Euler Backward Research Articles

Efficient time stepping for the multiplicative Maxwell fluid including the Mooney‐Rivlin hyperelasticity

SummaryA popular version of the finite‐strain Maxwell fluid is considered, which is based on the multiplicative decomposition of the deformation gradient tensor. The model combines Newtonian viscosity with hyperelasticity of the Mooney‐Rivlin type; it is a special case of the viscoplasticity model proposed by Simo and Miehe in 1992. A simple, efficient, and robust implicit time‐stepping procedure is suggested. Lagrangian and Eulerian versions of the algorithm are available, with equivalent properties. The numerical scheme is iteration free, unconditionally stable, and first order accurate. It exactly preserves the inelastic incompressibility, symmetry, and positive definiteness of the internal variables and w‐invariance. The accuracy of the stress computations is tested using a series of numerical simulations involving a nonproportional loading and large strain increments. In terms of accuracy, the proposed algorithm is equivalent to the modified Euler backward method with exact inelastic incompressibility; the proposed method is also equivalent to the classical integration method based on exponential mapping. Since the new method is iteration free, it is more robust and computationally efficient. The algorithm is implemented into MSC.MARC, and a series of initial boundary value problems is solved to demonstrate the usability of the numerical procedures.

CFD Modeling of Chamber Filling in a Micro-Biosensor for Protein Detection.

Tuberculosis (TB) remains one of the main causes of human death around the globe. The mortality rate for patients infected with active TB goes beyond 50% when not diagnosed. Rapid and accurate diagnostics coupled with further prompt treatment of the disease is the cornerstone for controlling TB outbreaks. To reduce this burden, the existing gap between detection and treatment must be addressed, and dedicated diagnostic tools such as biosensors should be developed. A biosensor is a sensing micro-device that consists of a biological sensing element and a transducer part to produce signals in proportion to quantitative information about the binding event. The micro-biosensor cell considered in this investigation is designed to operate based on aptamers as recognition elements against Mycobacterium tuberculosis secreted protein MPT64, combined in a microfluidic-chamber with inlet and outlet connections. The microfluidic cell is a miniaturized platform with valuable advantages such as low cost of analysis with low reagent consumption, reduced sample volume, and shortened processing time with enhanced analytical capability. The main purpose of this study is to assess the flooding characteristics of the encapsulated microfluidic cell of an existing micro-biosensor using Computational Fluid Dynamics (CFD) techniques. The main challenge in the design of the microfluidic cell lies in the extraction of entrained air bubbles, which may remain after the filling process is completed, dramatically affecting the performance of the sensing element. In this work, a CFD model was developed on the platform ANSYS-CFX using the finite volume method to discretize the domain and solving the Navier–Stokes equations for both air and water in a Eulerian framework. Second-order space discretization scheme and second-order Euler Backward time discretization were used in the numerical treatment of the equations. For a given inlet–outlet diameter and dimensions of an in-house built cell chamber, different inlet liquid flow rates were explored to determine an appropriate flow condition to guarantee an effective venting of the air while filling the chamber. The numerical model depicted free surface waves as promoters of air entrainment that ultimately may explain the significant amount of air content in the chamber observed in preliminary tests after the filling process is completed. Results demonstrated that for the present design, against the intuition, the chamber must be filled with liquid at a modest flow rate to minimize free surface waviness during the flooding stage of the chamber.

