Implicit Finite Element Formulation Research Articles

An implicit monolithic AFC stabilization method for the CG finite element discretization of the fully-ionized ideal multifluid electromagnetic plasma system

This study considers an implicit finite element formulation for an ideal fully-ionized multifluid electromagnetic plasma system. The formulation is based on fully-implicit Runge-Kutta time discretizations and a monolithic discrete algebraic flux corrected (AFC) continuous Galerkin (CG) spatial discretization of the coupled system. The AFC approach adds scalar artificial diffusion to the high-order, semi-discrete Galerkin method and uses mass lumping in the time derivative term. The result is a low-order method that attempts to enforce local-extremum-diminishing properties for the hyperbolic system. An element-based iterative limiter is applied to reduce the amount of artificial diffusion that is used in regions where the solution is smooth and the additional stabilization is not required. Two models are considered for the electromagnetics portion of the system: an electrostatic model, and a full Maxwell system with a parabolic divergence cleaning approach that enforces the required involutions on the electric and magnetic fields. Results are presented that demonstrate the accuracy and robustness of the formulation for smooth and discontinuous solutions to challenging plasma physics problems. This includes a demonstration that the solution of the full multifluid system yields the expected behavior in the ideal shock-MHD limit.

Enhanced finite element formulation incorporating stress jumps and interface behavior in structures with embedded linear inclusions

AbstractWhen linear inclusions are embedded into a matrix, a perturbation is introduced in the boundary value problem written at the macroscale. Such inhomogeneity can provoke sharp gradients in the solution and a correct evaluation of the interface stress state reveals necessary when localized material degradations can lead to global failure mechanisms. In order to account for such refined description in large‐scale structural problems, a novel enhanced implicit finite element formulation is presented. The standard displacement approximation is improved by means of a kinematic enrichment aimed at reproducing the local interaction between the matrix and the inclusion. The microscale interface behavior can then be directly evaluated from the stress jump arising in the bulk. From a computational point of view, this translates in solving an additional local equilibrium equation. The static boundary conditions at the inclusion ends are thus exactly verified and a good precision is achieved with minimal meshing effort and code modifications. The proposed model is applied to elementary and structural examples involving singularities and it is compared to other modeling strategies with focus both on global and local quantities as well on the convergence properties and corresponding CPU times of the approximated solution.

A staggered explicit–implicit finite element formulation for electroactive polymers

Electroactive polymers such as dielectric elastomers (DEs) have attracted significant attention in recent years. Computational techniques to solve the coupled electromechanical system of equations for this class of materials have universally centered around fully coupled monolithic formulations, which while generating good accuracy requires significant computational expense. However, this has significantly hindered the ability to solve large scale, fully three-dimensional problems involving complex deformations and electromechanical instabilities of DEs. In this work, we provide theoretical basis for the effectiveness and accuracy of staggered explicit–implicit finite element formulations for this class of electromechanically coupled materials, and elicit the simplicity of the resulting staggered formulation. We demonstrate the stability and accuracy of the staggered approach by solving complex electromechanically coupled problems involving electroactive polymers, where we focus on problems involving electromechanical instabilities such as creasing, wrinkling, and bursting drops. In all examples, essentially identical results to the fully monolithic solution are obtained, showing the accuracy of the staggered approach at a significantly reduced computational cost.

Modelling multiple cycles of static and dynamic recrystallisation using a fully implicit isotropic material model based on dislocation density

This paper presents the development and numerical implementation of a state variable based thermomechanical material model, intended for use within a fully implicit finite element formulation. Plastic hardening, thermal recovery and multiple cycles of recrystallisation can be tracked for single peak as well as multiple peak recrystallisation response. The numerical implementation of the state variable model extends on a $$J_{2}$$ isotropic hypo-elastoplastic modelling framework. The complete numerical implementation is presented as an Abaqus UMAT and linked subroutines. Implementation is discussed with detailed explanation of the derivation and use of various sensitivities, internal state variable management and multiple recrystallisation cycle contributions. A flow chart explaining the proposed numerical implementation is provided as well as verification on the convergence of the material subroutine. The material model is characterised using two high temperature data sets for cobalt and copper. The results of finite element analyses using the material parameter values characterised on the copper data set are also presented.

