Forward Euler Scheme Research Articles

Stress-update algorithms for Bigoni-Piccolroaz yield criterion coupled with a generalized function of kinematic hardening laws

Abstract A broad-ranging plasticity model is constructed by combining Bigoni-Piccolroaz yield function with a sweeping expression of kinematic hardening. Subsequently, two explicit and implicit update algorithms are developed to compute the stress from the rate equations of the plasticity in an incremental elastoplastic finite element analysis. The integrations are based on Forward and Backward Euler schemes. Considering a number of varying materials and the associated plasticity models, the proposed formulations are examined by means of a comprehensive set of numerical investigations. These include investigations to verify the suggested strategies along with two numerical simulations carried out by implementing the algorithms in a nonlinear finite element code. The versatility of the formulae allows for an expansive choice of elastoplastic constitutive models with a simple adjustment of a few parameters.

Numerical convergence in simulations of multiaxial ratcheting with directional distortional hardening

In this work, we investigate the numerical convergence of a set of plasticity models with different kinematic and directional distortional hardening rules under cyclic plastic loading. In particular, we revisit the results presented in Feigenbaum et al. (2012) in order to more robustly check for convergence during the numerical integration procedure, and show that the results presented in the previous work do not converge. We investigate the role of the step-size and numerical scheme on the convergence of these models when predicting ratcheting. By reducing step-sizes and using a forward Euler scheme during numerical integration, converged solutions are obtained. The new converged results lead to new conclusions. Results still suggest that directional distortional hardening can improve ratcheting predictions, however the addition of directional distortional hardening yields less improvements compared to kinematic hardening alone than previously thought. This new conclusion, strongly suggests the need for additional modeling developments in order accurately predict ratcheting strains under a wide variety of cyclic plastic loadings.

Numerical implementation of the microplane constitutive model for shape memory alloys

With the advent of shape memory alloys, several industrial applications were proposed due to their superior mechanical and biological properties. Since the fabrication and characterization of shape memory alloy devices is challenging and expensive, it is necessary to simulate their thermomechanical responses before fabrication. To do so, a powerful constitutive model capable of simulation of the important features of these materials is necessary. To be able to simulate a shape memory alloy device, it is vital to implement a suitable constitutive model in such a way to be used in finite element models. In this paper, an existing constitutive model based on microplane theory is numerically implemented and the effects of stress increment, different numerical integration formulas, and loading direction on the thermomechanical response of shape memory alloy is investigated through superelastic and shape memory proportional and nonproportional loadings. The obtained results show that the stress increment may have significant effect on the results if the forward Euler scheme is utilized. In addition, for the case of numerical integration over the surface of a unit hemisphere, 61 points integration formula without orthogonal symmetry provides the best results while 21 orthogonally symmetric one is the most inaccurate one. Also, the orthogonally symmetric numerical integration formulas predict the isotropic material response while those without orthogonal symmetry predict a little anisotropy.

Elasto-plastic Constitutive Model for Overconsolidated Clays

Constitutive model for saturated cohesive soils based on the bounding surface plasticity notion with anisotropic hardening law is presented in the paper. The model predicts inelastic behaviour of overconsolidated cohesive soils. The projection centre is the only point in the stress space which represents elastic soil behaviour. Approximation of the plastic modulus within the preconsolidation domain is made using the radial mapping rule between a projection centre and a reflecting point on the bounding surface. The projection centre changes its position each time when stress path turns rapidly of more than 90°. The configuration of the elliptic bounding surface is governed by preconsolidation effective pressure p′c which depends on change of plastic both volumetric and deviatoric strain. Associated flow rule has been assumed in the formulation. Integration of constitutive relations is done according to forward Euler scheme with error control proposed by Sloan. The effectiveness of the proposed model is illustrated in both monotonic and cyclic loading in the homogeneous triaxial drained and undrained conditions.

Discretization of an eco-epidemiological model and its dynamic consistency

In this paper, we discretize a continuous-time eco-epidemiological model by non-standard finite difference (NSFD) scheme as well as standard Euler forward scheme. Dynamical properties of both the systems are explored and compared with their continuous-time model. We show that the solution of NSFD system remains positive for all positive initial values. Fixed points and their local stability properties are shown to be identical with that of the continuous model, indicating its dynamic consistency. Dynamics of the Euler model, however, depend on the step–size and therefore dynamically inconsistent. Solutions in this latter method may be negative and allows numerical instabilities, leading to chaos. Extensive numerical simulations have been performed to validate the theoretical results.

