Relaxation Scheme Research Articles

Efficient numerical approximations for a nonconservative nonlinear Schrödinger equation appearing in wind‐forced ocean waves

AbstractWe consider a nonconservative nonlinear Schrödinger equation (NCNLS) with time‐dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper, we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic nonlinear Schrödinger equation. Both schemes are second‐order accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour–Fortin–Payre scheme, and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly nonconservative problems. We finally compare the two numerical schemes and discuss their performance.

A Relaxation Scheme for the Simulation of Two‐Phase Flows With Inaccessible Pore Volume in Polymer Flooding

ABSTRACTThe simulation of polymer injection in a reservoir is of paramount importance in enhanced oil recovery. Despite decades of research, the computation of polymer flows in porous media remains a challenging task. The main difficulty lies in the necessity to take into account the effect of inaccessible pore volumes (IPV), for which standard closure laws give rise to a weakly hyperbolic or even non‐hyperbolic system. In the latter case, exponential instabilities may appear at the continuous level, which must be addressed at the discrete level so as to prevent a premature stop of the numerical simulations. The notion of IPV was introduced by engineers in order to account for the following observation: when a polymer solution is injected into an initial core saturated with water, the breakthrough of the polymer at the exit occurs before that of the water in which it is injected. It seems that due to their large size, the polymer molecules cannot insinuate themselves into all pores as well as water. Having less volume to flood, the polymer molecules see their speed increased, hence the ad hoc acceleration factor associated with the polymer. In this work, we propose a relaxation method that guarantees some practical robustness for all IPV laws. This is achieved by replacing the original system by a relaxation model which is always hyperbolic. The designed relaxation model involves two parameters which enable us not only to adjust the correct amount of numerical dissipation, but also to ensure positivity for some critical quantities such as water saturation and polymer concentration. Extensive numerical tests are performed in order to compare the relaxation scheme to the more classical upwind scheme for several IPV laws.

Approach for multi-valued integer programming in multi-material topology optimization: Random discrete steepest descent (RDSD) algorithm

The present study models the multi-material topology optimization problems as the multi-valued integer programming (MVIP) or named as combinatorial optimization. By extending classical convex analysis and convex programming to discrete point-set functions, the discrete convex analysis and discrete steepest descent (DSD) algorithm are introduced. To overcome combinatorial complexity of the DSD algorithm, we employ the sequential approximate integer programming (SAIP) to explicitly and linearly approximate the implicit objective and constraint functions. Considering the multiple potential changed directions for multi-valued design variables, the random discrete steepest descent (RDSD) algorithm is proposed, where a random strategy is implemented to select a definitive direction of change. To analytically calculate multi-material discrete variable sensitivities, topological derivatives with material contrast is applied. In all, the MVIP is finally transferred as the linear 0–1 programming that can be efficiently solved by the canonical relaxation algorithm (CRA). Explicit nonlinear examples demonstrate that the RDSD algorithm owns nearly three orders of magnitude improvement compared with the commercial software (GUROBI). The proposed approach, without using any continuous variable relaxation and interpolation penalization schemes, successfully solves the minimum compliance problem, strength-related problem, and frequency-related optimization problems. Given the algorithm efficiency, mathematical generality and merits over other algorithms, the proposed RDSD algorithm is meaningful for other structural and topology optimization problems involving multi-valued discrete design variables.

