Euler Backward Research Articles

Asymptotic mean‐square boundedness of the numerical solutions for stochastic complex‐valued neural networks with jumps

The paper devoted to asymptotic mean‐square boundedness of several numerical methods for stochastic complex‐valued neural networks with Poisson jumps. The definition of asymptotic mean‐square boundedness of the numerical solution is presented, and some sufficient conditions for the underlying systems which are asymptotic mean‐square boundedness are derived. By taking the advantage of the compensated split‐step backward Euler (CSSBE) method and compensated backward Euler (CBE) method, sufficient criteria promising the asymptotic mean‐square boundedness of CVNNs without any restriction on time stepsize, while the split‐step backward Euler (SSBE) method and backward Euler (BE) method can derive asymptotic mean‐square boundedness under a time stepsize constraint. The obtained theoretical results show that the compensated numerical methods own an incredible advantage over the noncompensated numerical methods on the part of asymptotic mean‐square boundedness. Finally, an example is proposed and analyzed to demonstrate the effectiveness and feasibility of the proposed results.

SOLUTIONS AND STABILITY ANALYSIS OF BACKWARD-EULER METHOD FOR SIMPLIFIED MAGNETOHYDRODYNAMICS WITH NONLINEAR TIME RELAXATION

In this study, the solutions of Simplified Magnetohyrodynamics (SMHD) equations by finite element method are examined with nonlinear time relaxation term. The differential filter κ(|u-u ̅ |(u-u ̅ )) term is added to SMHD equations. Also SMHD Nonlinear Time Relaxation Model (SMHDNTRM) is introduced. The model is discretized by Backward-Euler (BE) method to obtain the finite element solutions. Moreover, the stability of the method is proved. The method is found unconditionally stable. The effectiveness of the method is exemplified by several cases with comparing different methods. FreeFem++ is used for all computations.

Numerical Solution Strategies in Permeation Processes

In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

A four-leg matrix converter based hybrid distribution transformer for smart and resilient grids

This paper proposes a novel four leg matrix converter based active distribution transformer (4LMC-ADT), which aims at empowering legacy distribution transformers with on-load voltage control, a fundamental requirement for the future grid. Resorting to a four-leg direct matrix converter and a series transformer, this work shows that a traditional transformer can be empowered with local on demand voltage regulation in the LV grid, series harmonic filtering, dynamic voltage restoration, and MV support through reactive power injection, therefore actively contributing to improve power quality in a LV micro-grid even under severely unbalanced load conditions. Non-linear Lyapunov based controllers are designed for the fast tracking of the LV voltage references combined with Euler Backward predictive controllers for the matrix converter currents. Experimental results show that the proposed solution is able to cope with balanced and unbalanced voltage sags, voltage swells and load step changes, guaranteeing fast and accurate dynamic response of the 4LMC-ADT.

Unconditional superconvergence of the fully-discrete schemes for nonlinear prey-predator model

In this paper, the linearized Backward-Euler (BE) and backward-differential-formula (BDF) schemes are developed for the nonlinear and coupled prey-predator system with finite element method (FEM). The stability of the discrete solution in H1-norm is proved and used to deal with the quadratic term appeared in the system. Thanks to the boundness of the discrete solution obtained from the stability, the superclose results without time-step restriction are derived, in which a novel splitting technique is adopted to cope with the estimations related to the discrete time derivatives. Then, based on the high accuracy analysis, the unconditional superconvergence results of order O(h2+τ) and O(h2+τ2) in H1-norm are derived for BE and BDF schemes through post-processing approach, respectively. Here, h is the subdivision parameter and τ, the time step.

Superconvergence analysis of a mixed finite element approximation for the nonlinear fourth-order Rosenau-RLW equation

The focus of this paper is on a implicit Backward-Euler (BE) scheme with the mixed finite element method (FEM) for the two-dimentional general Rosenau-RLW equation. In which the bilinear element is used to approximate the exact solution v and the variable p=−Δv, and the zero-order nédéléc's finite element to the variable q→=ϕ∇v, respectively. An important point is that with the proposed scheme we are able to bound the numerical solution in H1-norm, so that we can efficiently solve the nonlinear term. Moreover, the stability, existence and uniqueness of approximate solution are demonstrated. Based on the combination of the interpolation and projection technique, the superconvergence estimates of order O(h2+τ) for v in H1-norm and the introducing variable q→ in L2-norm, and optimal error estimate of order O(h+τ) for introducing variable p in H1-norm is proved. Finally, numerical example is done to certify our theoretical results.

