Finite Element Context Research Articles

Leaky mode analysis using complex infinite elements

We propose a novel numerically efficient model for accurate characterization of leaky modes in optical waveguides. Within the context of finite elements, the mesh boundaries are truncated using complex infinite elements (CIEs). The CIE is based on adopting the homogeneous solution of the wave equation with the shape function of infinite elements to accurately and physically model semi-infinite subdomains instead of placing an artificial layer such as PML. The proposed treatment can be easily implemented and requires less computational resources compared to other conventional mesh truncation methods. The accuracy and rigor of our approach are demonstrated through studying different leaky waveguides’ configurations.

Physics and chemistry-based phase-field constitutive framework for thermo-chemically aged elastomer

We propose a physics and chemistry-based constitutive framework to predict the stress and brittle failure responses of thermo-chemically aged elastomers. High-temperature aging in the presence of oxygen causes chemical reactions inducing significant changes in the elastomer macromolecular network. Experimental studies agree that the increase in effective chemical crosslinks is dominant, leading to increased material stiffness and ultimately causing brittle failure of aged elastomers. The main novelty of this work relies on directly connecting the evolution of the effective crosslink density to the mechanical properties which control stress and failure response of thermo-chemically aged elastomers. We equip the Arruda–Boyce hyperelastic constitutive equations to predict the stress–strain response until failure with phase-field to capture the induced brittle failure via a strain-based criterion for fracture. Four material properties associated with the stress and failure responses evolve to capture the detrimental effects of thermo-chemical aging. The evolution of the four material properties is characterized by the change in the effective crosslink density, obtained based on chemical characterization tests, for any given aging temperature and duration. The established constitutive framework is first solved analytically for the case of uniaxial tension in a homogeneous bar to highlight the interconnection between the four material properties. Then, the framework is numerically implemented within a finite element (FE) context via a user-element subroutine (UEL) in the commercial FE software Abaqus. The framework is validated versus a set of experimental results available in the literature. The comparison confirms that the proposed constitutive framework can accurately predict the stress–strain and failure responses of thermo-chemically aged elastomers. Further numerical examples are provided to demonstrate the effects of evolving material properties on the response of specimens containing pre-existing cracks exploiting the capabilities of the phase-field approach.

On a finite strain modeling of growth in budding yeast.

Cell's ability to proliferate constitutes one of the most defining features of life. The proliferation occurs through a succession of events; the cell cycle, whereby the cell grows and divides. In this paper, focus is made on the growth step and we deal specifically with Saccharomyces cerevisiae yeast that reproduces by budding. For this, we develop a theoretical model to predict the growth powered by the turgor pressure. This cell is herein considered as a thin-walled structure with almost axisymmetrical shape. Due to its soft nature, the large deformation range is a priori assumed through a finite growth modeling framework. The used kinematics is based on the multiplicative decomposition of the deformation gradient into an elastically reversible part and a growth part. Constitutive equations are proposed where use is made of hyperelasticity together with a local evolution equation, this latter to describe the way growth takes place. In particular, two essential parameters are involved: a stress-like threshold, and a characteristic time. The developed model is extended to a shell approach as well. In a finite element context, representative numerical simulations examining stress-dependent growth are given and a parametric study is conducted to show the sensitivity with respect to the above mentioned parameters. Finally, a suggestion for natural contractile ring modeling closes this study.

An edge-based smoothed finite element method for semi-implicit coupling of unsteady viscoelastic fluid–structure interaction

In this paper, a new partitioned semi-implicit coupling algorithm is developed for unsteady viscoelastic fluid–structure interaction in association with an edge-based smoothed finite element method (ESFEM). Stabilization techniques tailored for the viscoelastic fluid subproblem are naturally integrated into the hybrid explicit/implicit coupling framework to achieve the enhanced stability. In particular, a mass source term derived from the modified continuity equation in the finite-element context enables the ESFEM to fulfill the geometric conservation law on variable-node smoothing cells during the subiterations. Moreover, both multi-field equations and interface conditions are easily discretized with the ESFEM which enjoys high flexibility of smoothed Galerkin weak-form integral. Desirable numerical results are given for two benchmark problems.