Fast heuristics for the time‐constrained immobile server problem

AbstractWe propose easy‐to‐implement heuristics for time‐constrained applications of a problem referred to in the literature as the facility location problem with immobile servers, stochastic demand, and congestion, the service system design problem, or the immobile server problem (ISP). The problem is typically posed as one of allocating capacity to a set of M/M/1 queues to which customers with stochastic demand are assigned with the objective of minimizing a cost function composed of a fixed capacity‐acquisition cost, a variable customer‐assignment cost, and an expected‐waiting‐time cost. The expected‐waiting‐time cost results in a nonlinear term in the objective function of the standard binary programming formulation of the problem. Thus, the solution approaches proposed in the literature are either sophisticated linearization or relaxation schemes, or metaheuristics. In this study, we demonstrate that an ensemble of straightforward, greedy heuristics can rapidly find high‐quality solutions. In addition to filling a gap in the literature on ISP heuristics, new stopping criteria for an existing cutting plane algorithm are proposed and tested, and a new mixed‐integer linear model requiring no iterating algorithm is developed. In many cases, our heuristic approach finds solutions of the same or better quality than those found by exact methods implemented with expensive, state‐of‐the‐art mathematical programming software, in particular a commercial nonlinear mixed‐integer linear programming solver, given a five‐minute time limit.

Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations

This paper focuses on the study of robust two-stage quadratic multiobjective optimization problems. We formulate new necessary and sufficient optimality conditions for a robust two-stage multiobjective optimization problem. The obtained optimality conditions are presented by means of linear matrix inequalities and thus they can be numerically validated by using a semidefinite programming problem. The proposed optimality conditions can be elaborated further as second-order conic expressions for robust two-stage quadratic multiobjective optimization problems with separable functions and ellipsoidal uncertainty sets. We also propose relaxation schemes for finding a (weak) efficient solution of the robust two-stage multiobjective problem by employing associated semidefinite programming or second-order cone programming relaxations. Moreover, numerical examples are given to demonstrate the solution variety of our flexible models and the numerical verifiability of the proposed schemes.

Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations (PDEs) involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the inviscid Burgers equation in one-dimension (1D), nonlinear wave resonances lead to the formation of localised oscillatory structures called tygers in well-resolved areas of the flow, far from the shock. Recently, the author of [1] has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown in [1] that the novel SR and SP approximations of the solution converge strongly in L2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws, such as the 2×2 system of the shallow water equations and the standard 3×3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation [2]. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method (PPS), with the improved estimate obtained using the novel spectral relaxation scheme.

A symmetric multigrid-preconditioned Krylov subspace solver for Stokes equations

Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability and efficiency. The degree of success of such methods, however, is highly contingent on the design of key components of the multigrid scheme, including the hierarchy of discretizations, and the relaxation scheme used. Additionally, in many practical cases, it may be more effective to use a multigrid scheme as a preconditioner to an iterative Krylov subspace solver, as opposed to striving for maximum efficacy of the relaxation scheme in all foreseeable settings. In this paper, we propose an efficient symmetric multigrid preconditioner for the Stokes Equations on a staggered finite difference discretization. Our contribution is focused on crafting a preconditioner that (a) is symmetric indefinite, matching the property of the Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative scheme [1], and (c) has the requisite symmetry properties to be used in this context. In addition, our design is efficient in terms of computational cost and facilitates scaling to large domains.

Vibrational relaxation of HOD by collisions with Ar atoms.

Vibrational relaxation of HOD(v12, v3) molecules by collisions with Ar was studied at 298K (v12 denotes coupled bending, v2, and OD stretching, v1, vibrational modes and v3 denotes OH stretching mode). The vibrationally excited HOD molecules were generated by exothermic abstraction reactions of OD radicals with 13 different RH reactants and observed by infrared emission from a fast-flow reactor as a function of Ar pressure and reaction time. State-specific relaxation rate constants were obtained by comparison of the time evolution of the experimental vibrational distributions with numerical kinetic calculations for vibrational populations. The relaxation mechanism was based on the relaxation scheme of H2O studied earlier with the addition of specific channels for HOD(v12, v3). Unlike H2O, energy in stretching and bending vibrations of HOD cannot be separated due to close ν1 and 2ν2 energies, which leads to fast collisional equilibration between these Fermi-resonant levels. For relaxation of the only pure bending state (10), a rate constant of (1.5 ± 0.3) × 10-13cm3molecule-1s-1 was obtained. The relaxation rate of higher v12 states linearly increases with quantum number and very likely includes transfer of population from OD stretch levels, v1, to a lower energy bend level. The average rate constants for the loss of population from (01), (02), and (03) stretching states are (1.1 ± 0.3) × 10-14, (3.2 ± 1.0) × 10-14, and (5.6 ± 1.2) × 10-14cm3molecule-1s-1, respectively.