An isogeometrical approach to structural level set topology optimization

This paper aims to utilize the Isogeometric Analysis (IGA) for the level set structural topology optimization. The level set function is parametrized using the Non-Uniform Rational B-Spline (NURBS) basis functions in a higher dimension. The same basis functions are employed for approximating the unknown deformations, geometry modeling of the design domain and the level set function. In this research, three optimization problems including minimization of the mean compliance considering a certain amount of material, minimization of weight with avoiding local stress concentration as well as minimization of weight and strain energy under local stress constraints are dealt with. The sensitivity analyses for the optimization problems are carried out to obtain velocity functions on the boundaries. In order to move the boundaries towards optimum the Hamilton–Jacobi (H–J) equation is solved using the forward Euler scheme. In order to illustrate the performance of the method to obtain reasonable results with smooth boundaries, several numerical examples are presented and compared with well-known problems in literature.

BIFURCATION AND CHAOS IN A DISCRETE TIME PREDATOR-PREY SYSTEM OF LESLIE TYPE WITH GENERALIZED HOLLING TYPE Ⅲ FUNCTIONAL RESPONSE

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type Ⅲ functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R₊². Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

Runge–Kutta-like scaling techniques for first-order methods in convex optimization

It is well known that there is a strong connection between time integration and convex optimization. In this work, inspired by the equivalence between the forward Euler scheme and the gradient descent method, we broaden our analysis to the family of Runge–Kutta methods and show that they enjoy a natural interpretation as first-order optimization algorithms. The strategies intrinsically suggested by Runge–Kutta methods are exploited in order to detail novel proposal for either scaling or preconditioning gradient-like approaches, whose convergence is ensured by the stability condition for Runge–Kutta schemes. The theoretical analysis is supported by the numerical experiments carried out on some test problems arising from suitable applications where the proposed techniques can be efficiently employed.

Numerical Blow-Up of Nonlinear Parabolic Integro-Differential Equations on Unbounded Domain

The efficient numerical methods of the nonlinear parabolic integro-differential PDEs on unbounded spatial domains whose solutions blow up in finite time are considered. Based on the unified approach proposed in Zhang et al. (Phys Rev E 78:026709, 2008), Zhang et al. (Phys Rev E 79:046711, 2009), the nonlinear absorbing boundary conditions for one-dimensional and two-dimensional nonlinear parabolic integro-differential PDEs are derived. Thus the original problem on the unbounded spatial domain is reduced to an initial-boundary-value (IBV) problem on a bounded computational domain. Secondly, a simple but efficient adaptive time-stepping scheme for the reduced IBV problem is achieved by using the fixed point method to make the finite difference approximation stable at each time level. At each time level, we also prove that the lower bound and upper bound of the blow-up time can be bounded by the numerical blow-up times of the forward and backward Euler schemes. Finally, the theoretical results are illustrated by a broad range of numerical examples, including a problem with a circle line blow-up.

Microstructure based modeling of ductile fracture initiation in press-hardened sheet metal structures

The manufacturing of sheet metal components with spatially varying microstucture composition and mechanical properties using press-hardening technology is now an established practice in the automotive industry. To estimate the performance envelopes of such components, a multi-scale approach to ductile fracture prediction based on mean-field homogenization is proposed. Two non-interacting fracture criteria are formulated in terms of the local average stress field, referring to inter-phase and intra-phase fracture mechanisms. The overall ductility is governed by the weakest constituent or interface present in the multiphase material. Moreover, instabilities related to the strain localization problem at the macroscale are treated by embedding discontinuities in the element formulation. These are triggered by a localization criterion derived via bifurcation analysis of the homogenized material. Issues concerning numerical implementation include a forward Euler scheme for integrating the mean-field equations, suitable for explicit finite element analysis of heterogeneous materials. Tensile specimens with ten distinctly different microstructure compositions are evaluated, for which useful predictions of the overall force–displacement response and fracture elongations are demonstrated.