Fully Implicit Stress Update Algorithm for Distortion-Based Anisotropic Hardening with Cross-Loading Effect: Comparative Algorithmic Study and Application to Large-Size Forming Problem

A fully implicit stress integration algorithm is developed for the distortional hardening model, namely the e−HAH model, capable of simulating cross−hardening/softening under orthogonal loading path changes. The implicit algorithm solves a complete set of residuals as nonlinear functions of stress, a microstructure deviator, and plastic state variables of the constitutive model, and provides a consistent tangent modulus. The number of residuals is set to be 20 or 14 for the continuum or shell elements, respectively. Comprehensive comparison programs are presented regarding the predictive accuracy and stability with different numerical algorithms, strain increments, material properties, and loading conditions. The flow stress and r−value evolutions under reverse/cross−loading conditions prove that the algorithm is robust and accurate, even with large strain increments. By contrast, the cutting−plane method and partially implicit Euler backward method, which are characterized by a reduced number of residuals, result in unstable responses under abrupt loading path changes. Finally, the algorithm is implemented into the finite element modeling of large−size, S−rail forming and the springback for two automotive steel sheets, which is often solved by a hybrid dynamic explicit–implicit scheme. The fully implicit algorithm performs well for the whole simulation with the solely static implicit scheme.

Numerical Implementation of a Hydro-Mechanical Coupling Constitutive Model for Unsaturated Soil Considering the Effect of Micro-Pore Structure

An unsaturated soil constitutive model considering the influence of microscopic pore structure can more accurately describe the hydraulic–mechanical behavior of unsaturated soil, but its numerical implementation is more complicated. Based on the fully implicit Euler backward integration algorithm, the ABAQUS software is used to develop the established hydro-mechanical coupling constitutive model for unsaturated soil, considering the influence of micro-pore structure, and a new User-defined Material Mechanical Behavior (UMAT) subroutine is established to realize the numerical application of the proposed model. The developed numerical program is used to simulate the drying/wetting cycle process of the standard triaxial specimen. The simulation results are basically consistent with those calculated by the Fortran program, which verifies the rationality of the developed numerical program.

Quasi-uniform and unconditional superconvergence analysis of Ciarlet–Raviart scheme for the fourth order singularly perturbed Bi-wave problem modeling [formula omitted]-wave superconductors

In this paper, two implicit Backward Euler (BE) and Crank-Nicolson (CN) formulas of Ciarlet–Raviart mixed finite element method (FEM) are presented for the fourth order time-dependent singularly perturbed Bi-wave problem arising as a time-dependent version of Ginzburg-Landau-type model for d-wave superconductors by the bilinear element. The well-posedness of the weak solution and the approximation solutions of the considered problem are proved through Faedo-Galerkin technique and Brouwer fixed point theorem, respectively. The quasi-uniform and unconditional superconvergent estimates of O(h2+τ) and O(h2+τ2)(h, the spatial parameter, and τ, the time step) in the broken H1- norm are obtained for the above formulas independent of the negative powers of the perturbation parameter δ. Some numerical results are provided to illustrate our theoretical analysis.

A ROBUST SECOND-ORDER MULTIPLE BALANCE METHOD FOR TIME-DEPENDENT NEUTRON TRANSPORT SIMULATIONS

A second-order “Time-Dependent Multiple Balance” (TDMB) method for solving neutron transport problems is introduced and investigated. TDMB consists of solving two coupled equations: (i) the original balance equation (the transport equation integrated over a time step) and (ii) the “balance-like” auxiliary equation (an approximate neutron balance equation). Simple analysis shows that TDMB is second-order accurate and robust (unconditionally free from spurious oscillation). A source iteration (SI) method with diffusion synthetic acceleration (DSA) is formulated to solve these equations. A Fourier analysis reveals that the convergence rates of the proposed iteration schemes for TDMB are similar to those of the common (SI + DSA) schemes for Backward Euler (BE); however, TDMB requires about twice the computational effort per iteration. To demonstrate the theory—accuracy, robustness, and convergence rate—and investigate the efficiency of TDMB, we present results from a discrete ordinates (Sn) research code. Results are discussed, and future work is proposed.

Plasticity based interface model for failure modelling of unreinforced masonry under cyclic loading

In this work our objective is to understand the failure behaviour of unreinforced masonry under in-plane cyclic loading. For this purpose we proposed a plasticity based interface model consists of a single yield surface criteria which is a direct extension of Mohr-Coulomb criteria with a tension cut and compression cap and a back stress vector is introduced as a mixed hardening law variable in the adopted yield surface to capture the unloading/reloading behaviour of masonry under cyclic loading. A simplified micromechanical interface modelling approach is adopted to capture all the failure modes of masonry. The integration of the differential constitutive equation is done by using implicit Euler backward integration approach and the obtained non-linear set of equations are solved by a combined local/global Newton solver. The proposed constitutive model is implemented in ABAQUS by writing UMAT (user-defined subroutine) and the obtained numerical results are compared with experimental results available in the literature.