Computational cardiology: A modified Hill model to describe the electro-visco-elasticity of the myocardium

This contribution presents a novel three-dimensional constitutive model which describes the orthotropic electro-visco-elastic response of the myocardium. The model can be regarded as the viscoelastic extension of the recently proposed generalized Hill model for orthotropic active muscle cells. The formulation extends the recent contributions of Cansız et al. (CMBBE 18: 1160–1172, 2015) and Göktepe et al. (JMPS 72: 20–39, 2014) in a novel rheological description which incorporates the active (electrical) and mechanical (viscous and elastic) deformations in a multiplicative format. To this end, the stress response is additively decomposed into passive (purely mechanical) and visco-active (electro-visco-elastic) contributions in line with a rheological model in which the passive part is connected parallel to a branch that consists of an elastic spring, a dashpot and a contractile element in serial. The former branch is assumed to be a function of the total deformation gradient while the formulation of the latter one is based on a multiplicative decomposition of the total deformation gradient into a mechanical and an active part. The active deformation gradient is devised by means of the prescribed active stretch which arises from the electrical excitation of the myocardial tissue and is governed by the intracellular calcium concentration. Thanks to the proposed rheology, marked differences are observed in isometric and isotonic tests between viscoelastic and elastic cases that are performed on material level. Moreover, novel evolution equations for the description of active stretch and intracellular calcium concentration having superiorities over the existing approaches have been proposed. On the numerical side, a fully implicit finite element formulation along with surface elements accounting for blood pressure evolution in the ventricles during successive phases of the cardiac cycle is presented. The argument of the constitutive equation describing the ventricular blood pressure is specified as the associated ventricular cavity volume which leads to the mutual interaction of the non-adjacent surface elements. The performance of the theory and algorithms are demonstrated by means of representative multi-field initial–boundary value problems. The results indicate that viscous effects significantly alter the electromechanical response of the cardiac tissue and is thought to be crucial in the virtual assessment of the cardiac function.

Implicit finite element formulation of multiresolution continuum theory

Abstract The multiresolution continuum theory is a higher order continuum theory where additional kinematic variables account for microstructural inhomogeneities at several distinct length scales. This can be particularly important for localization problems. The strength of this theory is that it can account for details in the microstructure of a material without using an extremely fine mesh. The present paper describes the implementation and verification of a 3D elastic–plastic multiresolution element based on an implicit time stepping algorithm. It is implemented in the general purpose finite element program FEAP. The mesh independency associated with the length scale parameter is examined and the convergence rate of the element is also evaluated.

Development and Validation of a Fluid–Structure Solver for Transonic Panel Flutter

A partitioned fluid–structure coupling code for transonic panel flutter has been developed and validated. The Reynolds-averaged Navier–Stokes equations are solved numerically by means of an implicit finite volume method to account for nonlinear aerodynamics, as there are shock waves and a viscous boundary layer at the panel surface. An implicit finite element formulation of the structural equations, as well as a Galerkin solution of the von Kármán plate equation are employed to solve elastic panel deformations with respect to geometric nonlinearities. A detailed validation process has shown good agreement with results from the literature for high subsonic and low supersonic Mach numbers. Thereby, a recent comparison between theory and experiment is confirmed. The validated solver is then used for further studies focusing on the impact of turbulent boundary layers on aeroelastic stability boundaries and instabilities. An increase in aerodynamic damping due to a viscous boundary layer is identified by an increase of the aeroelastic stability boundary. Furthermore, a significant damping on high flutter frequencies and mode shapes is revealed. The application of either a one-equation or two-equation turbulence model did not cause any major deviations in the results.

Recent advances in EdgeCFD on wave-structure interaction and turbulence modeling

The computation of free surface flows is challenging since waves are highly nonlinear and commonly present merging, frag mentation and cusps, leading to the use of interface capturing ALE approaches. Turbulence is also important on several offshore applications. In this work we report recent advances of EdgeCFD to simulate these problems. EdgeCFD is a fully implicit 3D incompressible stabilized parallel edge-based finite element flow solver associated to the Volume-of-Fluid (VoF) method. The VoF marking function is also solved by a fully implicit parallel edge-based finite element formulation. Turbulence in EdgeCFD is treated within the Large Eddy Simulation (LES) framework. The performance and accuracy of EdgeCFD is tested in the simulation waves and in the interaction between waves and a semisubmersible structure. We also access the performance of several LES models on the simulation of the flow around a cylinder at Reynolds 3,900.