A New Mixed Formulation and Efficient Numerical Solution of Ginzburg--Landau Equations Under the Temporal Gauge

In this paper, we present a new numerical approach to the time-dependent Ginzburg--Landau (GL) equations under the temporal gauge (zero electric potential gauge). The approach is based on a mixed formulation of the GL equations, which consists of two parabolic equations for the order parameter $\psi$ and the magnetic field $\sigma = \mathrm{curl} \, \mathbf{A}$, respectively, and a vector ordinary differential equation for the magnetic potential $\mathbf{A}$. A fully linearized Galerkin finite element method is presented for solving the mixed GL system. The new approach offers many advantages on both accuracy and efficiency over existing methods. In particular, the equations for $\psi$ and $\sigma$ are uniformly parabolic and, therefore, the method provides optimal-order accuracy for the two physical components $\psi$ and $\sigma$. Since in the temporal direction, a fully linearized backward Euler scheme is used for $\psi$ and $\sigma$ and a forward Euler scheme is used for $\mathbf{A}$, respectively, the system is fully decoupled and at each time step, the three variables $\psi$, $\sigma$, and $\mathbf{A}$ can be solved simultaneously. Moreover, we present numerical comparisons with two commonly used Galerkin methods for the GL equations under the temporal gauge and the Lorentz gauge, respectively. Our numerical results show that the new approach requires fewer iterations for solving the linear systems arising at each time step and the computational cost for the vector ODE seems neglectable. Several numerical examples in both two- and three-dimensional spaces are investigated.

Asynchronous numerical scheme for modeling hyperbolic systems

We present an asynchronous method for the explicit integration of multi-scale partial differential equations. This method is restricted by a local CFL condition rather than the traditional global CFL condition. First, we developed an upwind asynchronous forward Euler scheme for the transport equation and we proved that the asynchronous scheme is first order convergent. To improve the convergence rate of the asynchronous scheme, we derived an asynchronous Runge–Kutta 2 scheme from a standard explicit Runge–Kutta method.

Stability and convergence of time-stepping methods for a nonlocal model for diffusion

A time-dependent nonlocal model for diffusion is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The explicit forward-Euler, implicit backward-Euler, and Crank-Nicolson methods are considered for discretizing the time derivative and piecewise-linear finite element methods are used for spatial discretization. The unconditional stability of the backward-Euler and Crank-Nicolson schemes and the conditional stability of the forward-Euler scheme are proved as are optimal error estimates for all three schemes. Comparisons with the analogous results for classical local diffusion problems, e.g., the heat equation, are provided as are the results of numerical experiments that illustrate the theoretical results.

Integral Step Size Makes a Difference to Bifurcations of a Discrete-Time Hindmarsh–Rose Model

A three-dimensional discrete-time Hindmarsh–Rose model obtained by the forward Euler scheme is investigated in this paper. When the integral step size is chosen as a bifurcation parameter, conditions of existence for the fold bifurcation, the flip bifurcation, and the Hopf bifurcation are derived by using the center manifold theorem, bifurcation theory and a criterion of Hopf bifurcation. Numerical simulations including time series, bifurcation diagrams, Lyapunov exponents, phase portraits show the consistence with the analytical analysis. Our research results demonstrate that the integral step size makes a difference corresponding to local and global bifurcations of the three-dimensional discrete-time Hindmarsh–Rose model. These results can supply a solid analytical basis to the study of Hindmarsh–Rose model, and it is necessary to illustrate how much the integral step size is adopted in advance when numerical solutions or approximate solutions of the original continuous-time model is concerned.

Bifurcation Analysis and Chaos Control in a Discrete Epidemic System

The dynamics of discreteSIepidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interiorR+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the systemx→x+δ[rN(1-N/K)-βxy/N-(μ+m)x],y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.

Numerical analysis of an adsorption dynamic model at the air–water interface

In this paper we deal with the numerical analysis of an adsorption dynamic model arising in a surfactant solution at the air–water interface; the diffusion model is considered together with the so-called Langmuir isotherm. An existence and uniqueness result is stated. Then, fully discrete approximations are introduced by using a finite element method and a hybrid combination of backward and forward Euler schemes. Error estimates are proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is derived assuming a dependence between both spatial and time discretization parameters. Finally, some numerical simulations are presented in order to demonstrate the accuracy of the algorithm and the behaviour of the solution for two commercially available surfactants.