Cryogenic liquid nitrogen stimulation-induced cracking in gas-enriched shale reservoirs: A numerical investigation

Cryogenic shock-induced brittle cracking has promising application prospects in reservoir stimulation. In this paper, a fully coupled thermo-hydro-mechanical (T-H-M) model combined with the statistical meso-damage theory is proposed to investigate the trans-scale progressive cracking process of shale gas reservoirs stimulated by liquid nitrogen (LN2). In this theoretical framework, shale meso-heterogeneity in both thermo-mechanical and hydraulic properties is characterized by two independent statistical distribution functions; temperature-dependent rock properties and temperature- and pressure-dependent fluid thermodynamic properties are described using some experimental and theoretical formulas. The numerical model is implemented in a finite element context and subsequently validated against existing classical coupling results. Numerical results demonstrate that LN2 fracturing accompanied by the unique cryogenic shock effect leads to a complex cracking process involving multiple-point fracture initiation, asymmetric macrofracture propagation, and primary fracture bifurcation. Sensitivity analysis indicates that approximately equidistant or equilength radial fractures may be generated at a lower stress difference, lower permeability, or higher elastic modulus; both fracture initiation pressure and induced fracture morphology are more sensitive to injected fluid temperature relative to initial reservoir temperature; a larger thermal expansion coefficient is beneficial for inducing more fractures, but not applicable to the thermal conductivity. Preferable candidates for LN2 stimulation to generate multiple fractures in shale reservoirs are recommended. It is also found that compared with H2O and supercritical CO2, fracturing using LN2 can generate fractures with lower initiation pressure and greater complexity. Preliminary results confirm the potential of using LN2 fracturing to improve the stimulation effect of shale reservoirs.

On rate dependence and anisotropy in phase field modeling of polycrystalline fracture

A phase field theory of fracture incorporating rate dependence (i.e., kinetic viscosity) and anisotropic elasticity is implemented to study strength and failure modes in polycrystalline materials. After presentation of the constitutive framework, a new analytical solution is derived for one-dimensional (1D) homogeneous loading at constant strain rate, wherein crack viscosity is enabled. The three-dimensional (3D) theory is implemented in a finite element (FE) context, with static and dynamic solutions verified up to peak loading for homogeneous microstructures. Extensive simulations of heterogeneous polycrystalline microstructures follow, where baseline material properties correspond to either boron carbide (B4C) or titanium diboride (TiB2). Results quantify effects of loading rate, elastic anisotropy, cleavage energy anisotropy, and grain boundary surface energy on tensile strength. Notable are effects of strong basal texture, which influences average strength via both elastic anisotropy and basal cleavage plane orientation. Effects of loading rate are naturally more prominent for materials that fail mostly intergranularly rather than transgranularly, in agreement with experimental observations.

Construction and Numerical Assessment of Local Absorbing Boundary Conditions for Heterogeneous Time-Harmonic Acoustic Problems

This article is devoted to the derivation and assessment of local absorbing boundary conditions (ABCs) for numerically solving heterogeneous time-harmonic acoustic problems. To this end, we develop a strategy inspired by the work of Engquist and Majda to build local approximations of the Dirichlet-to-Neumann operator for heterogeneous media, which is still an open problem. We focus on three simplified but characteristic examples of increasing complexity to highlight the strengths and weaknesses of the proposed ABCs: the propagation in a duct with a longitudinal variation of the speed of sound, the propagation in a nonuniform mean flow using a convected wave operator, and the propagation in a duct with a transverse variation of the speed of sound and density. For each case, we follow the same systematic approach to construct a family of local ABCs and explain their implementation in a high-order finite element context. Numerical simulations allow us to validate the accuracy of the ABCs and to give recommendations for the tuning of their parameters.

Connections between finite difference and finite element approximations

We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.

FFT‐based homogenization with mixed uniform boundary conditions

AbstractThe fast Fourier transform (FFT) based homogenization method of Moulinec and Suquet has been established as a fast, accurate, and robust tool for periodic homogenization in solid mechanics. In a finite element context, Pahr and Zysset have introduced nonperiodic boundary conditions (PMUBC) for homogenization problems. We show how to implement PMUBC efficiently in an FFT‐based code using discrete sine and cosine transforms. Compared with the domain mirroring approach, we reduce the runtime by a factor of 2 to 3, and the memory requirements by a factor of 8. We show that the use of periodic boundary conditions for nonperiodic geometries yields vastly different results than with PMUBC. Furthermore, we examine the influence of the discretization method by comparing the staggered grid discretization with a finite element discretization.

