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Modeling deformation processes of materials under high strain rate and impact conditions in which the strain rates vary spatiotemporally over several orders of magnitude is challenging, especially in terms of constitutive description. Visco-plastic power-law flow rule, commonly used within crystal plasticity constitutive models, usually introduces superfluous strain rate effects, entering numerically via the slip activation criterion, which result in artificially high flow stress predictions under high strain rate deformation conditions. This paper presents a novel, implicit finite element implementation of a visco-plastic self-consistent polycrystal model that is implicit in dislocation-based hardening with removed strain rate sensitivity introduced by the power-law flow rule. In the model, the strain rate dependence of the flow stress is defined solely by the rate sensitivity of Peierls stress and the evolution of dislocation density within the hardening law, enabling the prediction of high strain rate and impact deformation. As a result, the predicted flow stress is accurate in its magnitude. The physically based strain rate and temperature sensitive model also features, in addition to Schmid, the non-Schmid activation contribution for slip, which is necessary for understanding and modeling the plastic deformation of body-centered cubic metals. The model is calibrated to simulate the strain rate- and temperature-sensitive monotonic deformation of tantalum and is subsequently applied to a Taylor impact test of the same material. Since the impact simulation required remeshing and interpolation of state variables, the spherical linear interpolation algorithm in the space of quaternions is implemented to facilitate the interpolation of texture. Predictions of the simulation were found to compare favorably with experimental measurements of post-test geometrical changes and texture evolution. The implementation and insights from these predictions are presented and discussed in the paper. [ABSTRACT FROM AUTHOR]

The paper develops a finite element method for the Navier–Stokes equations of incompressible viscous fluid in a time-dependent domain. The method builds on a quasi-Lagrangian formulation of the problem. The paper provides stability and convergence analysis of the fully discrete (finite-difference in time and finite-element in space) method. The analysis does not assume any CFL time-step restriction, it rather needs mild conditions of the form Δ t ≤ C , where C depends only on problem data, and h 2 m u + 2 ≤ c Δ t , m u is polynomial degree of velocity finite element space. Both conditions result from a numerical treatment of practically important non-homogeneous boundary conditions. The theoretically predicted convergence rate is confirmed by a set of numerical experiments. Further we apply the method to simulate a flow in a simplified model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced computed tomography images. [ABSTRACT FROM AUTHOR]

This paper introduces a preconditioned variational multiscale stabilization (P-VMS) method for compressible flows. In this introductory paper we focus on inviscid flow and steady state problems. The Euler equations are solved on fully unstructured grids and discretized using the finite element method. The P-VMS method can be decomposed into three parts. First, a local preconditioner is applied to the continuous equations to reduce the stiffness while covering a wide range of Mach numbers. Then, the resulting preconditioned system is discretized in space using finite elements and stabilized with a variational multiscale stabilization method adapted for the preconditioned equations. In this paper, the solution is advanced in time using a fully explicit time discretization, although P-VMS is general and can be applied to fully implicit solvers. The proposed method is assessed by comparing convergence and accuracy of the solutions between the non-preconditioned and preconditioned cases, in particular for van Leer–Lee–Roe ’s (1991) and Choi-Merkle ’s (1993) preconditioners, in some selected examples covering a large range of Mach numbers. [ABSTRACT FROM AUTHOR]

The paper develops a finite element method for partial differential equations posed on hypersurfaces in R N , N = 2 , 3 . The method uses traces of bulk finite element functions on a surface embedded in a volumetric domain. The bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures. The cartesian structure of the bulk mesh leads to easy and efficient adaptation process, while the trace finite element method makes fitting the mesh to the surface unnecessary. The number of degrees of freedom involved in computations is consistent with the two-dimension nature of surface PDEs. No parametrization of the surface is required; it can be given implicitly by a level set function. In practice, a variant of the marching cubes method is used to recover the surface with the second order accuracy. We prove the optimal order of accuracy for the trace finite element method in H 1 and L 2 surface norms for a problem with smooth solution and quasi-uniform mesh refinement. Experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom, if grid adaptation is based on an appropriate error indicator. The paper shows results of numerical experiments for a variety of geometries and problems, including advection–diffusion equations on surfaces. Analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces. [ABSTRACT FROM AUTHOR]