Bifurcation and chaos in a discrete-time predator–prey system of Holling and Leslie type

A discrete-time predator–prey system of Holling and Leslie type with a constant-yield prey harvesting obtained by the forward Euler scheme is studied in detail. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, phase portraits display new and rich nonlinear dynamical behaviors. More specifically, when the integral step size is chosen as a bifurcation parameter, this paper presents the finding of period-1,2,11,17,19,22 orbits, attracting invariant cycles, and chaotic attractors of the discrete-time predator–prey system of Holling and Leslie type with a constant-yield prey harvesting. These results demonstrate that the integral step size plays a vital role to the local and global stability of the discrete-time predator–prey system with the Holling and Leslie type after the original continuous-time predator–prey system is discretized.

A robust three-dimensional phenomenological model for polycrystalline SMAs: Analytical closed-form solutions

This paper presents a robust three-dimensional phenomenological model and analytical closed-form solutions to simulate self-accommodation, martensitic transformation and orientation/reorientation of martensite in polycrystalline shape memory alloys (SMAs). The model is developed within the classical framework of thermo-dynamics of irreversible processes and utilizes the volume fractions of self-accommodated and oriented martensite as scalar internal variables and the preferred direction of oriented martensite variants as tensorial internal variable. Linear and exponential interpolation functions are introduced which respectively result in coarse and smooth transitions in stress-induced martensitic transformation. A unified constitutive model is presented for both stress and strain control modes that has the property of completely decoupling the reorientation mechanism from the martensitic transformation mechanism. The time-discrete counterpart of the unified constitutive model is introduced, integrating the evolution equation of martensite reorientation using both implicit backward Euler and explicit forward Euler schemes. Analytical closed-form solutions are derived for the preferred direction of oriented martensite variants and the volume fractions of self-accommodated and oriented martensite. In order to examine capabilities of the developed SMA model as well as the proposed closed-form solutions, two boundary value problems are solved including a thin NiTi wire under combined tension–torsion non-proportional loadings and a thin-walled NiTi tube subjected to combined internal pressure-tension/compression/torsion-heating paths. In the first problem, the model predictions are compared with the experimental data that shows good correlations. Due to simplicity and accuracy, the model can be used as an efficient and analytic computational tool to analyze structures made of SMAs under multi-axial non-proportional loading histories.

Structure-preserving properties of three differential schemes for oscillator system

A numerical method for the Hamiltonian system is required to preserve some structure-preserving properties. The current structure-preserving method satisfies the requirements that a symplectic method can preserve the symplectic structure of a finite dimension Hamiltonian system, and a multi-symplectic method can preserve the multi-symplectic structure of an infinite dimension Hamiltonian system. In this paper, the structure-preserving properties of three differential schemes for an oscillator system are investigated in detail. Both the theoretical results and the numerical results show that the results obtained by the standard forward Euler scheme lost all the three geometric properties of the oscillator system, i.e., periodicity, boundedness, and total energy, the symplectic scheme can preserve the first two geometric properties of the oscillator system, and the Stormer-Verlet scheme can preserve the three geometric properties of the oscillator system well. In addition, the relative errors for the Hamiltonian function of the symplectic scheme increase with the increase in the step length, suggesting that the symplectic scheme possesses good structure-preserving properties only if the step length is small enough.

Preservation of Takens–Bogdanov Bifurcations for Delay Differential Equations by Euler Discretization

In this paper we study the discretization effects on the Takens–Bogdanov bifurcation of delay differential equations by forward Euler scheme. We show that the Takens–Bogdanov point is inherited by the discretization without any shift and turns into a 1:1 resonance point. The normal form on the center manifold near this singular point of the forward Euler method is calculated by applying a new technique, which is developed in this work for a general class of parameterized maps. The local dynamical behaviors are investigated in detail through this normal form. We show that the bifurcated Hopf point branch and the homoclinic branch of the numerical method are $$O(h)$$ close to their continuous counterparts, where $$h$$ is stepsize. A numerical experiment is presented to illustrate the theoretical results.