Linear time complexity sorting algorithms for electromagnetic transient simulation of MMC‐HVdc system

In the electromagnetic transient (EMT) simulation programs, the candidate integration methods backward Euler (BE) method and Trapezoidal rule (TR) are normally used to discretise the submodule capacitors in modular multilevel converter (MMCs). This study proposes linear time complexity sorting algorithms for nearest level control-based BE and TR MMC models to further accelerate the EMT simulation of the equivalent MMC-HVdc models. First, the capacitor voltage increments in the charging and discharging processes are investigated from an EMT point of view. Second, for both BE and TR methods, although the actual capacitor voltage increments are not identical at each time step, the voltage ranking of the capacitors from the same inserted/bypassed category at each control period are proved to be unchanged. Taking this advantage, when preparing the entire ranked voltage table for the next control period, the sorting can be significantly simplified since one only needs to compare the capacitor voltages from capacitor categories of ascending orders. Third, when implementing the sorting algorithms, at most ( N -1) comparisons for BE model and (2 N -3) comparisons for TR model are required, showing that the proposed sorting algorithms have linear time complexities. Finally, all the proposed approaches are validated by EMT simulations on MATLAB/Simulink.

Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose–Einstein condensates (BEC), modeled by the Gross–Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank–Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned steepest descent (PSD) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.

Finite element analysis and approximation of Burgers’‐Fisher equation

In this article, the authors present finite element analysis and approximation of Burgers’‐Fisher equation. Existence and uniqueness of weak solution is proved by Galerkin's finite element method for non‐smooth initial data. Next, a priori error estimates of semi‐discrete solution in norm, are derived and the convergence of semi‐discrete solution is established. Then, fully discretization of the problem is done with the help of Euler's backward method. The nonlinearity is removed by lagging it to previous known level. The scheme is found to be convergent. Positivity of fully discrete solution is discussed, and bounds on time step are discovered for which the solution preserves its positivity. Finally, numerical experiments are performed on some examples to demonstrate the effectiveness of the scheme. The proposed scheme found to be fast, easy and accurate.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1652–1677, 2017

Numerical solution of one-dimensional transient, two-phase flows with temporal fully implicit high order schemes: Subcooled boiling in pipes

An extension for 1-D transient two-phase flows of the SIMPLE-ESDIRK method, initially developed for incompressible viscous flows by Ijaz is presented. This extension is motivated by the high temporal order of accuracy demanded to cope with fast phase change events. This methodology is suitable for boiling heat exchangers, solar thermal receivers, etc. The methodology of the solution consist in a finite volume staggered grid discretization of the governing equations in which the transient terms are treated with the explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) method. It is suitable for stiff differential equations, present in instant boiling or condensation processes. It is combined with the semi-implicit pressure linked equations algorithm (SIMPLE) for the calculation of the pressure field. The case of study consists of the numerical reproduction of the Bartolomei upward boiling pipe flow experiment. The steady-state validation of the numerical algorithm is made against these experimental results and well known numerical results for that experiment. In addition, a detailed study reveals the benefits over the first order Euler Backward method when applying 3rd and 4th order schemes, making emphasis in the behaviour when the system is subjected to periodic square wave wall heat function disturbances, concluding that the use of the ESDIRK method in two-phase calculations presents remarkable accuracy and computational advantages.

Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

The main aim of this paper is to apply the conforming bilinear finite element to solve the nonlinear Schrödinger equation (NLSE). Firstly, the stability and convergence for time discrete scheme are proved. Secondly, through a new estimate approach, the optimal order error estimates and superclose properties in H1-norm are obtained with Backward Euler (B-E) and Crank-Nicolson (C-N) fully-discrete schemes, the global superconvergence results are deduced with the help of interpolation postprocessing technique. Finally, some numerical examples are provided to verify the theoretical analysis.

Enhanced high-speed electromagnetic transient simulation of MMC-MTdc grid

This paper introduces a very fast electromagnetic transient (EMT) simulation model for the HVdc modular multilevel converter (MMC) that maintains the identity of each switching level, but achieves computation speeds comparable to the much simplified averaged-value models (AVMs) when simulating the multi-terminal dc grid. Speedup is achieved by representing the off state of a MMC sub-module (SM) with ideal zero conductance, and representing the converter with a companion model using the A-stable Backward Euler (BE) method. Often, the user may wish to use the nearest level control (NLC) based voltage balancing algorithm. Then additional speedup can be obtained by using an efficient sorting algorithm which is integrated into the Thévenin equivalent circuit. This achieves a linear speedup (i.e. order O(N)) with system size. When compared with a fully detailed simulation (no simplifications), the method shows one to two orders of magnitude speed improvement with earlier reported fast MMC models, with negligible loss in accuracy.

