Euler Backward Research Articles

Study of a Numerical Integral Interpolation Method for Electromagnetic Transient Simulations

In the fixed time-step electromagnetic transient (EMT)-type program, an interpolation process is applied to deal with switching events. The interpolation method frequently reduces the algorithm’s accuracy when dealing with power electronics. In this study, we use the Butcher tableau to analyze the defects of linear interpolation. Then, based on the theories of Runge–Kutta integration, we propose two three-stage diagonally implicit Runge–Kutta (3S-DIRK) algorithms combined with the trapezoidal rule (TR) and backward Euler (BE), respectively, with TR-3S-DIRK and BE2-3S-DIRK for the interpolation and synchronization processes. The proposed numerical integral interpolation scheme has second-order accuracy and does not produce spurious oscillations due to the size change in the time step. The proposed method is compared with the critical damping adjustment method (CDA) and the trapezoidal method, showing that it does not produce spurious numerical oscillations or first-order errors.

Superconvergence analysis of an anisotropic nonconforming FEM for parabolic equation

In this paper, the constrained nonconforming rotated Q1 (CNQ1rot) element is employed to study the superconvergence behavior of the backward-Euler (B-E) and Crank–Nicolson (C–N) fully-discrete schemes for the parabolic equation. By applying the mean-value and integral identities techniques, a novel high-accuracy estimate for the CNQ1rot element is proved on anisotropic meshes rigorously, which is essential to derive the superclose result. Then, the superconvergence results for the above two schemes are deduced with the help of the interpolation post-processing approach, respectively. Eventually, some numerical experiments are carried out to verify the rationality of the theoretical analysis.

Optimal error estimates of nonconforming Wilson element for nonlinear nonlocal parabolic problems

The penalty backward Euler (BE) and 2-step backward differential formula (BDF2) fully discrete schemes are proposed for the nonlinear nonlocal parabolic problems with nonconforming Wilson element. Optimal convergence results with order O(h2+τ) and order O(h2+τ2) in a modified norm, larger than the usual broken H1-norm, are obtained directly which improve the results of order O(h+τ) and order O(h+τ2) with respect to h in the previous studies, respectively. Here, h and τ denote the mesh size and time step, respectively. Finally, some numerical results are provided to confirm the theoretical analysis.

A finite deformation formulation for amorphous glassy polymers under moderate and impact strain rates: Application to adhesive films

This work describes a novel finite strain hypo-elastic formulation for amorphous thermoset polymers working in the glassy regime. The constitutive formulation is able to describe time and pressure-dependent behaviour under loading rates ranging from quasi-static to impact-type loading conditions and to account for thermo-mechanical coupling effects. A non-linear pressure dependent potential function is introduced to capture the non-associative visco-plastic potential flow for volumetric plastic strain control. The formulation also incorporates an internal state or deformation resistance variable to enable the description of the polymer hardening — softening behaviour, as well as a novel relation for the dissipative fraction of the plastic work in terms of the equivalent accumulated plastic strain. The constitutive model is formulated within a finite strain kinematics framework, and full details of its numerical implementation into the finite element method using a fully implicit Euler Backward algorithm are given. Model calibration is carried out for three different thermosetting resins (PR520 and RTM6 Epoxies, and MMA adhesives).To illustrate the model capabilities, two case studies are investigated: (i) the de-formation behaviour of epoxy under moderate and impact-type loading conditions under isothermal and fully adiabatic conditions, and (ii) the fracture behaviour of adhesive layers used to bond stiff metallic substrates. Results on RTM6 epoxy show that at, e.g., 60% true strain, 48.5% of the rate of plastic work is dissipative, and that the corresponding predicted increase in temperature due to the locally dissipated heat is consistent with published calorimetry data on a similar thermoset polymer. It was also found that, by coupling the proposed constitutive model with a cohesive zone model, it was possible to predict accurately the effect of an MMA adhesive layer thickness and pressure-dependency on the growth of an interfacial crack between the MMA adhesive and the metallic substrates. The proposed model should constitute a generic phenomenological formulation for the mechanical behaviour prediction of a wide range of thermoset polymers under confined and quasi-adiabatic conditions such as in composites and adhesive joints.

