Implicit Finite Element Scheme Research Articles

A Conservative Finite Element Scheme for the Kirchhoff Equation

This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively.

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Two-grid fully discrete finite element algorithms on temporal graded meshes for nonlinear multi-term time-fractional diffusion equations with variable coefficient

In this paper, we are concerned with two-grid fully discrete finite element algorithms for solving the nonlinear multi-term time-fractional diffusion equation with a variable coefficient. Taking into account the weak singularity of the solution at the initial time t=0, we employ graded meshes in time, which is an effective method to compensate for the lack of smoothness. A fully discrete implicit finite element scheme is established with the L1 method on temporally graded meshes; the stability and optimal error estimates are theoretically proved. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete scheme, and then the second algorithm is constructed to further improve the optimal convergence order. Similar properties as standard FEM are presented for the two algorithms. A significant fact is that this technique reduces the computation time without loss of numerical precision. Several numerical examples are provided to support our theoretical findings and verify the performance of the proposed algorithms.

Implementation and verification of a user‐defined element (UEL) for coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in saturated geological media

AbstractEfficient and accurate modeling of the coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in various rock formations is indispensable for designing energy geo‐structures such as underground repositories for high‐level nuclear wastes. This work focuses on developing and verifying an implicit finite element solver for generic coupled THMC problems in geological settings. Starting from the mass, momentum, and energy balance laws, a specialized set of governing equations and a thermoporoelastic constitutive model is derived. This system is then solved by an implicit finite element (FE) scheme. Specifically, the residuals and the Jacobians are scripted in a user‐defined element (UEL) subroutine which is then combined with the general‐purpose FE software Abaqus Standard to solve initial‐boundary value problems. Considering the complexity of the system, the UEL development follows a stepwise manner by first solving the coupled hydraulic‐mechanical (HM) and thermal‐hydraulic‐mechanical (THM) equations before moving on to the full THMC problem. Each implementation step consists of at least one verification test by comparing computed results with closed‐form analytical solutions to ensure that the various coupling effects are correctly realized. To demonstrate the robustness of the algorithm and to validate the UEL, a three‐dimensional case study is performed with reference to the in‐situ heating test of ATLAS at Belgium in 1980s. A hypothetical radionuclide leakage event is then simulated by activating the chemical‐concentration degree of freedom and prescribing a constant high concentration at the heater's surface. The model predicts a limited contaminated regime after six years considering both diffusion and advection effects on species transport.

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

An FCT finite element scheme for ideal MHD equations in 1D and 2D

This paper presents an implicit finite element (FE) scheme for solving the equations of ideal magnetohydrodynamics in 1D and 2D. The continuous Galerkin approximation is constrained using a flux-corrected transport (FCT) algorithm. The underlying low-order scheme is constructed using a Rusanov-type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed. The accuracy of the low-order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity-preserving iterative manner. At the FCT correction step, the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure. Divergence-free magnetic fields are extracted using projections of the FCT predictor into staggered finite element spaces forming exact sequences. In the 2D case, the magnetic field is projected into the space of Raviart–Thomas finite elements. Numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal MHD flows.

Progressive damage modeling in carbon fibers/carbon nanotubes reinforced polymer composites

Abstract Damage in carbon fibers/Epoxy laminate composites containing carbon nanotubes (CNTs) is described. This investigation presents the third part of paper series on composite materials based CNTs. Experimental characterization and numerical homogenizations were published in order to obtain effective elastic orthotropic properties. Therefore, the homogenized properties are used as input data for investigation of damage initiation and propagations in the specimens for Open Hole tensile tests. The numerical intralaminar damage model is created under ABAQUS software with Hashin's criteria. The model is formulated according to damage mechanics. The prediction of ply delamination in laminated composites containing CNTs is modeled by using interface elements. The numerical approach is based on the cohesive zone model, provides an efficient description of the delamination growth. The numerical study is performed in the quasi-static regime and an implicit finite element scheme is used. The force-displacement curves obtained numerically shows a good agreement with the experimental.