Efficient implicit integration for finite-strain viscoplasticity with a nested multiplicative split

An efficient and reliable stress computation algorithm is presented, which is based on implicit integration of the local evolution equations of multiplicative finite-strain plasticity/viscoplasticity. The algorithm is illustrated by an example involving a combined nonlinear isotropic/kinematic hardening; numerous backstress tensors are employed for a better description of the material behavior.The considered material model exhibits the so-called weak invariance under arbitrary isochoric changes of the reference configuration, and the presented algorithm retains this useful property. Even more: the weak invariance serves as a guide in constructing this algorithm. The constraint of inelastic incompressibility is exactly preserved as well. The proposed method is first-order accurate.Concerning the accuracy of the stress computation, the new algorithm is comparable to the Euler Backward method with a subsequent correction of incompressibility (EBMSC) and the classical exponential method (EM). Regarding the computational efficiency, the new algorithm is superior to the EBMSC and EM. Some accuracy tests are presented using parameters of the aluminum alloy 5754-O and the 42CrMo4 steel. FEM solutions of two boundary value problems using MSC.MARC are presented to show the correctness of the numerical implementation.

Dual estimation: Constructing building energy models from data sampled at low rate

Estimation of energy models from data is an important part of advanced fault detection and diagnosis tools for smart energy purposes. Estimated energy models can be used for a large variety of management and control tasks, spanning from model predictive building control to estimation of energy consumption and user behavior. In practical implementation, problems to be considered are the fact that some measurements of relevance are missing and must be estimated, and the fact that other measurements, collected at low sampling rate to save memory, make discretization of physics-based models critical. These problems make classical estimation tools inadequate and call for appropriate dual estimation schemes where states and parameters of a system are estimated simultaneously. In this work we develop dual estimation schemes based on Extended Kalman Filtering (EKF) and Unscented Kalman Filtering (UKF) for constructing building energy models from data: in order to cope with the low sampling rate of data (with sampling time 15min), an implicit discretization (Euler backward method) is adopted to discretize the continuous-time heat transfer dynamics. It is shown that explicit discretization methods like the Euler forward method, combined with 15min sampling time, are ineffective for building reliable energy models (the discrete-time dynamics do not match the continuous-time ones): even explicit methods of higher order like the Runge–Kutta method fail to provide a good approximation of the continuous-time dynamics which such large sampling time. Either smaller time steps or alternative discretization methods are required. We verify that the implicit Euler backward method provides good approximation of the continuous-time dynamics and can be easily implemented for our dual estimation purposes. The applicability of the proposed method in terms of estimation of both states and parameters is demonstrated via simulations and using historical data from a real-life building.

A Rate Independent Cohesive Zone Model for Modeling Failure in Quasi-Brittle Materials

Computational modeling of failure in quasi-brittle materials at various length scales is important. In this work we present a rate independent cohesive zone model for modeling failure in quasi-brittle materials. The proposed model can simulate cracking, slipping, and crushing of planes through a traction-separation law. A single surface hyperbolic failure criterion, which naturally comes as a direct extension of Coulomb friction criterion with cut-off in tension and cap-off in compression, has been developed. A Euler backward integration scheme together with a global-local Newton solver compatible with a substepping strategy has been used in numerical computations. The proposed model is then used for modeling of shear wall panels. The numerical results obtained are validated by comparing them with experimental results available in literatures.