Control design for thrust generators with application to wind turbine wave-tank testing: A sliding-mode control approach with Euler backward time-discretization

The control design for a propeller-based thrust generator used in a wind turbine testing platform is studied in this work. A mathematical model has been developed for the system including the motors and propeller. Subsequently, a continuous-time sliding-mode controller is designed based on the developed model and its stability and robustness have been addressed. An Euler backward time discretization method has been developed for the continuous-time sliding-mode controller to achieve a chattering-free implementation. The properties of the sliding-mode controller under the developed time discretization method e.g., finite-time convergence, and gain insensitivity have been studied analytically. In order to evaluate the developed sliding-mode control law under the discretization method, three known control strategies, i.e., gain-scheduling proportional-integral control, fuzzy control, and feedforward compensator strategies have been designed for the system. Some remarks are also given for the differentiator selection used to estimate the velocity. The experiments under different scenarios are then conducted and the results corresponding to all four controllers are provided along with a comparative analysis to identify the properties of each control configuration.

A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple

This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Numerical investigation of generalized tempered-type integrodifferential equations with respect to another function

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct a semi-discrete BE scheme. Second, the backward differentiation formula (BDF) and second-order interpolation quadrature rule are adopted to establish a semi-discrete second-order BDF (BDF2) scheme. Additionally, the stability and convergence of two semi-discrete methods are deduced in detail. To further demonstrate the effectiveness of proposed techniques, fully discrete BE and BDF2 finite difference schemes are formulated. Subsequently, the theoretical results of two fully discrete difference schemes are presented. Finally, the numerical results demonstrate the accuracy and competitiveness of the theoretical analysis.

An improved high-accuracy interpolation method for switching devices in EMT simulation programs

This paper presents an interpolation with Backward Euler (BE) method that enhances the switching simulations accuracy in an electromagnetic transient (EMT) simulation with a fixed time-step. Because of the switching operations on discrete instants, the simulations can show artificial voltage spikes and numerical oscillations (chatter), compromising the accuracy and producing misleading results. We thus present a method that eliminates the spurious switching losses and improves the accuracy by combining interpolation and extrapolation with a half-time-step BE solution over existing interpolation-based approaches to resolve the issues above. In addition, this improved method requires the same calculation steps as the industry-accepted instantaneous interpolation. Analytical discussion reinforced by the simulation studies for a simple and complex power electronic circuits demonstrates the accuracy and efficacy of the proposed interpolation with BE method.

Nonconforming modified Quasi-Wilson finite element method for convection–diffusion-reaction equation

The nonconforming modified Quasi-Wilson finite element method (FEM) is employed to investigate the unconditional superconvergence behavior of the convection–diffusion-reaction equation for the linearized energy stable Backward-Euler (BE) and the second order Backward differentiation formula (BDF2) fully discrete schemes on arbitrary quadrilateral meshes. Then, the energy stabilities of the discrete solutions for the proposed schemes are demonstrated through the mathematical induction, which is crucial to proving the unique solvabilities with Brouwer fixed point theorem. Based on the above achievements and the two typical characteristics of the element: one is that its consistency error in the broken H1-norm can be estimated as order O(h2) when the exact solution belongs to H3(Ω), which is one order higher than its interpolation error, and the other is that the nonconforming part of the function in the finite element space is orthogonal to the bilinear polynomials on each element of the subdivision in the integration sense, the superclose estimates without any restriction between spatial partition parameter h and the time step τ are obtained. Furthermore, the superconvergence properties for the aforementioned schemes are derived by the interpolation post-processing technique. Finally, the theoretical analysis is justified by the provided numerical experiments.