A parallel implementation of an implicit discontinuous Galerkin finite element scheme for fluid flow problems

The discontinuous Galerkin (DG) method is frequently used in computational fluid dynamics for its stability and high order of accuracy. A disadvantage of the DG method is its high computational demands. The aim of this paper is to weaken this drawback by means of parallelization of the DG algorithm. The computation is performed on a network of computers with distributed memory using the Java Remote Method Invocation, which is included in the Java programming language. The partition of the boundary value problem into n subproblems, which is then solved by n computers separately, is based on the overlapping Schwarz method. On basis of physical nature of the problem, the present paper proposes minimal size of the overlap that allows for only one Schwarz iteration thereby increasing efficiency of parallelization. The scalability and efficiency of the presented parallelization approach is demonstrated on several test problems. In order to stabilize the DG method in presence of shocks, a recently developed technique by Huerta et al. (Int. J. Numer. Meth. Fluids 69(10), 2012, 1614–1632), which introduces discontinuities in basis functions in regions with a shock, is adopted here. A modification of this approach, which lowers the computational and implementational demands, is presented here.

Modelling of the post yield response of amorphous polymers under different stress states

In this contribution, an elasto-viscoplastic constitutive model based on the single mode EGP (Eindhoven Glassy Polymer) model is proposed to describe the deformation behaviour of solid polymers subjected to finite deformations under different stress states. The polymeric material examined in this work is a specific commercial grade of Bisphenol, a polycarbonate called Makrolon 2607, for which there were experimental results available in the open literature for: uniaxial compression, plane strain compression and tensile test on a dumbbell shape specimen. The material properties of the original model are determined and calibrated from a uniaxial compression-loading test. Then, several numerical examples under different stress states are presented to illustrate the limitations of the single mode EGP model. A more general elasto-viscoplastic model is proposed, which preserves the isotropy of the original model, using the lode angle parameter to distinguish shear-dominated stress states and capture the material post yield response. The numerical treatment of the model, including the state update procedure and also the consistent tangent operator, required for the finite implementation of the model within an implicit finite element scheme, is presented. A comprehensive set of numerical examples is employed to compare the predictions of the original and new models against experimental results and to investigate the effect of the proposed modifications. The numerical results show that the proposed model provides a closer agreement with experimental evidence and opens the possibility for computational simulations of amorphous polymers under different stress states.

Compact Modified Implicit Finite Element Schemes for Wave Propagation Problems with Superior Dispersive Properties

In this paper, we study in detail dispersive properties of the widely used implicit scheme called Newmark trapezoidal rule in conjunction with modified mass matrix to enjoy superior dispersive properties keeping the finite element stencil compact. We call such schemes compact modified implicit finite element scheme. In case of one-dimensional propagation, following contributions are made: (a) for modified finite element (MFE) scheme, we find an optimal value of dispersion controlling parameter depending on Courant–Friedrichs–Lewy (CFL) number which provides exact solutions at the nodes of spatial gird; (b) for standard finite element (SFE) scheme optimal value of CFL number is obtained which provides fourth-order accurate solutions. Moreover, in case of two-dimensional propagation following contributions are made: (c) we have found optimal value of CFL number for all angles in case of both SFE and MFE schemes; (d) superior dispersive behaviour is evident in case of MFE scheme in comparison with SFE scheme. Furthermore, the MFE scheme can be efficiently implemented using non-standard quadrature rules or just updating mass matrix which does not require to write brand new code and makes it computationally very attractive. Also for specific value of parameter, i.e. τ = 0, the MFE scheme leads back to the SFE scheme.

Hybrid numerical method with adaptive overlapping meshes for solving nonstationary problems in continuum mechanics

Techniques that improve the accuracy of numerical solutions and reduce their computational costs are discussed as applied to continuum mechanics problems with complex time-varying geometry. The approach combines shock-capturing computations with the following methods: (1) overlapping meshes for specifying complex geometry; (2) elastic arbitrarily moving adaptive meshes for minimizing the approximation errors near shock waves, boundary layers, contact discontinuities, and moving boundaries; (3) matrix-free implementation of efficient iterative and explicit–implicit finite element schemes; (4) balancing viscosity (version of the stabilized Petrov–Galerkin method); (5) exponential adjustment of physical viscosity coefficients; and (6) stepwise correction of solutions for providing their monotonicity and conservativeness.