Implicit solution of the material transport in Stokes flow simulation: Toward thermal convection simulation surrounded by free surface

We present implicit time integration schemes suitable for modeling free surface Stokes flow dynamics with marker in cell (MIC) based spatial discretization. Our target is for example thermal convection surrounded by deformable surface boundaries to simulate the long term planetary formation process. The numerical system becomes stiff when the dynamical balancing time scale for the increasing/decreasing load by surface deformation is very short compared with the time scale associated with thermal convection. Any explicit time integration scheme will require very small time steps; otherwise, serious numerical oscillation (spurious solutions) will occur. The implicit time integration scheme possesses a wider stability region than the explicit method; therefore, it is suitable for stiff problems. To investigate an efficient solution method for the stiff Stokes flow system, we apply first (backward Euler (BE)) and second order (trapezoidal method (TR) and trapezoidal rule—backward difference formula (TR-BDF2)) accurate implicit methods for the MIC solution scheme. The introduction of implicit time integration schemes results in nonlinear systems of equations. We utilize a Jacobian free Newton Krylov (JFNK) based Newton framework to solve the resulting nonlinear equations. In this work we also investigate two efficient implicit solution strategies to reduce the computational cost when solving stiff nonlinear systems. The two methods differ in how the advective term in the material transport evolution equation is treated. We refer to the method that employs Lagrangian update as “fully implicit” (Imp), whilst the method that employs Eulerian update is referred to as “semi-implicit” (SImp). Using a finite difference (FD) method, we have performed a series of numerical experiments which clarify the accuracy of solutions and trade-off between the computational cost associated with the nonlinear solver and time step size. In comparison with the general explicit Euler method, the second order accurate Imp methods reduce total computational cost successfully through the utilization of a large time step without sacrificing accuracy and stability. Moreover, the proposed SImp method is effective in reducing the computational cost associated with evaluating the nonlinear residual while obtaining a solution similar to the Imp method.

回路分割法によるパワーエレクトロニクス回路の並列化シミュレーション

This paper proposes a fast parallel circuit simulation method for a power electric system by applying an explicit integration formula to selected energy storage elements such as inductors and capacitors. The forward Euler (FE) formula is applied to selected series inductors and parallel capacitors. Then an implicit formula such as the backward Euler (BE) formula is applied for stable numerical integration, and the optimum step size is selected for each subcircuit. A parallel processing method that utilizes thread processing techniques based on OpenMP application program interfaces with a multicore CPU and shared-memory system reduces its processing time. A new general algorithm based on a thread processing technique, a subcircuit memory processing technique, and a binary variable step size method is described to utilize numerical processing at each time step synchronously and efficiently. Then a new automatic partitioning method of a system into subcircuits is described. This paper also proposes how to process a PE system with a DSP-based digital control principle. The proposed method is applied to power electronic systems. It is investigated how the proposed method works and shown for effectiveness of parallel simulation by the circuit partitioning.

A backward euler orthogonal spline collocation method for the time-fractional Fokker-Planck equation

A backward euler orthogonal spline collocation method for the time-fractional Fokker-Planck equation

Convergence analysis for Backward-Euler and mixed discontinuous Galerkin methods for the Vlasov-Poisson system

We construct and analyze a numerical scheme for the two-dimensional Vlasov-Poisson system based on a backward-Euler (BE) approximation in time combined with a mixed finite element method for a discretization of the Poisson equation in the spatial domain and a discontinuous Galerkin (DG) finite element approximation in the phase-space variables for the Vlasov equation. We prove the stability estimates and derive the optimal convergence rates depending upon the compatibility of the finite element meshes, used for the discretizations of the spatial variable in Poisson (mixed) and Vlasov (DG) equations, respectively. The error estimates for the Poisson equation are based on using Brezzi-Douglas-Marini (BDM) elements in L2 and H?s, s>0, norms.