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

Analysis of a Filtered Time-Stepping Finite Element Method for Natural Convection Problems

In this paper, we present, analyze, and test a novel low-complexity time-stepping finite element method for natural convection problems utilizing a time filter (TF). First, via a TF to postprocess the solutions of backward Euler (BE) schemes, we make a minimally intrusive modification to the existing codes to improve the time accuracy by one order. This also provides, at no extra complexity, an estimate of the temporal error,which is easy to construct a novel adaptive algorithm. Additionally, the TF can remove the overdamping of the BE scheme while remaining unconditionally energy stable. Hence, this paper addresses the question, how can one improve the time accuracy without increasing computational and cognitive complexity? Then long time stability and error estimates of BE plus time filter (BETF) with constant time stepsize are proved. Moreover, we construct adaptive algorithms by extending the approach to variable time stepsize, and we extend the methods to higher order algorithms. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.

The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics

In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on the backward Euler (BE) method and Crank-Nicolson (CN) method, respectively, from which, the Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The $ L^2 $-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.

Modeling of masonry structures using a new 3D cohesive interface material model considering dilatancy softening

This study presents a new 3D multi-yield surfaces material model for masonry mortar joints, which can be used with interface elements for finite element (FE) modeling of masonry structures to capture various failure mechanisms. This model is featured with (1) two hyperbolic yield surfaces, capable of capturing various failure modes of mortar joints, including tensile cracking, shear sliding, and compressive crushing, (2) an unassociated flow rule to capture the ‘dilatancy’ phenomenon in the mortar joints, and (3) the dilatancy softening and variation of mode II fracture energy under different normal stress levels. An implicit Euler backward integration algorithm, combined with local-global Newton-Raphson (NR) solver, is adopted to achieve the predictor-corrector return mapping in the numerical formulation. The error-based auto-adaptive sub-stepping algorithm is employed to achieve the robustness and efficiency in the integration procedure. The developed model is implemented in the general-purpose FE software ABAQUS with the specialized capability of modeling masonry structures. The developed model is validated through three brick-mortar-brick assemblages and three un-reinforced masonry (URM) walls under in-plane (IP) loading or out-of-plane (OOP) loading. The importance of appropriate modeling of dilatancy on simulating the IP and OOP behavior of URM walls is highlighted. The validation results show that the developed constitutive model is capable of modeling masonry structures at various scales with improved accuracy by considering dilatancy softening.

Unconditional superconvergence analysis for the nonlinear Bi-flux diffusion equation

The fourth-order Bi-flux diffusion equation with a nonlinear reaction term is studied by the linear finite element method (FEM). First, a novel important property of high accuracy of this element is proved by the Bramble-Hilbert (B-H) lemma, which is essential to the superconvergence analysis. Then, the Backward-Euler (B-E) and Crank-Nicolson (C-N) fully discrete schemes are developed, and the stabilities of their numerical solutions and the unique solvabilities are demonstrated. Furthermore, by applying a splitting argument to dealing with the nonlinear term, the superconvergence results in H1-norm are derived without any restriction between the mesh size h and the time step τ. Finally, numerical results are presented to verify the rationality of the theoretical analysis.

Unconditional superconvergence analysis of two modified finite element fully discrete schemes for nonlinear Burgers' equation

In this paper, two modified nonconforming finite element fully discrete schemes are considered for solving the nonlinear Burgers' equation. The schemes are constructed by using the nonconforming Wilson element approximation in space combining with the backward Euler(BE) and 2-step backward differentiation formula (BDF2) respectively in time. We present the superconvergence results of the two proposed schemes in modified energy norm of order O(h2+τ) and O(h2+τ2), where h and τ are the space subdivision parameter and time step respectively. The proof involves the time-space error splitting technique, the discrete embedding inequality, the technique of taking difference between two time, the discrete derivative transfer technique and a new modified projection operator. Our analysis not only gets rid of the restriction between temporal and spatial step sizes, but also provides one order higher error estimate results than that of corresponding traditional finite element fully discrete schemes. Finally, numerical experiments are carried out to demonstrate the effective of theoretical analysis.