Hybrid asymptotic-numerical modeling of thin layers for dynamic thermal analysis of structures

Purpose – The purpose of this paper is to propose a computational model for thin layers, for problems of linear time-dependent heat conduction. The thin layer is replaced by a zero-thickness interface. The advantage of the new model is that it saves the need to construct and use a fine mesh inside the layer and in regions adjacent to it, and thus leads to a reduction in the computational effort associated with implicit or explicit finite element schemes. Design/methodology/approach – Special asymptotic models have been proposed for linear heat transfer and linear elasticity, to handle thin layers. In these models the thin layer is replaced by an interface with zero thickness, and specific jump conditions are imposed on this interface in order to represent the special effect of the layer. One such asymptotic interface model is the first-order Bövik-Benveniste model. In a paper by Sussmann et al., this model was incorporated in a FE formulation for linear steady-state heat conduction problems, and was shown to yield an accurate and efficient computational scheme. Here, this work is extended to the time-dependent case. Findings – As shown here, and demonstrated by numerical examples, the new model offers a cost-effective way of handling thin layers in linear time-dependent heat conduction problems. The hybrid asymptotic-FE scheme can be used with either implicit or explicit time stepping. Since the formulation can easily be symmetrized by one of several techniques, the lack of self-adjointness of the original formulation does not hinder an accurate and efficient solution. Originality/value – Most of the literature on asymptotic models for thin layers, replacing the layer by an interface, is analytic in nature. The proposed model is presented in a computational context, fitting naturally into a finite element framework, with both implicit and explicit time stepping, while saving the need for expensive mesh construction inside the layer and in its vicinity.

Comparison of the Johnson-Cook and VPSC Models to Describe the Constitutive Behaviour of Ti-6Al-4V in an Implicit Finite Element Scheme

The use of finite element simulations has become one of the main tools of the mechanical engineer. The method is applied to the analysis and design of engineering structures, the study of manufacturing processes and even to perform virtual experiments. Traditionally, the constitutive laws chosen for finite element analysis have been as simple as possible, mainly due to the limitation imposed by the available computing power. However, the development of more powerful computers and more efficient methods is opening the possibility of using more elaborated (and, most often, more accurate) material models. In particular, polycrystal models capable of predicting not only the mechanical behaviour of the material, but also of the evolution of properties with increasing strain, are particularly well suited for the simulation of forming processes, for which a precise knowledge of the properties of the resulting product is of paramount importance.The present work studies how the Visco Plastic Self-Consistent model (VPSC) can be used in combination with the implicit finite element package Abaqus/Standard to simulate the behaviour of Ti-6Al-4V sheet, and compares it with the more common (and much simpler) Johnson-Cook model. More specifically, the goal of this study is to determine whether or not, with using similar experimental calibration data, the use of the much more complex polycrystal model, justifies the increased complexity and execution time. Using standard tensile experiments at different strain rates, the parameters of the VPSC and Johnson-Cook models are fitted using a minimization method. Then, both models are used in finite element simulations and the results given by both models are compared.

Cauchy Solution of the Kinematic Wave Shallow-Water Equations Using Square-Grid Finite-Element Method

AbstractThe kinematic wave system of shallow-water equations is posed as an initial value problem, or Cauchy problem, to investigate error analysis using the square-grid finite-element method. The consistent, lumped, and upwind models of the square finite-element method are analyzed using the Fourier (or von Neumann) stability method as separate initial value problems for two-dimensional shallow-water equations. The small difference between the Cauchy solution and integer (or half) multiples of the eigenvalue solution of amplification factors using the eigenvalue method indicates that the Cauchy solution is a possible stable numerical solution of the kinematic wave shallow-water equations within the reasonable limits of solution accuracy. The nodal amplification factors are less than or equal to unity for implicit finite-element schemes for all of the three formulations—consistent, lumped, and upwind for all of the wave numbers, implying unconditional stability. For explicit and semi-implicit finite-eleme...

