Implicit numerical integration of highly nonlinear plasticity models

The issue of appropriate time discretization methods for the unsteady incompressible Navier-Stokes equations is considered from a practical perspective. Conventional implicit time-stepping algorithms are not feasible for long-time simulations since they inherit the quadratic nonlinearity of the steady-state equations. As a result, two new linearized versions of the 'pure' algorithms are analyzed herein. These have similar stability properties and comparable accuracy to the underlying nonlinear methods.

FE simulation of asymmetric creep-ageing behaviour of AA2050 and its application to creep age forming

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.

This paper reports on further results concerning nonlinear equations of the form F(x) + Ax = B, in which F(·) is a “diagonal nonlinear mapping” of real Euclidean n-space En into itself, A is a real n × n matrix, and B is an element of En. Such equations play a central role in the dc analysis of transistor networks, the computation of the transient response of transistor networks, and the numerical solution of certain nonlinear partial-differential equations. Here a nonuniqueness result, which focuses attention on a simple special property of transistor-type nonlinearities, is proved; this result shows that under certain conditions the equation F(x) + Ax = B has at least two solutions for some B ∊ En. The result proves that some earlier conditions for the existence of a unique solution cannot be improved by taking into account more information concerning the nonlinearities, and therefore makes more clear that the set of matrices denoted in earlier work by P 0 plays a very basic role in the theory of nonlinear transistor networks. In addition, some material concerned with the convergence of algorithms for computing the solution of the equation F(x) + Ax = B is presented, and some theorems are proved which provide more of a theoretical basis for the efficient computation of the transient response of transistor networks. In particular, the following proposition is proved. If the dc equations of a certain general type of transistor network possess at most one solution for all B ∊ En for “the original set of α's as well as for an arbitrary set of not-larger α's”, then the nonlinear equations encountered at each time step in the use of certain implicit numerical integration algorithms possess a unique solution for all values of the step size, and hence then for all step-size values it is possible to carry out the algorithms.

Efficient implicit algorithm for the equations of 2-D viscous compressible flow: application to shock-boundary layer interaction

This paper presents an efficient mathematical algorithm for a class of non‐linear kinematic hardening models with multiple back stresses, as an extension of the implicit integration algorithm for a single back stress hardening model. Explicit formulations for general three‐dimensional stress states as well as plane stress and plane strain are given. The new formulation is implemented in a general‐purpose finite element code, ABAQUS, and is verified by comparison with the existing formulation for the single back‐stress constitutive model. Comparison is also made with the experimental results obtained from a plate containing a circular hole subjected to cyclic loading, demonstrating the validity of new method. Copyright © 2001 John Wiley & Sons, Ltd.

A visco-plastic constitutive model for ratcheting of cyclically stable materials and its finite element implementation

An implicit time integration algorithm for a non-local, state-based, peridynamics plasticity model is developed. The flow rule was proposed in [3] without an integration strategy or yield criterion. This report addresses both of these issues and thus establishes the first ordinary, state-based peridynamics plasticity model. Integration of the flow rule follows along the lines of the classical theories of rate independent J{sub 2} plasticity. It uses elastic force state relations, an additive decomposition of the deformation state, an elastic force state domain, a flow rule, loading/un-loading conditions, and a consistency condition. Just as in local theories of plasticity (LTP), state variables are required. It is shown that the resulting constitutive model does not violate the 2nd law of thermodynamics. The report also develops a useful non-local yield criterion that depends upon the yield stress and horizon for the material. The modulus state for both the ordinary elastic material and aforementioned plasticity model is also developed and presented.

Modelling of plastic anisotropy requires the definition of stress potentials (coinciding with the yield criteria in case of the associated flow rules) or, alternatively, plastic strain‐rate potentials. The latter approach has several advantages whenever material parameters are determined by means of texture measurements and crystal plasticity simulations. This paper deals with a phenomenological description of anisotropy in elastoplastic rate‐insensitive models by using strain‐rate potentials. A fully implicit time integration algorithm is developed in this framework and implemented in a static‐implicit finite element code. Algorithmic details are discussed, including the derivation of the consistent (algorithmic) tangent modulus and the numerical treatment of the yield condition. Typical sheet forming applications are simulated with the proposed implementation, using the recent non‐quadratic strain‐rate potential Srp2004‐18p. Numerical simulations are carried out for materials that exhibit strong plastic anisotropy. The numerical results confirm that the presented algorithm exhibits the same generality, robustness, accuracy, and time efficiency as state‐of‐the‐art yield criterion‐based algorithms. Copyright © 2009 John Wiley & Sons, Ltd.

