of the effects in the contact. The second important feature is the fact that the method is based on the analysis of appropriate loops on the energy phase plane leading to an easy estimation of the rail strength through the evaluation of the loop’s area. That model based simulation in the applied dynamics relies on advanced methods for model setup, robust and efficient numerical solution techniques and powerful simulation tools for practical applications. Fundamental properties of contact displacements of the rail surface have been considered on the basis of the newly established method. The contact zone between railway wheels and the rail surfaces made of bulk materials is perceived as strong enough to resist the normal (vertical) forces introduced by heavy loads and the dynamic response induced by track and wheel irregularities. The analysis is carried out for a wheel running on an elastic rail rested on sleepers arranged on completely rigid foundation. The equations of displacement motion are established through the application of the Lagrange equations approach. The established model of the wheel-rail contact dynamics has been applied to that same roll plane but with taking into account a nonlinear characteristic of the sleeper with respect to the ground. Attention then is focused completely on the modeling of the energy absorbed by the rail. The applied method employs the energy state variables as time functions leading to determine the susceptibility of a given contact on the strength induced by the rail roll.

Inexact Newton methods for model simulation

Robust and efficient solution techniques for solving macroeconometric models are increasingly becoming a key factor in developing models employed by policy-making institutions for policy simulations and forecasting. Traditionally, when solved in the presence of forward-looking variables, these models are nonlinear, large-scale and sparse and give rise to large and highly structured nonlinear systems. This paper proposes a Newton-GMRES method obtained tuning up the basic algorithm by properly choosing the forcing terms sequence and the preconditioning strategy. In addition, the Newton–GMRES method is wrapped into a globalization strategy based on a nonmonotone linesearch technique in order to enlarge its convergence basin and to enhance its robustness. The combination of these ingredients yields a reliable method with low memory requirements. Numerical experiments using the MULTIMOD model and a basic real business cycle model are presented. A Matlab code based on this approach is provided.

Uniform stability analysis of Austin, Manteuffel and McCormick finite elements and fast and robust iterative methods for the Stokes‐like equations

AbstractIn this paper, we discuss uniformly stable discretizations and fast solvers for the steady and semi‐discrete time‐dependent Stokes equations, which will be called the Stokes‐like equations. We study two pairs of conforming finite elements that are uniformly accurate with respect to all relevant parameters in the equations, one of which is the Scott–Vogelius (RAIRO Modél. Math. Anal. Numér. 1985; 19:111–143) element and the other of which is the finite element pair by Austin et al. (Numer. Linear Algebra Appl. 2004; 11:115–140). We prove that the finite element pair by Austin, Manteuffel and McCormick is uniformly accurate and we design and analyze the fast and robust solution techniques for the linear algebraic systems arising from the discretizations of the Stokes‐like equations as well. Our method is based on the augmented Lagrangian Uzawa iterative methods (Augmented Lagrangian Methods. North‐Holland: Amsterdam, 1983) and the multigrid methods designed by Austin et al. (Numer. Linear Algebra Appl. 2004; 11:115–140) for the linear algebraic equations from the discretizations of the differential operator ρ2I−κ2Δ−µ2∇div. We prove our methods are robust with respect to all relevant parameters that appear in the equations and also the mesh size, thereby, achieve the fast and robust methods for the solutions to the Stokes‐like equations. In particular, the convergence analysis for the multigrid methods designed by Austin et al., has been posed as an open question (Numer. Linear Algebra Appl. 2004; 11:115–140), which is resolved in this paper. Copyright © 2009 John Wiley & Sons, Ltd.

A variable neighborhood search for job shop scheduling with set-up times to minimize makespan

Production Scheduling problems are typically named bases on the processing routes of their jobs on different processors and also the number of processors in each stage. In this paper, we consider the problem of scheduling a job shop (JSS) where set-up times are sequence-dependent (SDST) to minimize the maximum completion times of operations or makespan. Our problem is generally formulated as J / S T s d / C max . To tackle such an NP-hard problem, a recent effective metaheuristic algorithm known as variable neighborhood search (VNS) is employed. VNS algorithms have shown excellent capability to solve scheduling problems to optimal or near-optimal schedule. Our proposed VNS is readily intelligible yet is a robust solution technique for the problem of SDST JSS. VNS is categorized as a local search-based algorithm armed with systematic neighborhood search structures. Our proposed VNS obviates the notorious myopic behavior of local search-based metaheuristic algorithms by the means of several systematic insertion neighborhood search structures. An experimental design based on Taillard’s benchmark is conducted to evaluate the efficiency and effectiveness of our proposed algorithm against some effective algorithms in the literature. The obtained results strongly support the high performance of our proposed algorithm with respect to other well-known heuristic and metaheuristic algorithms.