Effects of notch geometry on bursting fracture of a V-notched rectangular cup for Li-Ion secondary batteries

A V-notched rectangular cup produced by a multi-stage deep drawing process is presented as a primary safety structure of the lithium-ion (Li-Ion) secondary batteries. The V-notched rectangular cup is required to sustain an inner pressure resulting from the elevated temperature induced by the chemical reaction of the electrolyte during normal operation, and it has to fail when the inner pressure reaches a critical pressure. The inner pressure below approximately 0.500 MPa is considered to be safe, and the battery that reaches the pressure above the critical limit must leak the excess inner pressure, without any explosion. In this study, a V-notched structure on the outer surface of a rectangular cup is proposed as the primary safety device. Predictions with respect to the bursting fracture pressure are presented, where geometric parameters such as the notch angle and depth are considered. The bursting behavior is predicted by adopting an implicit elasto-plastic finite element scheme combined with Cockcroft and Latham’s ductile fracture criterion. The predicted bursting fracture pressure is compared to the results from experimental investigations. It is confirmed that the bursting pressure of the V-notched rectangular cup is well matched with the variation of 5.88% between the predicted and experimentally measured results.

An improved algorithm for the polycrystal viscoplastic self-consistent model and its integration with implicit finite element schemes

A new algorithm for the solution of the deformation of a polycrystalline material using a self-consistent scheme, and its integration as part of the finite element software Abaqus/Standard are presented. The method is based on the original VPSC formulation by Lebensohn and Tomé and its integration with Abaqus/Standard by Segurado et al. The new algorithm has been implemented as a set of Fortran 90 modules, to be used either from a standalone program or from Abaqus subroutines. The new implementation yields the same results as VPSC7, but with a significantly better performance, especially when used in multicore computers.

A well‐balanced explicit/semi‐implicit finite element scheme for shallow water equations in drying–wetting areas

SUMMARYNumerical solutions of the shallow water equations can be used to reproduce flow hydrodynamics occurring in a wide range of regions. In hydraulic engineering, the objectives include the prediction of dam break wave propagation, fluvial floods and other catastrophic flooding phenomena, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. In this paper, a well‐balanced explicit and semi‐implicit finite element scheme for shallow water equations over complex domains involving wetting and drying is proposed. The governing equations are discretized by a fractional finite element method using a two‐step Taylor–Galerkin scheme. First, the intermediate increment of conserved variable is obtained explicitly neglecting the pressure gradient term. This is then corrected for the effects of pressure once the pressure increment has been obtained from the Poisson equation. In order to maintain the ‘well‐balanced’ property, the pressure gradient term and bed slope terms are incorporated into the Poisson equation. Moreover, a local bed slope modification technique is employed in drying–wetting interface treatments. The proposed model is well validated against several theoretical benchmark tests. Copyright © 2014 John Wiley & Sons, Ltd.

Finite element simulation of three-dimensional particulate flows using mixture models

In this paper, we discuss the numerical treatment of three-dimensional mixture models for (semi-)dilute and concentrated suspensions of particles in incompressible fluids. The generalized Navier–Stokes system and the continuity equation for the volume fraction of the disperse phase are discretized using an implicit high-resolution finite element scheme, and maximum principles are enforced using algebraic flux correction. To prevent the volume fractions from exceeding the maximum packing limit, a conservative overshoot limiter is applied to the converged convective fluxes at the end of each time step. A numerical study of the proposed approach is performed for 3D particulate flows over a backward-facing step and in a lid-driven cavity.

Implicit iterative finite element scheme for a strain gradient crystal plasticity model based on self-energy of geometrically necessary dislocations

In this study, an implicit iterative finite element scheme is developed for the strain gradient theory of single-crystal plasticity that accounts for the self-energy of geometrically necessary dislocations (GNDs). This strain gradient theory belongs to the Gurtin framework for viscoplastic single-crystals. The self-energy of GNDs gives a specific form of energetic higher-order stresses. An implicit finite element equation is obtained for solving a set of homogenization equations. The developed scheme is employed to analyze a model grain, and is verified by comparison with the analytical estimation derived by Ohno and Okumura (2007) [4]. The computational efficiency of the scheme and the incremental stability are discussed. Furthermore, it is shown that the developed scheme is available and applicable to different types of higher-order stresses including energetic and dissipative terms.

The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

AbstractOn the basis of our previous work, we introduce novel fully discrete, fully practical parametric finite element approximations for geometric evolution equations of curves in the plane. The fully implicit approximations are unconditionally stable and intrinsically equidistribute the vertices at each time level. We present iterative solution methods for the systems of nonlinear equations arising at each time level and present several numerical results. The ideas easily generalize to the evolution of curve networks and to anisotropic surface energies. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010