The effect of size of Cu precipitation on the mechanical properties of microalloyed steel

The effect of size of nanoscale Cu precipitate on the mechanical response of microalloyed steel was investigated computationally and experimentally. A phenomenological constitutive description is adopted to build the computational crystal model. The material is envisaged as a composite; the Cu precipitate is modeled as a monocrystalline core surrounded with a lower yield stress and higher work hardening rate response. Both a quasi-isotropic and crystal plasticity approaches are used to simulate the matrix. The nanoscale precipitate is modeled as ellipsoidal inclusion with different Yound's modulus to matrix. Elastic and plastic anisotropy are incorporated into this simulation. An implicit Lagrangian finite element formulation with von Mises plasticity or rate dependent crystal plasticity is used to study the nonuniform deformation and localized plastic flow. The computational predictions are compared with the experimentally determined mechanical response of HSLA-100 steel with average size of nanoscale precipitates of 2.02±1.89nm. The tendency of the calculated yield strength attributed to Cu precipitates is in good agreement with experimental result.

An improved strain gradient plasticity formulation with energetic interfaces: theory and a fully implicit finite element formulation

A fully implicit backward-Euler implementation of a higher order strain gradient plasticity theory is presented. A tangent operator consistent with the numerical update procedure is given. The implemented theory is a dissipative bulk formulation with energetic contribution from internal interface to model the behavior of material interfaces at small length scales. The implementation is tested by solving some examples that specifically highlight the numerics and the effect of using the energetic interfaces as higher order boundary conditions. Specifically, it is demonstrated that the energetic interface formulation is able to mimic a wide range of plastic strain conditions at internal boundaries. It is also shown that delayed micro-hard conditions may arise under certain circumstances such that an interface at first offers little constraints on plastic flow, but with increasing plastic deformation will develop and become a barrier to dislocation motion.

Elastic-plastic constitutive model for accurate springback prediction in hot press sheet forming

The present study investigates the effect of phase transformation plasticity on the springback of hot press formed parts. Employing an implicit finite element formulation to take phase transformations during rapid cooling from fully austenitic phase and their related thermo-mechanical behavior into account, two validations — (1) the loaded dilatometry problem which induces significant plastic deformation even at lower stress levels than the classical yield stress, and (2) the 2-D draw bending problem, which introduces high temperature forming and subsequent quenching — are presented and compared with known experimental data. The study showed that the magnitude of springback predicted by considering the phase transformation plasticity and the temperature change induced volumetric strain agreed well with date from experiments, while the results determined with the conventional elastic-plasticity theory in which only plastic deformation by external load was used, had considerable deviation for the springback profile after hot press forming. The negligible springback amount after hot press forming, which has been frequently reported in many experimental observations, is due to the stress relaxation by the addition of abnormal transformation plasticity.

DYNAMICAL FINITE ELEMENT MODELING OF SOFT TISSUES AS CHEMOELECTRIC POROUS MEDIA

An implicit mixed finite element formulation of hydrated soft biological tissues, based on the Simon model, is presented that incorporates the coupling of solid, fluid, and ion phases as well as the viscoelasticity of soft tissue in the dynamical process. The tissues are modeled as a multi-field viscoelastic body subject to finite deformation. In addition to a three-field (u-w-p) modeling of the porous matrix, the study also includes an ion phase for the ionic solution. After presenting the formulation, an efficient staggered solution scheme is presented: within each time step, the ion charge equation is solved first to give the distribution of the charge concentration, the charge induced osmotic water pressure is then employed in solving the u-w-p equations. The resulting u field becomes a forcing term to the solution of the ion charge concentration equations for iteration. This methodology and codes developed for the study have been verified with one-dimensional (1D) analytical solutions. A 2D chemical electric swelling model illustrates the important role of viscoelasticity. A brain tissue impact example demonstrates the potential application of the model.

Computational Techniques for Stabilized Edge-Based Finite Element Simulation of Nonlinear Free-Surface Flows