Transient stability analysis of composite hydrogel structures based on a minimization-type variational formulation

We employ a canonical variational framework for the predictive characterization of structural instabilities that develop during the diffusion-driven transient swelling of hydrogels under geometrical constraints. The variational formulation of finite elasticity coupled with Fickian diffusion has a two-field minimization structure, wherein the deformation map and the fluid-volume flux are obtained as minimizers of a time-discrete potential involving internal and external energetic contributions. To analyze the structural stability of a certain equilibrium state of the gel satisfying the minimization principle, we apply the local stability criterion on the incremental potential, which is based on the idea that a stable equilibrium state has the lowest potential energy among all possible states within an infinitesimal neighborhood. Using this criterion in a finite-element context, it is understood that bifurcation-type structural instabilities are activated when the coupled global finite-element stiffness matrix loses its positive definiteness. This concept is then applied to determine the onset and nature of wrinkling instabilities occurring in a pair of representative film-substrate hydrogel systems. In particular, we analyze the dependencies of the critical buckling load and mode shape on the system geometry and material parameters.

One-dimensional Eikonal Non-Local (ENL) damage models: Influence of the integration rule for computing interaction distances and indirect loading control on damage localization

The Eikonal Non-Local approach models damage dependent non-local interactions by considering that interaction distances between material points are computed from solving an eikonal equation with an isotropic/anisotropic damage dependent metric Riemann function. In the finite element context, such a formulation exhibits good regularization properties and naturally allows one to model strain localization when non-local interactions vanish. In an isotropic damage mechanics framework, the damage variable tends to unity on a single integration point, and no damage diffusion occurs. In a one-dimensional context, evaluating the interaction distance between two points comes into computing an integral where the integrated function depends on the damage field between them. Desmorat et al. [9] performed the integration using a trapezoidal rule (under the assumption that the damage field is constant between adjacent integration points). Conversely, Jirásek and Desmorat [19] assumed that the damage field is linear between nearby integration points. Mainly motivated by improved integration accuracy, this method leads to a more gradual damaging process and makes it easier to follow the structural response when damage localizes. In this contribution, we show that despite these advantages, such a choice induces parasite damage diffusion issues, making the use of a standard trapezoidal rule more relevant. Moreover, once this choice is made, the structural response during damage localization can be conveniently described using a path-following method based on controlling the non-local strain variation of the localizing element instead of the local strain as in the works cited above.

Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements

Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the modal expansion is observed when applying the method with 3D elements, because of nonlinear couplings occurring with very high frequency modes involving 3D thickness deformations. Focusing on the case of flat structures, we first show by computing all the modes of the structure that a converged solution can be exhibited by using either static condensation or normal form theory. We then show that static modal derivatives provide the same solution with fewer calculations. Finally, we propose a modified STEP, where the prescribed displacements are imposed solely on specific degrees of freedom of the structure, and show that this adjustment also provides efficiently a converged solution.

Criteria evaluation for the transition of cracking modes in a single-crystal nickel-base superalloy

Single-crystal nickel-base superalloys frequently experience two distinct fatigue crack growth modes. It has been observed that, under certain conditions, cracks transition from a path perpendicular to the loading direction to a crystallographic slip plane. As crystallographic cracking is associated with an increased fatigue crack growth rate, it is important to be able to predict when this transition occurs. In this work three different criteria for crystallographic cracking based on resolved anisotropic stress intensity factors are evaluated in a three-dimensional finite element context. The criteria were calibrated and evaluated using isothermal fatigue experiments on two different specimen geometries. It is suggested by the results, that a threshold value of a resolved shear stress intensity factor can act as a conservative criterion indicating cracking mode transition. Further, a trend hinting towards a loading frequency dependency could be observed.

A new perspective on hierarchical spline spaces for adaptivity

We introduce a framework for spline spaces of hierarchical type, based on a parent–children relation, which is very convenient for the analysis as well as the implementation of adaptive isogeometric methods. This framework exploits the innate refinement by functions in the B-splines context, rather than by elements or cells, which is more natural in the finite element context. Furthermore, it entails a new language to handle hierarchical spline spaces, which allows to tackle fundamental questions in a very simple manner. For example, it makes it simple to create hierarchical bases with several desired properties with a refinement procedure which has linear complexity, i.e., the resulting bases have cardinality bounded by the number of initially marked functions.