This paper presents a novel computational formulation for large strain polyconvex elasticity. The formulation, based on the original ideas introduced by Schröder et al. (2011), introduces the deformation gradient (the fibre map), its adjoint (the area map) and its determinant (the volume map) as independent kinematic variables of a convex strain energy function. Compatibility relationships between these variables and the deformed geometry are enforced by means of a multi-field variational principle with additional constraints. This process allows the use of different approximation spaces for each variable. The paper extends the ideas presented in Schröder et al. (2011) by introducing conjugate stresses to these kinematic variables which can be used to define a generalised convex complementary energy function and a corresponding complementary energy principle of the Hellinger–Reissner type, where the new conjugate stresses are primary variables together with the deformed geometry. Both compressible and incompressible or nearly incompressible elastic models are considered. A key element to the developments presented in the paper is the new use of a tensor cross product, presented for the first time by de Boer (1982), page 76, which facilitates the algebra associated with the adjoint of the deformation gradient. For the numerical examples, quadratic interpolation of the displacements, piecewise linear interpolation of strain and stress fields and piecewise constant interpolation of the Jacobian and its stress conjugate are considered for compressible cases. In the case of incompressible materials two formulations are presented. First, continuous quadratic interpolation for the displacement together with piecewise constant interpolation for the pressure and second, linear continuous interpolation for both displacement and pressure stabilised via a Petrov–Galerkin technique. [ABSTRACT FROM AUTHOR]

The paper presents a frequency-domain h p -adaptive Finite Element (FE) Method for a class of coupled acoustics/elasticity/poroelasticity problems with application to modeling of acoustic logging measurements in complex borehole environments. The paper extends methodology, software, and results presented in Matuszyk et al. (2012). We derive for poroelastic media a mixed weak formulation, which is closely related to the enhanced ( u , p ) formulation developed by Atalla et al. (2001). The formulation is extended to the transversely anisotropic (VTI) case, and makes use of the viscodynamic operator, which enables more accurate calculations for higher frequencies. In this approach, a special split of the differential operator results in boundary and coupling conditions that can be easily implemented within the FE framework. The appropriate PML technique is employed to truncate the computational domain. Accordingly, the original h p -FE code is modified by the implementation of the appropriate scaling of the independent variables to make efficient the automatic adaptation algorithm. Solutions of non-trivial examples involving formations with permeable layers are presented. They positively verify the consistency and accuracy of the presented method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to accurately solve short wave scattering problems governed by the Helmholtz equation using the Bernstein–Bézier Finite Element method (BBFEM), combined with a conformal perfectly matched layer (PML). Enhanced PMLs, where curved geometries are represented by means of the blending map method of Gordon and Hall, are numerically investigated. In particular, the performance of radial and elliptical shaped PMLs, with a parabolic absorption function, are assessed and compared in terms of accuracy against second order Bayliss–Gunzberger–Turkel (BGT 2) based local absorbing boundary conditions. Numerical results dealing with problems of Hankel source radiation and wave scattering by a rigid cylinder show that the radial shaped PML, with the h and p versions of BBFEM, enables the recovery of the predicted algebraic and exponential convergence rates of a high order finite element method (FEM). Furthermore, radial shaped BGT 2 and PML have a similar performance, as long as the wave is not sufficiently well resolved. But, BGT 2 performs poorly as the wave resolution increases. Additionally, the effect of harmonics of higher modes on accuracy is examined. The study reveals that the PML outperforms BGT 2 for almost all propagating modes. However, a similar performance is achieved with both methods either with higher modes or a low wave resolution. Results from a multiple scattering benchmark problem provide evidence of the good performance of the proposed PMLs and the benefit of elliptical shaped PMLs in reducing significantly the size of the computational domain, without altering accuracy. The choice of the PML parameters ensuring optimal performance is also discussed. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a coupled peridynamic-finite element method for modeling mechanical behavior and damage growth in materials is developed. Peridynamics is a nonlocal continuum model which, in contrast to other continuum models, uses integration instead of spatial derivations in its governing equations. Utilizing integration instead of derivatives is advantageous in modeling fracture since the governing equations remain valid even after the initiation or growth of discontinuities. Although peridynamics can capture material fracture effectively, however due to its nonlocal formulation peridynamics is computationally expensive. To reduce the computational costs, we propose to couple peridynamics with finite elements and use peridynamics only in small zones where higher accuracy is needed. The main challenge in developing such a coupling method is to eliminate the artifacts introduced by the interface of the two subdomains. One of the main issues is spurious wave reflections which occurs because high frequency waves traveling from peridynamic region cannot enter the finite element zone and spuriously reflect back into the peridynamic zone. This will lead to an increase in the energy of the peridynamic zone and will drastically reduce the computational accuracy. The main feature of the proposed method is eliminating the spurious wave reflections such that the coupled method is as accurate as the pure peridynamics. Highlights • A coupled peridynamic-finite element method is developed to model dynamic fracture. • Spurious wave reflections are effectively suppressed. • Arlequin method is used to impose the mechanical compatibility between finite element and peridynamic sub regions. [ABSTRACT FROM AUTHOR]