Free-surface flows occur in several problems in hydrodynamics, such as fuel or water sloshing in tanks, waves breaking in ships, offshore platforms, harbors, and coastal areas. The computation of such highly nonlinear flows is challenging, since free-surfaces commonly present merging, fragmentation, and breaking parts, leading to the use of interface-capturing Eulerian approaches. In such methods the surface between two fluids is captured by the use of a marking function, which is transported in a flow field. In this work we discuss computational techniques for efficient implementation of 3D incompressible streamline-upwind/Petrov–Galerkin (SUPG)/pressure-stabilizing/Petrov–Galerkin finite element methods to cope with free-surface problems with the volume-of-fluid method (Elias, and Coutinho, 2007, “Stabilized Edge-Based Finite Element Simulation of Free-Surface Flows,” Int. J. Numer. Methods Fluids, 54, pp. 965–993). The pure advection equation for the scalar marking function was solved by a fully implicit parallel edge-based SUPG finite element formulation. Global mass conservation is enforced, adding or removing mass proportionally to the absolute value of the normal velocity of the interface. We introduce parallel edge-based data structures, a parallel dynamic deactivation algorithm to solve the marking function equation only in a small region around the interface. The implementation is targeted to distributed memory systems with cache-based processors. The performance and accuracy of the proposed solution method is tested in the simulation of the water impact on a square cylinder and in the propagation of a solitary wave.

Implicit finite element formulations for multi-phase transformation in high carbon steel

An anomalous plastic deformation observed during the phase transformation of steels was implemented into the finite element modeling. The constitutive equations for the transformation plasticity originally proposed by Greenwood and Johnson [Greenwood, G.W., Johnson, R.H., 1965. The deformation of metals under small stresses during phase transformation. Proc. Roy. Soc. A 283, 403] and further extended by Leblond et al. [Leblond, J.B., Mottet, G., Devaux, J.C., 1986a. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, I. Derivation of general relations. J. Mech. Phys. Solids 34, 395–409; Leblond, J.B., Mottet, G., Devaux, J.C., 1986b. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34, 411–432; Leblond, J.B., Devaux, J., Devaux, J.C., 1989a. Mathematical modeling of transformation plasticity in steels, I: case of ideal-plastic phases. Int. J. Plasticity 5, 511–572; Leblond, J.B., 1989b. Mathematical modeling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int. J. Plasticity 5, 573–591] were modified to consider the thermo-mechanical response of generalized multi-phase steel during phase transformations from austenite at high temperature. An implicit numerical solution procedure to calculate the plastic deformation of each constituent phase was newly proposed and implemented into the general purpose implicit finite element program via user material subroutine. The new algorithms include efficient calculation of consistent tangent modulus for the transformation plasticity and application of general anisotropic yield functions without limitation to the isotropic yield function. Besides the thermo-elastic–plastic constitutive equations, non-isothermal transformation kinetics was characterized by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation and additivity relationship for the diffusional transformation, while the model proposed by Koistinen and Marburger was used for the diffusionless transformation. Numerical verifications for the continuous cooling experiments under various loading conditions were conducted to demonstrate the applicability of the developed numerical algorithms to the high carbon steel SK5.

Influence of stress path on tunnel excavation response – Numerical tool selection and modeling strategy

The actual stress path in a rock mass during tunnel excavation is complex. To capture the correct tunnel excavation response, it is important to correctly resemble the stress path in situ in the numerical tools. FLAC and Phase 2 are two powerful two-dimensional continuum codes for modeling soil, rock, and structural behavior, in the fields of geotechnical, geomechanics and in civil and mining engineering. FLAC is based on explicit finite difference formulation while Phase 2 is based on implicit finite element formulation. When the two codes are applied to the analysis of tunnel excavation problems, difference in results might occur simply due to the different formulation methodologies used in these codes. It is shown that for linear elastic tunnel excavation problems, both codes provide the same result because stress path is unimportant. For tunnel excavation in elasto-plastic materials using long-round drill and blast method, there is significant difference in terms of yielding zone distribution by the two codes if conventional modeling approach is used, especially when the rock strength is low relative to the in situ stress magnitude. The mechanism of the difference is investigated and recommendation provided for choosing appropriate tools and modeling strategies for tunnel excavation problems. The importance of honoring the true stress path in tunnel excavation response simulation is illustrated using a few examples.

High-order time integration applied to metal powder plasticity

The present article treats two objectives. In the first investigation attention is focused on the application of time-adaptive finite elements formulated on the basis of a high-order time integration procedure on a constitutive model for compressible finite strain viscoplasticity for metal powder. In this connection, it has to be emphasized that the integration procedure is not only applied to the evolution equations on Gauss-point level but on the total system of differential–algebraic equations resulting from the application of the vertical line method on the quasi-static finite element equations. The specific application emerges from the field of metal powder compaction. Particular studies are carried out using stiffly accurate, diagonally implicit Runge–Kutta methods in combination with the Multilevel-Newton algorithm for solving the DAE-system. In this respect, the effort vs. accuracy behavior is investigated which is also related to order reduction known in elastoplasticity. The second topic treats the local stress algorithm for taking into account the yield function based finite strain viscoplasticity model, where the classical Newton–Raphson method fails. This is the reason why most constitutive models of powder materials are implemented into explicit finite element codes. Thus, the proposed investigations compare different methods in view of a stable and efficient integration process in implicit finite element formulations.