A variable neighborhood search for the capacitated arc routing problem with intermediate facilities

The capacitated arc routing problem (CARP) focuses on servicing edges of an undirected network graph. A wide spectrum of applications like mail delivery, waste collection or street maintenance outlines the relevance of this problem. A realistic variant of the CARP arises from the need of intermediate facilities (IFs) to load up or unload the service vehicle and from tour length restrictions. The proposed Variable Neighborhood Search (VNS) is a simple and robust solution technique which tackles the basic problem as well as its extensions. The VNS shows excellent results on four different benchmark sets. Particularly, for all 120 instances the best known solution could be found and in 71 cases a new best solution was achieved.

A two-level discretisation algorithm for the efficient solution of higher-dimensional population balance models

The increasing range of complex applications that call for population balances based models motivates the development of efficient and robust solution techniques. Several contemporary applications mandate the consideration of multi-dimensional population balances, accounting for the non-homogeneous and/or wide distribution of particle populations among several state variables (internal coordinates). This article presents an efficient and novel solution technique for multi-dimensional population balance models, inspired by the underlying physics of the problem and the dictates of the relevant rate processes in the population balance in terms of the accuracy of the solutions.

A Model Predictive Control Strategy for Stabilization of Oscillating Yeast Cultures

Continuous cultures of budding yeast are known to exhibit autonomous oscillations that adversely affect bioreactor stability and productivity. This phenomenon is modeled by coupling the population balance equation (PBE) for the cell mass distribution to the mass balance for the rate limiting substrate. An efficient and robust numerical solution technique is developed using orthogonal collocation on finite elements. The controller design model is obtained by linearizing and temporally discretizing the ordinary differential equations resulting from spatial discretization of the PBE model. The resulting linear state-space model is used to design model predictive controllers that regulate the cell number distribution by manipulating the dilution rate and the feed substrate concentration. The ability of the controllers to stabilize steady-state and periodic solutions is evaluated via simulation.

Load stepping schemes for critical state models

This paper investigates the performance of various load-stepping schemes for finite element analysis with critical state soil models. The accuracy of simple incremental schemes is found to be strongly influenced by the load increment size, the type of flow rule, and the overconsolidation ratio. Similarly, these factors are shown to have a pronounced effect on the efficiency and stability of some classical iterative schemes. Unless they are performed with small load steps, critical state analyses with fixed increment sizes frequently exhibit non-convergent behaviour or lead to inaccurate solutions. The automatic incrementation schemes developed by Abbo and Sloan (International Journal for Numerical Methods in Engineering 1996; 39:1737–1759; Proceedings of 5th International Conference, Owen DRJ, Onate E, Hinton E. International Center for Numerical Methods in Engineering, Barcelona, 1997; 1:325–333), which are based on standard methods for integrating systems of ordinary differential equations, are shown to be efficient, accurate and robust solution techniques for a wide variety of critical state problems. Copyright © 2001 John Wiley & Sons, Ltd.

An algorithm competition: First-order iterations versus Newton-based techniques

Abstract The development and use of forward-looking macromodels in policy making institutions has proceeded at a much slower pace than what was predicted in the early 1980s. An important reason for this is that researchers have not had access to robust and efficient solution techniques for solving nonlinear forward-looking models. This paper discusses the properties of alternative algorithms for solving MULTIMOD, the IMF's multicountry model of the world economy. Relative to traditional first-order algorithms in use today, we find that Newton-based techniques are considerably faster and much less prone to simulation failure.

A Robust and Efficient Method for Solving Nonlinear Rational Expectations Models

The development and use of forward-looking macro models in policymaking institutions has proceeded at a pace much slower than predicted in the early 1980s. An important reason is that researchers have not had access to robust and efficient solution techniques for solving nonlinear forward-looking models. This paper discusses the properties of a new algorithm that is used for solving MULTIMOD, the IMF’s multicountry model of the world economy. This algorithm is considerably faster and much less prone to simulation failures than to traditional algorithms and can also be used to solve individual country models of the same size.

A Robust and Efficient Method for Solving Nonlinear Rational Expectations Models

The development and use of forward-looking macro models in policymaking institutions has proceeded at a pace much slower than predicted in the early 1980s. An important reason is that researchers have not had access to robust and efficient solution techniques for solving nonlinear forward-looking models. This paper discusses the properties of a new algorithm that is used for solving MULTIMOD, the IMF`s multicountry model of the world economy. This algorithm is considerably faster and much less prone to simulation failures than to traditional algorithms and can also be used to solve individual country models of the same size.