A Semi-Implicit Fully Exactly Well-Balanced Relaxation Scheme for the Shallow Water System

A Semi-Implicit Fully Exactly Well-Balanced Relaxation Scheme for the Shallow Water System

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

Improvements for matrix-type implicit temporal scheme regarding entropy fix and local time step

The matrix version of Symmetric Successive Over Relaxation (matrix-SSOR) scheme has been proved to be more efficient than the standard Lower-Upper Symmetric Gauss-Seidel (LU-SGS), but less robust for high-speed flows. In order to ulteriorly improve the convergence rate as well as numerical stability of matrix-SSOR, two improvements regarding entropy fix and local time step have been proposed and validated. Firstly, an augmented entropy fix method is imposed on the inviscid Jacobian matrix and proved to be effective in two high-speed flows, in which the key parameter in entropy fix is discussed and found to be insensitive within appropriate range of values. Since the time step also has great effects on the numerical stability and convergence rate, a modified cell residual adapted local time step method with consideration of the residual history is developed, which is found to be effective for increasing the convergence rate when the matrix-SSOR is applied, but invalid when the LU-SGS is used. The proposed modified local time step method is also insensitive to the key parameter within appropriate range of values. The two modifications can be conveniently implanted into analogous matrix-type implicit schemes to improve the numerical performance.

Hydrodynamics for Asymmetric Simple Exclusion on a Finite Segment with Glauber-Type Source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in Bardos et al (Commun Partial Differ Equ 4(9):1017–1034, https://doi.org/10.1080/03605307908820117, 1979) and Otto (C R Acad Sci Paris 322(1):729–734, 1996), discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme.

Anderson acceleration for PWR whole-core pin-by-pin Nu-TH coupling calculation in NECP-Bamboo2.0

For PWR whole-core pin-by-pin neutronics/thermal-hydraulics (Nu-TH) coupling analysis, the Picard iteration strategy is widely employed benefitting from the convenience of implementation. Nevertheless, it is limited to the global multi-physics convergence rate regarding its oscillatory convergence behaviors in the local solution of each single-physics field. In this paper, the Anderson Acceleration (AA) scheme is employed and investigated to enhance the convergence performance of the pin-by-pin Nu and subchannel TH coupling analysis code NECP-Bamboo2.0. Different from the classical relaxation scheme where the relaxation factor is predetermined and fixed, the AA scheme utilizes the linear combination of several previous solutions to correct the current solution. The coefficients of this linear combination can be obtained by solving the residual equation, making the approach more robust. In addition, the selection of the accelerated solution vector is investigated based on the tightly coupling strategy of Nu-TH in NECP-Bamboo2.0. It is illustrated that the neutron-transport solution vector could have superior theoretical acceleration performance compared to the commonly used TH solution vector in the practical PWR analysis scenario. Finally, the AA scheme in NECP-Bamboo2.0 is validated using the VERA#6 3D single-assembly transport/TH coupling benchmark problem and the VERA#7 3D whole-core multi-physics coupling benchmark. It is demonstrated by the numerical results that the AA scheme possesses good robustness and computational efficiency. For the 3D whole-core VERA#7 benchmark, compared with the classic Picard iteration with a relaxation factor of 0.5, the AA scheme based on the neutron-transport solution vector can reduce the number of global iterations and calculation time by around 80%.

