Static condensation within the context of multifield elastoplasticity

In the present paper a multifield approach towards ideal elastoplasticity is presented and its numerical implementation is touched. Furthermore, the strategy of static condensation is applied to this ansatz. With the help of an axisymmetric model problem its realization is demonstrated and compared to the pure multifield approach. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Tikhonov regularization method in elastoplasticity

The numerical simulation of the mechanical behavior of industrial materials is widely used for viability verification, improvement and optimization of designs. Elastoplastic models have been used to forecast the mechanical behavior of different materials. The numerical solution of most elastoplastic models comes across problems of ill-condition matrices. A complete representation of the nonlinear behavior of such structures involves the nonlinear equilibrium path of the body and handling of singular (limit) points and/or bifurcation points. Several techniques to solve numerical problems associated to these points have been disposed in the specialized literature. Two examples are the load-controlled Newton–Raphson method and displacement controlled techniques. However, most of these methods fail due to convergence problems (ill-conditioning) in the neighborhood of limit points, specially when the structure presents snap-through or snap-back equilibrium paths. This study presents the main ideas and formalities of the Tikhonov regularization method and shows how this method can be used in the analysis of dynamic elastoplasticity problems. The study presents a rigorous mathematical demonstration of existence and uniqueness of the solution of well-posed dynamic elastoplasticity problems. The numerical solution of dynamic elastoplasticity problems using Tikhonov regularization is presented in this paper. The Galerkin method is used in this formulation. Effectiveness of Tikhonov’s approach in the regularization of the solution of elastoplasticity problems is demonstrated by means of some simple numerical examples.     

Finite strain inelasticity for isotropy,a simple and efficient finite element formulation

We consider a formulation of associative isotropic J₂‐elastoplasticity at finite inelastic strains and aspects of its numerical implementation. The essential ingredients include the multiplicative decomposition of the deformation gradient in elastic and inelastic parts, the definition of a convex elastic domain in stress space and a material representation of the constitutive equations for general non‐Cartesian coordinate charts. On the numerical side we propose a stress update algorithm for elasto‐plastic response, including isotropic hardening. The finite element formulation is based on assumed strain and enhanced strain variational principles, for a complete outline see [3]. Remarkably the formulation is very similar to the case of infinitesimal plasticity: (i) The scheme of linear return mapping algorithm takes the form of standard return mapping of the infinitesimal theory for the case of isotropic elastic response. (ii) The algorithmic elastoplastic moduli have a similar structure as in the linear case. Together with an exact fulfillment of plastic incompressibility by means of a simple correction one achieves an advantageously efficient finite element formulation. Its performance is documented by a numerical example.     

A semi-smooth NEWTON method for dynamic multifield plasticity

In the present paper a multifield approach towards isotropic dynamic small strain elastoplasticity is depicted. Its solution procedure is based on a semi-smooth NEWTON method coupled to a higher order accurate time integration scheme in the context of the finite element method. The appearing interactions are partly analyzed by means of an axisymmetric example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity

Using the interpolating moving least-squares (IMLS) method to obtain the shape function, we present a novel interpolating element-free Galerkin (IEFG) method to solve two-dimensional elastoplasticity problems. The shape function of the IMLS method satisfies the property of Kronecker δ function, then in the meshless methods based on the IMLS method, the essential boundary conditions can applied directly. Based on the Galerkin weak form, we obtain the formulae of the IEFG method for solving two-dimensional elastoplasticity problems. The IEFG method has some advantages, such as simpler formulae and directly applying the essential boundary conditions, over the conventional element-free Galerkin (EFG) method. The results of three numerical examples show that the computational precision of the IEFG method is higher than that of the EFG method.     

Application and evaluation of hyper- and hypoelastic-based plasticity models in the FE simulation of metal forming processes

Metal forming processes are usually accompanied by large plastic strains and rotations of the material elements which emphasizes the need for reliable finite strain elastoplasticity models in corresponding FE simulations. In this work, two specific finite strain hyper- and hypoelastic-based plasticity models with combined nonlinear isotropic and kinematic hardening are presented and compared in numerical FE simulations. Although both models led to remarkably different results in a shear-dominated single element deformation test, the structural simulation of a standard deep drawing process delivered nearly congruent results which suggests that both models are equally well-suited for modeling metals in common forming processes. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison between a recursive method based on an inverse mechanical formulation and shape optimization for solving inverse form finding problems in isotropic elastoplasticity

In this paper we compare two methods, a recursive method based on an inverse mechanical formulation and a method based on a recursive shape optimization formulation, in order to solve inverse form finding problems in isotropic elastoplasticity. Both methods are succinctly presented and a numerical example is given. It was found that no difference could be found between the node coordinates on the undeformed configurations computed with both methods. However the convergence to the solution is faster with the recursive method based on an inverse mechanical formulation than with the method based on shape optimization. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of hydroforming processes