An Adaptive Time Filter Based Finite Element Method for the Velocity-Vorticity-Temperature Model of the Incompressible Non-Isothermal Fluid Flows

This paper studies a velocity-vorticity-temperature (VVT) model of the Boussinesq equations and introduces a numerical method for solving that. The proposed numerical method adds separate three minimally intrusive steps, one for each fluid variable, except pressure, to the standard semi-implicit backward-Euler (BE) approximation of VVT-model. The key idea in these intrusive steps is to post-process the BE approximate solutions with 2-step, second order, linear time filters. The paper provides full mathematical analysis of the proposed numerical method, and two numerical experiments for that. The first numerical experiment verifies the predicted convergence rates while the second one shows the effectiveness of the method on a benchmark problem.

Kalman and UFIR state estimation with coloured measurement noise using backward Euler method

Under coloured noise, known modifications of the Kalman filter (KF) exist only for discrete-time state-space models produced by the forward Euler (FE) method, which fits with feedback control. In this study, the authors modify the KF and unbiased finite impulse response (UFIR) filter using the backward Euler (BE) method for models with coloured measurement noise (CMN), which better fits systems without feedback. The FE- and BE-based models differ by time indexes in the system input and noise that is essential for time-varying and Markov jump systems. Employing measurement differencing, two KF algorithms and a unique UFIR algorithm are derived for time-correlated and de-correlated noise. An equivalence of the KF algorithms is proved analytically and confirmed by simulations. Numerical examples are given for target tracking and experimental verification is provided for visual object tracking. The high efficiency of the designed algorithms in removing CMN is demonstrated experimentally.

Superconvergence analysis of two-grid methods for bacteria equations

In this paper, two-grid methods (TGMs) are developed for a system of reaction-diffusion equations of bacterial infection with initial and boundary conditions. The backward Euler (B-E) and Crank–Nicolson (C-N) fully discrete schemes are established, and the existence and uniqueness of the solutions of these schemes are proved. Moreover, based on the combination technique of the interpolation and Ritz projection and derivative transfer trick which are important ingredients in the TGMs, the superclose estimates of order O(h2 + H4 + τ) and O(h2 + H4 + τ2) in H1-norm are deduced for the above schemes, respectively, where h is fine mesh size, H is coarse mesh size, and τ is time step size. Then, by the interpolated postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some other popular finite elements are discussed and numerical results are provided to verify the theoretical analysis, which show that the computing cost of TGMs are only half of Galerkin finite element methods (FEMs) for the test problem.

3D simulations of solidification with liquid phase flow

The work discusses issues related to 3D coagulation simulations including liquid phase flow. The approach chosen by the authors is based on the numerical solution of the enthalpy solidification formula and the Navier-Stokes equation, in which the buoyancy force approximation was added. Both of these equations are discretized in space using Finite Element Method, and in both cases time integration utilizes the Theta scheme, which is a generalized form of Euler Backward approach. The presence of the Navier-Stokes equation in the model forces the use of special techniques to ensure the convergence of the solution. From a number of available methods, the method of adding stabilizing elements in the FEM formulation was chosen, i.e. the stabilization formulas PSPG/SUPG (Pressure Stabilized Petrov Galerkin/Streamline Upwind Stabilized Petrov Galerkin). The occurrence of large temperature gradients, however, made it necessary to use the so-called lumped mass matrix in the energy equation. The work presents the results obtained using a model implemented by the authors in the form of a computer program written in C++ programming language. The use of this language and appropriate libraries made it possible to obtain sufficient performance to perform 3D solidification simulations.