Inclusion of Hall and Ion slip consequences on inclined magnetized cross hybrid nanofluid over a heated porous cone: Spectral relaxation scheme

The cone geometry has a great significant for heat transmission in many industrial processes due to its ability to induce turbulence, enable directional flow, promote uniform temperature distribution, and offer versatility in applications. The current study aims to investigate the heat transport process of an inclined magnetized cross hybrid nanofluid over a heated porous cone under the influence of Hall and Ion slip consequences. Additionally, porous medium the flow is past under the effect of inclined uniform magnetic field and porosity of the medium is used to enhance heat transfer. The framed set of governing equations took the form of dimension free structure through appropriate transformations and then finally solved by an effective spectral relaxation method. Thermal impacts and heat transport mechanism associated with the hybrid flow is seen through different values of emerging parameters. Heat transport is seen higher with higher radiation parameters, as radiation and rising temperature are similar. Augmented values of Fr causes pressure drop in fluids which reduces the fluid motion and brings depreciation in velocity field. Eckert number also boosts the temperature of the fluid stirring via a porous rotating cone.

Shell analysis by different methods of dynamic relaxation

Shell analysis is a complex process. So far, many equations and theories have been proposed. With the increasing processing power of computers, numerical and approximate solutions were developed. The dynamic relaxation scheme is one of these methods. This procedure is an explicit technique for solving simultaneous equations. The fictitious mass and damping are added to the system of static structural equations in the dynamic relaxation technique to obtain a fictitious dynamic system. In this paper, 12 well-known dynamic relaxation processes were used for the linear analysis of shells. It is found that so far, the ability of many solutions of dynamic relaxation in the linear analysis of shells has not been shown. In each approach, the mass, damping and time step matrices are obtained based on the researchers’ work. Twelve samples are analysed. Most of these are examples of well-known benchmarks in non-linear analysis. The number of iterations and the analysis time gives the score of each technique. Then, the final rank of each method was obtained. Numerical solutions show that minimising the residual energy was the best process for linear shell analysis. Also, minimising the error of displacement, Zhang, Qiang and the power iteration methods behaved very similarly to each other.

M‐AGC tangent visco‐plastic operator with hardening/softening, and application to the visco‐plastic relaxation analysis of stable and unstable problems using fracture‐based geomechanical interfaces

AbstractA previous perfect visco‐plastic constitutive formulation of the Perzyna type incorporating the concepts of prescribed stress increments and m‐AGC tangent operator (m‐Assumed algorithmic generalized compliance tangent operator) is extended to the case of Hardening/Softening (H/S). This extension is possible thanks to the closed‐form solution developed for the evolution of the loading function during a visco‐plastic time step. The formulation is then applied to constitutive modeling of zero‐thickness interfaces on the basis of a well‐established fracture‐based elasto‐plastic formulation, which in this manner is extended to visco‐plasticity. The resulting model is implemented in the finite element (FE) and small‐strain context by using both a standard Newton Raphson scheme for physical visco‐plasticity in which visco‐plastic time corresponds to physical time, and a Visco‐Plastic Relaxation (VPR) scheme, in which time corresponds to a fictitious time governing the transition from the elastic to the inviscid elasto‐plastic response. The numerical implementation is verified satisfactorily for common loading cases at interfaces such as pure tension (mode I) opening and shear‐compression (mixed‐mode) cracking/sliding, showing that the visco‐plastic results match the predictions of the fracture mechanics inviscid model in the long term. In addition, it is also shown that the VPR strategy developed is capable of providing temporary stability while the equilibrium path is recovered during instability events such as a snap‐back in the load‐displacement curve. This feature opens the door to a number of potential advanced applications of the formulation developed in the geomechanical context.

Toward static and transient stress-constrained topology optimization for shell-infill structures

This paper contributes to a stress-constrained topology optimization approach for the design of shell-infill structures. Based on the density method, we utilize the two-step filtering and projection scheme and local volume constraint for generating the shell and non-uniform infills, respectively. The topology optimization formulation is defined as a Robust Minimum Compliance problem with Volume and Stress constraints (RMCVS), where the robust method is used to eliminate undesired topological characteristics. We globalize the static and transient stress constraints through the p-norm function, and tailor material interpolation and stress relaxation schemes for both cases, respectively, to avoid numerical difficulties. Sensitivity analysis is provided, and the derivatives of the objective function and constraints with respect to the design variable fields are derived. We then validate the suggested method through several 2D and 3D benchmarks. The results illustrate that the method is robust to generate various shell-infill structures while limiting the maximum static and transient stress.