Finite element formulation for modeling particle debonding in reinforced elastomers subjected to finite deformations

Interfacial damage nucleation and evolution in reinforced elastomers is modeled using a three-dimensional updated Lagrangian finite element formulation based on the perturbed Petrov–Galerkin method for the treatment of nearly incompressible behavior. The progressive failure of the particle–matrix interface is modeled by a cohesive law accounting for mode mixity. The meso-scale is characterized by a unit cell, which contains particles dispersed in a homogenized blend. A new, fully implicit and efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting the non-homogeneous meso-fields and damage nucleation and propagation along the particle–matrix interface. Simple deformations involving an idealized solid rocket propellant are considered to demonstrate the algorithm.

Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations

AbstractInterfacial damage nucleation and evolution in reinforced elastomers subjected to finite strains is modelled using the mathematical theory of homogenization based on the asymptotic expansion of unknown variables. The microscale is characterized by a periodic unit cell, which contains particles dispersed in a blend and the particle matrix interface is characterized by a cohesive law. A novel numerical framework based on the perturbed Petrov–Galerkin method for the treatment of nearly incompressible behaviour is employed to solve the resulting boundary value problem on the microscale and the deformation path of a macroscale particle is predefined as in the micro‐history recovery procedure. A fully implicit and efficient finite element formulation, including consistent linearization, is presented. The proposed multiscale framework is capable of predicting the non‐homogeneous micro‐fields and damage nucleation and propagation along the particle matrix interface, as well as the macroscopic response and mechanical properties of the damaged continuum. Examples are considered involving simple unit cells in order to illustrate the multiscale algorithm and demonstrate the complexity of the underlying physical processes. Copyright © 2005 John Wiley & Sons, Ltd.

Finite element analysis of thermoplastic composite plates in forming temperature

An implicit mixed finite element formulation is developed based on the plate theory to analyze thermoforming process of thermoplastic plates reinforced with unidirectional continuous fibers. The plate thickness variations and the rotation of transverse planes are considered using Reissner–Mindlin approximation in order to model the predominated deformation mechanisms of the transverse fiber flow and intra-ply shear in these highly kinematical constrained materials. The kinematical constrains due to material incompressibility and fiber inextensibility are considered in the plate theory formulations and taken into account in the finite element method by introducing penalty numbers. A new analytical approach is developed to analyze the forming procedure of a composite plate on the single curvature geometry to compare the numerical results with analytical approach. The forming of thermoplastic plates to the hemispherical configuration is also studied by means of finite element technique. The geometry of panel in blank holder region is estimated considering both the effect of blank holder normal pressure and the viscous layers in die contact surfaces. The finite element results show that the geometry and defects due to wrinkling during the shaping procedure may be conventionally predicted using the derived formulations.

Computational description of nanocrystalline deformation based on crystal plasticity

The effect of grain size on the mechanical response of polycrystalline metals was investigated computationally and applied to the nanocrystalline domain. A phenomenological constitutive description is adopted to build the computational crystal model. Two approaches are implemented. In the first, the material is envisaged as a composite; the grain interior is modeled as a monocrystalline core surrounded by a mantle (grain boundary) with a lower yield stress and higher work hardening rate response. Both a quasi-isotropic and crystal plasticity approaches are used to simulate the grain interiors. The grain boundary is modeled either by an isotropic Voce equation (Model I) or by crystal plasticity (Model II). Elastic and plastic anisotropy are incorporated into this simulation. An implicit Eulerian finite element formulation with von Mises plasticity or rate dependent crystal plasticity is used to study the nonuniform deformation and localized plastic flow. The computational predictions are compared with the experimentally determined mechanical response of copper with grain sizes of 1 μm and 26 nm. Shear localization is observed during work hardening in view of the inhomogeneous mechanical response. In the second approach, the use of a continuous change in mechanical response, expressed by the magnitude of the maximum shear stress orientation gradient, is introduced. It is shown that the magnitude of the gradient is directly dependent on grain size. This gradient term is inserted into a constitutive equation that predicts the local stress–strain evolution.