Thermo‐Mechanical Analysis of a Steam Turbine rotor

AbstractIn power plants, high temperatures prevail during long holding times. Furthermore, power plants are often started and shut‐down in order to account for gaps or oversupplies in energy production. These loading conditions induce both creep and fatigue loads. Due to their excellent thermo‐mechanical properties, such as high tensile strength and elevated corrosion resistance, heat‐resistant steels are established materials for power plant components. Nevertheless, these steels tend to soften under deformation, which should be accounted for by a constitutive model.The contribution at hand analyses the thermo‐mechanical behavior of a steam turbine rotor. For this purpose, a unified phase mixture model is introduced. This constitutive model accounts for rate‐dependent inelasticity, hardening, as well as softening by employing an iso‐strain approach with a soft and a hard constituent. While the soft constituent represents areas with a low dislocation density, such as the interior of subgrains, the hard constituent refers to regions with a high dislocation density, i.e. the subgrain boundaries. Furthermore, two internal variables are introduced: a backstress tensor of Armstrong‐Frederick type and a scalar softening variable. The model results in a coupled system of three evolution equations with respect to the inelastic strain, the backstress, and the softening variable.To allow for the analysis of real power plant components, the model is implemented into the finite element method such that the evolution equations are integrated based on the backward EULER method. The applicability of the model is demonstrated by conducting a thermo‐mechanical analysis of a steam turbine rotor with complex geometry under realistic boundary conditions. In a first step, the instationary temperature field in the rotor is computed in a heat transfer analysis. Thereby, typical steam temperatures in power plants and the corresponding heat transfer coefficients are prescribed. As a next step, the structural analysis is conducted based on the phase mixture model and the obtained temperature field as input. In addition, the time‐dependent rotational frequency and steam pressure are taken into account. Note that the influence of different start‐up procedures such as a cold or a hot start is examined in detail. As a result, the structural analysis provides the stress‐strain hystereses, which constitute the basis for further fatigue and damage assessment.

Numerical analysis of a 4th-order time parallel algorithm for the time-dependent Navier-Stokes equations

This work studies the stability and convergency properties for a time parallel algorithm: RIDC4 with reduced stencils, namely reduced 4th revisionist integral deferred correction method. For the spatial approximation of the velocity and the pressure, a finite element space discretization is applied. We proved that the scheme is almost unconditionally stable and is of 4th-order accuracy in time. Finally, numerical experiments are carried out to verify its time and space convergence orders and the formidable parallel efficiency demonstrated by the time speed-up compared with the classical Backward Euler (BE) methods. In conclusion, within almost same time to BE, the (reduced) RIDC4 scheme could achieve 4th order accuracy in time direction, provided four cores are available.

Spectral-element method with arbitrary hexahedron meshes for time-domain 3D airborne electromagnetic forward modeling

Mainstream numerical methods for 3D time-domain airborne electromagnetic (AEM) modeling, such as the finite-difference (FDTD) or finite-element (FETD) methods, are quite mature. However, these methods have limitations in terms of their ability to handle complex geologic structures and their dependence on quality meshing of the earth model. We have developed a time-domain spectral-element (SETD) method based on the mixed-order spectral-element (SE) approach for space discretization and the backward Euler (BE) approach for time discretization. The mixed-order SE approach can contribute an accurate result by increasing the order of polynomials and suppress spurious solutions. The BE method is an unconditionally stable technique without limitations on time steps. To deal with the rapid variation of the fields close to the AEM transmitting loop, we separate a secondary field from the primary field and simulate the secondary field only, for which the primary field is calculated in advance. To obtain a block diagonal mass matrix and hence minimize the number of nonzero elements in the system of equations to be solved, we apply Gauss-Lobatto-Legendre integral techniques of reduced order. A direct solver is then adopted for the system of equations, which allows for efficient treatment of the multiple AEM sources. To check the accuracy of our SETD algorithm, we compare our results with the semianalytical solution for a layered earth model. Then, we analyze the modeling accuracy and efficiency for different 3D models using deformed physical meshes and compare them against results from 3D FETD codes, to further show the flexibility of SETD for AEM forward modeling.

Implementation of a Phase Mixture Model for Rate‐Dependent Inelasticity

AbstractPower plant components have to withstand complex thermo‐mechanical loads, i.e. frequent start‐ups and shut‐downs in combination with long holding times at high temperatures induce creep‐fatigue loads. Tempered martensitic steels with high chromium content feature high creep strength and corrosion resistance among other properties such that these alloys are often used for power plant components.This contribution focuses on the simulation of the complex mechanical behavior of the alloy X20CrMoV12‐1 based on a multi‐axial phase mixture model. Within this model, we distinguish two phases: a hard and a soft phase. Both phases are connected via an iso‐strain approach, and nonlinear kinematic hardening as well as microstructural softening effects are taken into account by introducing a backstress and a softening variable as internal variables. The model is implemented into the commercial finite element code ABAQUS with a user material subroutine. For this purpose, the stress update algorithm and the consistent tangent operator are implemented based on the backward EULER method in combination with NEWTON‐RAPHSON iteration. Next, a wide range of benchmarks for uni‐axial as well as multi‐axial stress states is taken into account in order to verify the numerical implementation. Furthermore, a thermo‐mechanical fatigue test based on a typical sequence of start‐ups and shut‐downs of power plants is simulated.

An exponential time-integrator scheme for steady and unsteady inviscid flows

An exponential time-integrator scheme of second-order accuracy based on the predictor–corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge–Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.