Thermomechanical finite element simulations of selective electron beam melting processes: performance considerations

The present contribution is concerned with the macroscopic modelling of the selective electron beam melting process by using the finite element method. The modelling and simulation of the selective electron beam melting process involves various challenges: complex material behaviour, phase changes, thermomechanical coupling, high temperature gradients, different time and length scales etc. The present contribution focuses on performance considerations of solution approaches for thermomechanically coupled problems, i.e. the monolithic and the adiabatic split approach. The material model is restricted to nonlinear thermoelasticity with temperature-dependent material parameters. As a numerical example a straight scanning path is simulated, the predicted temperatures and stresses are analysed and the performance of the two algorithms is compared. The adiabatic split approach turned out to be much more efficient for linear thermomechanical problems, i.e. the solution time is three times less than with the monolithic approach. For nonlinear problems, stability issues necessitated the use of the Euler backward integration scheme, and therefore, the adiabatic split approach required small time steps for reasonable accuracy. Thus, for nonlinear problems and in combination with the Euler backward integration scheme, the monolithic solver turned out to be more efficient.

Convergence and stability of a numerical method for nonlinear stochastic pantograph equations

Stochastic pantograph equations (SPEs) are a special case of stochastic delay differential equations (SDDEs) with unbounded memory. Here we examine analytical and numerical solutions of nonlinear SPEs. Some sufficient conditions for the mean-square (MS) stability of the analytical solutions are obtained. We consider the Euler–Maruyama (EM) method and develop a fundamental analysis concerning its strong convergence and MS stability. We show that the numerical solution produced by the EM method converges to the exact solution under the local Lipschitz condition. We find that the stability conditions for the EM method are somewhat stronger than those for the analytical solution. Therefore, we consider an implicit method, the backward Euler (BE) method. By constructing an Ft-adaptive continuous approximate solution, we obtain an analogous result of convergence under the local Lipschitz condition, that is, the BE solution converges to the exact solution. Furthermore, we prove that if the nonlinear SPEs are stable, then so is the BE method applied to the systems for any step size. Some numerical examples are presented to demonstrate the MS stability of the numerical methods.

Accelerating convergence for Backward-Euler method by adding time relaxation term

In this study, we introduce an algorithm to numerically solve any initial value problem (IVP). This algorithm depends on the Backward-Euler (BE) time discretization and time relaxation model (TRM). That is, a time relaxation term is added into IVP and discretized by the BE method. The algorithm increases the order of convergence from \(O(t)\) to \(O(t^2)\) for IVPs which admits \(c_0 + c_1.e^\alpha.x\) as exact solution where \(c_0\); \(c_1\) and constants. Hence more accurate results are obtained. To show the effectiveness, it is applied not only some stiff problems but also nonlinear and linear IVP. All results obtained by the algorithm are better than the results of the BE method.

Block-centered finite difference methods for parabolic equation with time-dependent coefficient

Two block-centered finite difference schemes are introduced and analyzed to solve parabolic equation with time-dependent diffusion coefficient. One scheme is Euler backward scheme with first order accuracy in time increment while the other is Crank–Nicolson scheme with second order accuracy in time increment. Second-order error estimates in spacial meshsize both for the original unknown and its derivatives in discrete L2 norms are established on non-uniform rectangular grid. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.

Modeling of induced anisotropy at large deformations for polymers

AbstractIn this work we develop a model to describe the induced plasticity of polymers at large deformations. Polymers such as stretch films exhibit a pronounced strength in the loading direction. The undeformed state of the films is isotropic, whereas after the uni‐axial loading the material becomes anisotropic. In order to consider this induced anisotropy during the stretch process, a spectral decomposition of the inelastic right CAUCHY‐GREEN tensor is done. Therefore, the yield function can be formulated as a function of the anisotropic tensor, where again the anisotropic tensor is a function of the maximum eigenvalue. A backward EULER scheme is used for updating the evolution equations, and the algorithmic tangent operator is derived. The numerical implementation of the resulting set of constitutive equations is used in a finite element program and for parameter identification. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)