Thermal diffusivity of inclined magnetized Cross fluid with temperature dependent thermal conductivity: Spectral Relaxation scheme

Understanding and controlling the thermal transport phenomena are crucial in numerous applications. The current research emphasizes thermal diffusivity of an inclined magnetized Cross fluid with temperature-dependent thermal conductivity with a computational iterative spectral relaxation scheme. Cross mathematical model is employed to characterizes non-Newtonian behavior and to uncover viscoelastic properties of fluid. Flow is incorporated under temperature thermal influence and external inclined magnetic strength is considered for thermal variations. Various prominent factors, including cross index, magnetic field, inclination angle, temperature-dependent thermal conductivity are analyzed on the fluid's thermal diffusivity. The flow governing PDEs are converted into system of ODEs by using suitable transformation. Spectral relaxation computation scheme is then used for controlling the new set equations. SRM algorithm controlling subsystems is built through MATLAB. Numerical results are illustrated by MATLAB graphs. Physical quantities such as Sherwood numbers, Nusselt and skin friction coefficient are visually taken place through statistical graphs with two cases of imposed magnetic field. The results of this investigation shed light on how non-Newtonian fluids behave when exposed to temperature changes and magnetic fields and useful in understanding and leverage these effects for specific applications.

Local sparse discriminative feature selection

Feature selection has been widely used in machine learning for a long time. In this paper, we propose a supervised local sparse discriminative feature selection method named LSDFS to obtain sparse features by imposing ℓ2,0-norm constraint on transformation matrix. Differently from traditional approaches, our method does not require approximation or relaxation schemes, such as ℓ2,p-norm to solve long-standing challenge. Our method is based on the trace difference form of Linear Discriminant Analysis (LDA), which can efficiently obtain discriminative information in low-dimensional space. In order to explore the local structure of data which contains more discriminative information, we adopt a sparse connections graph between anchor points and data points instead of fully-connected graph with time-consuming, and add a decay parameter to avoid trivial solutions, making the model more precisely. Extensive experiments conducted on synthetic datasets and several real-world datasets have demonstrated the advantages of our method.

Eigenstate calculation in the state-averaged (multi-layer) multi-configurational time-dependent Hartree approach.

A new approach for the calculation of eigenstates with the state-averaged (multi-layer) multi-configurational time-dependent Hartree (MCTDH) approach is presented. The approach is inspired by the recent work of Larsson [J. Chem. Phys. 151, 204102 (2019)]. It employs local optimization of the basis sets at each node of the multi-layer MCTDH tree and successive downward and upward sweeps to obtain a globally converged result. At the top node, the Hamiltonian represented in the basis of the single-particle functions (SPFs) of the first layer is diagonalized. Here p wavefunctions corresponding to the p lowest eigenvalues are computed by a block Lanczos approach. At all other nodes, a non-linear operator consisting of the respective mean-field Hamiltonian matrix and a projector onto the space spanned by the respective SPFs is considered. Here, the eigenstate corresponding to the lowest eigenvalue is computed using a short iterative Lanczos scheme. Two different examples are studied to illustrate the new approach: the calculation of the vibrational states of methyl and acetonitrile. The calculations for methyl employ the single-layer MCTDH approach, a general potential energy surface, and the correlation discrete variable representation. A five-layer MCTDH representation and a sum of product-type Hamiltonian are used in the acetonitrile calculations. Very fast convergence and order of magnitude reductions in the numerical effort compared to the previously used block relaxation scheme are found. Furthermore, a detailed comparison with the results of Avila and Carrington [J. Chem. Phys. 134, 054126 (2011)] for acetonitrile highlights the potential problems of convergence tests for high-dimensional systems.