Filtered time-stepping method for incompressible Navier-Stokes equations with variable density

In this paper, we present a filtered time-stepping method for incompressible Navier-Stokes equations with variable density. The method increases the time accuracy from first order to second order that requires only one backward Euler (BE) solve at each time step followed by adding a time filter. The added time filter was expressed by the linear combinations of the solutions at previous time levels without extra complexity. We proved the stability of density and velocity for fully implicit BE algorithm and backward Euler plus time filter (BETF) algorithm. Moreover, we extend the approach to a variable time stepsize BETF algorithm, and construct a new adaptive BE algorithm and a novel variable stepsize variable order algorithm with low cost error estimators. Finally, numerical experiments show the stability and efficiency of our methods.

A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation

AbstractIn this paper, we investigate the numerical solution of the three‐dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first‐order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three‐dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non‐linear term. Thereafter we prove a positive‐type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.

An integrated unified elasto-viscoplastic fatigue and creep damage model with characterization method for structural analysis of nickel-based high-temperature structure

This paper proposes an integrated unified elasto-viscoplastic fatigue and creep damage model with modified hardening equations to simulate the elasto-viscoplasticity, primary-secondary and tertiary creep, relaxation, cyclic softening, temperature-dependency, and creep-fatigue interaction (CFI) damage. For this purpose, we integrate the creep damage and fatigue damage model into the viscoplasticity-damage model to predict tertiary creep and cyclic softening behavior caused by creep and fatigue damage modes. Interaction effects between each damage mode are considered by the creep-fatigue interaction (CFI) damage scheme. This paper also introduces a genetic algorithm-based integrated characterization method that can effectively optimize material input parameters of the unified elasto-viscoplastic-damage material model with limited experimental data. Implicit Euler's backward iteration scheme is adopted to improve the convergence and accuracy of solutions. The proposed material model is implemented in UMAT for finite element (FE) analysis of high-temperature structures under cyclic-dwell loading. Experimental data of tensile, dwell, creep, and cyclic loading at various temperatures are used to verify the model. Finally, full-scale turbine blade FE analysis is performed under anisothermal conditions. The proposed material modeling framework can also be utilized for other high-temperature structures, such as power plants, leading-edge nose cones, etc.

Fast 3D transient electromagnetic forward modeling using BEDS-FDTD algorithm and GPU parallelization

The time-step sizes of the conventional finite-difference time-domain (FDTD) method are severely restricted by the Courant-Friedrichs-Lewy stability condition. This may result in excessive iterations, high cost and large runtime, and increasing late-stage cumulative errors in 3D transient electromagnetic (TEM) forward modeling. At present, forward modeling speed is a major obstacle in the development of 3D TEM inversion. To address this issue, a new 3D TEM forward modeling algorithm is presented. In the new algorithm, the backward Euler (BE) scheme is used with a view toward relaxing the stability constraint. However, after using the BE approach, a huge sparse matrix needs to be inverted in each iteration. Directly solving such matrices by Gaussian elimination or an iterative method is still time-consuming and requires large amounts of computer memory. To improve computational efficiency, the direct-splitting (DS) strategy is adopted to transform the large sparse matrices into a series of small diagonally dominant tridiagonal matrices, which are easier to solve. Then, the complex-frequency-shifted perfectly matched layer boundary condition is implemented using the bilinear transform method to reduce the size of the model. To further speed up the calculation process, the codes are then programmed to run on graphics processing unit (GPU) devices. Three models are used for calculations, and the results are compared with other numerical methods to verify the accuracy of our algorithm. After that, to demonstrate the BEDS-FDTD algorithm’s ability to model complex earth structures, a model with a very complex shape is used for the calculation via the conformal mesh technique. In addition, the modeling speed is tested using different GPU devices. We find that an NVIDIA Tesla A100 is able to calculate the 50 × 50 × 50 cell model in only 8 s.