Wall‐modeled large‐eddy simulation in a finite element framework

SummaryThis work studies the implementation of wall modeling for large‐eddy simulation in a finite element context. It provides a detailed description of how the approach used by the finite volume and finite differences communities is adapted to the finite element context. The new implementation is as simple and easy to implement as the classical finite element one, but it provides vastly superior results. In the typical approach used in finite elements, the mesh does not extend all the way to the wall, and the wall stress is evaluated at the first grid point, based on the velocity at the same point. Instead, we adopt the approach commonly used in finite differences, where the mesh covers the whole domain and the wall stress is obtained at the wall grid point, with the velocity evaluated at the first grid point off the wall. The method is tested in a turbulent channel flow at Reτ=2003, a neutral atmospheric boundary layer flow, and a flow over a wall‐mounted hump, with significant improvement in the results compared to the standard finite element approach. Additionally, we examine the effect of evaluating the input velocity further away from the wall, as well as applying temporal filtering on the wall‐model input.

A quasi-optimal non-overlapping domain decomposition method for two-dimensional time-harmonic elastic wave problems

This article presents the construction of a new non-overlapping domain decomposition method (DDM) for two-dimensional elastic scattering problems. The method relies on a high-order Transmission Boundary Condition (TBC) between sub-domains, which accurately approximates the exact Dirichlet-to-Neumann map. First, we explain the derivation of this new TBC in the context of a non-overlapping DDM. Next, a mode-by-mode convergence study for a model problem is presented, which shows the new method to be quasi-optimal, i.e. with an optimal convergence rate for evanescent modes and an improved convergence rate for the other modes compared to the standard low-order Lysmer-Kuhlemeyer TBC. Finally, the effectiveness of the new DDM is demonstrated in a finite element context by analyzing the behavior of the method on high-frequency elastodynamic simulations.

Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

We propose an extension of the embedded boundary method known as “shifted boundary method” to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded.

Raviart–Thomas finite elements of Petrov–Galerkin type

Finite volume methods are widely used, in particular because they allow an explicit and local computation of a discrete gradient. This computation is only based on the values of a given scalar field. In this contribution, we wish to achieve the same goal in a mixed finite element context of Petrov–Galerkin type so as to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart–Thomas finite elements. Our purpose is to define test functions that are in duality with these shape functions: precisely, the shape and test functions will be asked to satisfy some orthogonality property. This paradigm is addressed for the discrete solution of the Poisson problem. The general theory of Babuška brings necessary and sufficient stability conditions for a Petrov–Galerkin mixed problem to be convergent. In order to ensure stability, we propose specific constraints for the dual test functions. With this choice, we prove that the mixed Petrov–Galerkin scheme is identical to the four point finite volume scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Convergence is proven with the usual techniques of mixed finite elements.

A partitioned two-step solution algorithm for concurrent fluid flow and stress–strain numerical simulation in solidification processes

One of the critical challenges encountered when modeling solidification processes is to achieve a concurrent and efficient computation of fluid flow and solid mechanics. Several detrimental casting defects justify this development: cracks, either as a result of stresses built at surface or sub-surface in solidified regions during the filling stage of ingot casting, or due to hot tears deep in the mushy zone during solidification; macrosegregation, as a result of thermo-solutal convection flows and possible deformation of solid. It is therefore of crucial importance to provide for a global and synthetic analysis of casting processes considering a single numerical modeling that includes coupling between fluid flow and solid mechanics. A two-step solution strategy combining fluid flow and solid mechanics has been developed. A partitioned formulation is used, performing at each time increment, separately a solid-oriented resolution and a fluid-oriented resolution. Liquid flow (natural convection or forced flow during ingot filling stage), solidification shrinkage as well as thermally induced deformation of the solid regions are taken into account. The paper presents the numerical formulation in a level set finite element context, and associated validation tests. Application in a practical case corresponding to an ingot filling is proposed in order to investigate the solidification process and associated fluid flow and stress evolutions. Some discussions on computation time and other numerical aspects are also developed at the end in order to show the potential improvements of this methodology.