Risers are critical structures for the offshore oil and gas industry connecting floating production platforms to seabed equipment for production, injection and export functions, often through catenary configurations. The effect of external flow induced vibrations (VIV) and the occurrence of buckling are critical factors to lifespan of these structures. Therefore, the consistent evaluation of these factors is a strategic issue. Software for riser structural analysis usually employs Morison’s equation [1] to evaluate hydrodynamic forces along riser structure. Although this methodology is well established, their results are potentially conservative due to simplifications, besides not including lift forces. The present work employs an alternative methodology to calculate hydrodynamic loading on risers, based on the discrete vortex method (DVM) [2]. The DVM uses the Lagrangian approach in the vortex modeling, for incompressible, two-dimensional flows with regions of high vorticity and with dominant convective effect over the viscous one. The method creates and moves vortices along the riser wall perimeter, updates wake vortices at every time step considering the Biot-Savart law and calculates circulation by imposing the zero normal velocity condition on the riser wall. The structural analysis software, based on finite element method (FEM), Anflex [3], drives the DVM algorithm. Anflex applies an implicit time integration algorithm based on Newmark method together with three-dimensional nonlinear Euler beam elements with large displacements. This coupling occurs by modeling the flow in two-dimensional domains, called DVM planes, associated with each structural finite element along riser structure. The flow through DVM planes is responsible for the hydrodynamic forces on the structure, which in turn interferes with the fluid flow by structure displacements, performing a two-way coupling process [4]. This work presents experimental model results of a free catenary riser subjected to top-end displacements in still water, and compares them to numerical results obtained by Anflex/DVM. The experimental riser model is 41.45 m long under a 14.5 m water depth. Two-dimensional AMTI load cells [5] measured the catenary top force values and MCS Qualisys camera system [6] captured displacements on 40 points along the riser model. LabOceano hydrodynamic tank [7] performed the tests. The results comprised out-of-plane catenary vibration, catenary top force, and deformed configuration with global buckling at touch down point region (TDP). The experimental results showed significant self-induced vibrations due to the vertical movements applied at top connection, which indicated that the global buckling at TDP is highly influenced by these vibrations. Numerical analysis using Anflex/DVM showed good agreement with the experimental results. Anflex/DVM satisfactorily captured TDP buckling, which did not occur for the model based on Morison’s equation. These results indicated that the DVM-based method leads to more realistic dynamic responses, when compared to Morison’s equation. This paper defines global buckling as the dynamic wavy configuration that takes place at the TDP region of the riser under compression loading due to the vertical movement imposed at the riser top.

For reliable stability analysis of geotechnical structures, accurate calibration of constitutive model parameters is essential. These parameters govern the simulated material response under load, and any uncertainty can lead to significant deviations in predicted behavior. Conventional parameter determination relies on experimental testing of soil samples, whereby stress–strain measurements are used to infer individual model parameters according to their constitutive definitions. However, this manual fitting process is complex and time-consuming, particularly for soils, which exhibit highly nonlinear behavior. To address these challenges, we propose an automated parameter identification procedure for constitutive models. The methodology is developed for the Modified Cam-Clay model (MCC), a widely adopted framework for simulating the mechanical response of soft clays and normally consolidated soils. The first part of this paper presents the theoretical foundations and the implicit stress integration algorithm for the MCC model, implemented within the PAK finite element software. For automated identification, the same integration scheme is translated into Python and applied at the level of single integration point. This point-wise approach is justified for homogeneous stress states, such as those encountered in standard laboratory tests (e.g., oedometer and triaxial tests). In the second part of the paper, we describe the parameter identification program, which interfaces with the Python integration routine to perform optimization against experimental data. Finally, the developed identification algorithm is verified through a comparison of parameter estimates obtained from the PAK-based finite element implementation and those produced by the automated procedure.

Stress integration schemes for novel homogeneous anisotropic hardening model

Construction of minimum energy high-order Helmholtz bases for structured elements

Piezoelectric modelling using a time domain finite element program

Undrained dynamic behaviour of peaty organic soil under long-term cyclic loading, Part II: Constitutive model and simulation