We consider an optimal control problem for sheet metal hydroforming. As a first step derivative free optimization algorithms are used to control the time dependent blank holder force and the fluid pressure, which are typical control variables. Our goal is to obtain a desired final configuration. We present numerical examples for 2D and 3D ABAQUS simulations for the hydroforming process of complexly curved sheet metals with bifurcated cross-sections. Since a single ABAQUS simulation takes a long time, optimization algorithms based on reduced models are under investigation. The reduced order model is based on a Galerkin solver for the elastoplasticity problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards variational constitutive updates for finite strain plasticity models based on non-associative evolution equations

In this contribution, first steps towards variational constitutive updates for finite strain plasticity theory based on non-associative evolution equations are presented. These schemes allow to compute the unknown state variables such as the plastic part of the deformation gradient, together with the deformation mapping, by means of a fully variational minimization principle. Therefore, standard optimization algorithms can be applied to the numerical implementation leading to a very robust and efficient numerical implementation. Particularly, for highly non-linear, singular or nearly ill-posed physical models like that corresponding to crystal plasticity showing a large number of possible active slip planes, this is a significant advantage compared to standard constitutive updates such as the by now classical return-mapping algorithm. While variational constitutive updates have been successfully derived for associative plasticity models, their extension to more complex constitutive laws, particularly to those featuring non-associative evolution equations, is highly challenging. In the present contribution, a certain class of non-associative finite strain plasticity models is discussed and recast into a variationally consistent format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the Sinc method to a dynamic elasto-plastic problem

This paper presents the application of Sinc bases to simulate numerically the dynamic behavior of a one-dimensional elastoplastic problem. The numerical methods that are traditionally employed to solve elastoplastic problems include finite difference, finite element and spectral methods. However, more recently, biorthogonal wavelet bases have been used to study the dynamic response of a uniaxial elasto-plastic rod [Giovanni F. Naldi, Karsten Urban, Paolo Venini, A wavelet-Galerkin method for elastoplasticity problems, Report 181, RWTH Aachen IGPM, and Math. Modelling and Scient. Computing, vol. 10, 2000]. In this paper the Sinc–Galerkin method is used to solve the straight elasto-plastic rod problem. Due to their exponential convergence rates and their need for a relatively fewer nodal points, Sinc based methods can significantly outperform traditional numerical methods [J. Lund, K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992]. However, the potential of Sinc-based methods for solving elastoplasticity problems has not yet been explored. The aim of this paper is to demonstrate the possible application of Sinc methods through the numerical investigation of the unsteady one dimensional elastic-plastic rod problem.     

Materialmodelling of shape memory alloys in range of large deformations

In this contribution we present a finite deformation material model for SMA which includes the effect of pseudoelasticity. The model's structure is similiar to a Frederick-Armstrong type hardening model for elastoplasticity. A special algorithm has been developed to incorporate the concept into a FE code. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Index 2 DAEs in Plasticity Theory

The equations of rate‐independent elastoplasticity form a differential‐algebraic equation (DAE) with discontinuities. For the numerical solution, implicit Runge‐Kutta methods are applied and combined with the return mapping strategy of computational plasticity. It turns out that the convergence order depends crucially on the switching point detection. Further, it is shown that algebraically stable Runge‐Kutta methods preserve the contractivity of the elastoplastic flow.     

An efficient time-step-based self-adaptive algorithm for predictor-corrector methods of Runge-Kutta type

Finding an efficient implementation variant for the numerical solution of problems from computational science and engineering involves many implementation decisions that are strongly influenced by the specific hardware architecture. The complexity of these architectures makes it difficult to find the best implementation variant by manual tuning. For numerical solution methods from linear algebra, auto-tuning techniques based on a global search engine as they are used for ATLAS or FFTW can be used successfully. These techniques generate different implementation variants at installation time and select one of these implementation variants either at installation time or at runtime, before the computation starts. For some numerical methods, auto-tuning at installation time cannot be applied directly, since the best implementation variant may strongly depend on the specific numerical problem to be solved. An example is solution methods for initial value problems (IVPs) of ordinary differential equations (ODEs), where the coupling structure of the ODE system to be solved has a large influence on the efficient use of the memory hierarchy of the hardware architecture. In this context, it is important to use auto-tuning techniques at runtime, which is possible because of the time-stepping nature of ODE solvers.In this article, we present a sequential self-adaptive ODE solver that selects the best implementation variant from a candidate pool at runtime during the first time steps, i.e., the auto-tuning phase already contributes to the progress of the computation. The implementation variants differ in the loop structure and the data structures used to realize the numerical algorithm, a predictor-corrector (PC) iteration scheme with Runge-Kutta (RK) corrector considered here as an example. For those implementation variants in the candidate pool that use loop tiling to exploit the memory hierarchy of a given hardware platform we investigate the selection of tile sizes. The self-adaptive ODE solver combines empirical search with a model-based approach in order to reduce the search space of possible tile sizes. Runtime experiments demonstrate the efficiency of the self-adaptive solver for different IVPs across a range of problem sizes and on different hardware architectures.     

An efficient time-step-based self-adaptive algorithm for predictor–corrector methods of Runge–Kutta type

Finding an efficient implementation variant for the numerical solution of problems from computational science and engineering involves many implementation decisions that are strongly influenced by the specific hardware architecture. The complexity of these architectures makes it difficult to find the best implementation variant by manual tuning. For numerical solution methods from linear algebra, auto-tuning techniques based on a global search engine as they are used for ATLAS or FFTW can be used successfully. These techniques generate different implementation variants at installation time and select one of these implementation variants either at installation time or at runtime, before the computation starts. For some numerical methods, auto-tuning at installation time cannot be applied directly, since the best implementation variant may strongly depend on the specific numerical problem to be solved. An example is solution methods for initial value problems (IVPs) of ordinary differential equations (ODEs), where the coupling structure of the ODE system to be solved has a large influence on the efficient use of the memory hierarchy of the hardware architecture. In this context, it is important to use auto-tuning techniques at runtime, which is possible because of the time-stepping nature of ODE solvers.In this article, we present a sequential self-adaptive ODE solver that selects the best implementation variant from a candidate pool at runtime during the first time steps, i.e., the auto-tuning phase already contributes to the progress of the computation. The implementation variants differ in the loop structure and the data structures used to realize the numerical algorithm, a predictor–corrector (PC) iteration scheme with Runge–Kutta (RK) corrector considered here as an example. For those implementation variants in the candidate pool that use loop tiling to exploit the memory hierarchy of a given hardware platform we investigate the selection of tile sizes. The self-adaptive ODE solver combines empirical search with a model-based approach in order to reduce the search space of possible tile sizes. Runtime experiments demonstrate the efficiency of the self-adaptive solver for different IVPs across a range of problem sizes and on different hardware architectures.     

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

The numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient-extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot-type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi-field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A general FE computer program for 3D incremental analysis of frictional contact problems of elastoplasticity

The paper presents an original computer program PLAST for incremental analysis of 3D geometrically (unknown geometry of contact) and physically (frictional) non-linear contact problems of elasticity (thermoelasticity) or elastoplasticity (thermoelastoplasticity ) with or without strain hardening. Linearized contact problems of elastoplasticity and elasticity can also be analyzed. The program is based on the general algorithm combining different procedures for Coulomb and elasto-Coulombian friction models with or without hardening. All the friction terms for both the models are consistently defined as global (on the level of structure nodes). The program is assigned for large moving or stationary technological structures consisting of two bodies in contact with or without manufacturing or assemblage deviations. The capabilities of the program include strain and stress, stress concentration and contact analyses. The program can cooperate with commercial pre- and postprocessors. Verification of the program is based on calculations of test examples that can be easily checked analytically. Usefulness of the program for technological problems is supported with real turbomachinery blade attachment calculations.     

An Iterative method for nonlinear variational inequalities arising in elastoplasticity ∗

In this paper, we study an iterative method, which includes the Ka?anov method as a particular case, for solving nonlinear variational inequalities of the second kind. A convergence result is also proved under suitable assumptions. We apply our iterative method to solve an elastoplasticity problem.     

A Three-dimensional Continuum Theory of Dislocation Plasticity - Modelling and Application to a Composite

The increasing demand for materials with well defined microstructure, which is accompanied by the advancing miniaturization of devices calls for physically motivated, dislocation-based continuum theories of plasticity. Only recently rigorous techniques have been developed for performing meaningful averages over systems of moving, curved dislocations, yielding evolution equations for a higher order dislocation density tensor. Our continuum dislocation theory allows for generalizing the planar system towards a three-dimensional system, where dislocations may have arbitrary orientation and curvature. With the inclusion of curvature, the theory naturally takes into account a deformation-induced increase in the overall dislocation density without having to invoke ad-hoc assumptions about dislocation sources. A numerical implementation and some benchmark tests of this continuum theory for dislocation dynamics has already been discussed in the literature. In this paper, we apply this continuum theory to composite materials, where we analyze a plastically deforming matrix with an elastic inclusion. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)