This paper presents a simple and effective strain smoothing method (the strain-smoothed element method) for 3-node triangular and 4-node tetrahedral solid finite elements. While piecewise constant strain fields are constructed through smoothing domains in previous strain smoothing methods, in the proposed method, linear strain fields are constructed within finite elements using constant strains of neighboring finite elements. When the new strain smoothing method is adopted, the triangular and tetrahedral solid finite elements pass all the basic tests; patch, isotropy and zero energy mode tests, and show improved predictive capability. The formulation of the proposed method is presented in detail. Through various numerical examples, we demonstrate the accuracy improvement achieved. [ABSTRACT FROM AUTHOR]

This paper deals with the introduction of mesh refinement techniques within the non-intrusive patch process. For this, an ad hoc residual based explicit error estimator is built, which is adapted to a multi-scale solution, associated with those non-intrusive mesh refinement technique. Moreover, to reduce the global cost of the process, one introduces an estimate of the convergence error of the non-intrusive algorithm, which allows to reduce the number of iterations. This method is discussed and illustrated in various numerical examples. [ABSTRACT FROM AUTHOR]

Typical probabilistic problems in an engineering context include rare event probability estimation for physical models where spatial autocorrelation of material property parameters is significant. Subset Simulation, a Markov Chain Monte Carlo technique, can be used to estimate rare event probabilities in physical models more efficiently than Monte Carlo Simulation. This efficiency gain is important when the sampling operation is computationally demanding, as is the case in the solution of stochastic Partial Differential Equations. In high dimensional spaces where Polynomial Chaos or other direct integration techniques become intractable, sampling methods may be the only way to compute integral functions in probabilistic analysis. In this paper, Subset Simulation is applied to probability of failure estimation in nonlinear elasto-plastic finite element problems. Further, a derivation of confidence intervals for Subset Simulation relative errors is presented. This new technique allows for vastly improved efficiency in the computation of error estimates for Subset Simulation. Significantly, the numerical studies presented indicate that for the tested finite element problems, Metropolis–Hastings sampling can outperform Componentwise Metropolis–Hastings and Gibbs sampling. This result is relevant to the design of efficient Subset Simulation methodologies. [ABSTRACT FROM AUTHOR]

This paper presents a novel two-level finite element method for convection–diffusion–reaction problems. The proposed scheme consists of a global problem and an ensemble of local problems. The boundary conditions of the local problems are provided by a global approximation of the same problem. The stability is ensured with an artificial diffusion mechanism that acts on the difference between local and global approximation, and leads to an a priori error estimate for the global solution. For the computation of the solutions, an efficient fixed-point algorithm is proposed as an alternative to the well established static condensation technique. The local solutions can be computed in parallel and enter in a communication step. This step takes place in the entire domain and consists of a mass matrix inversion only. Therefore, the overall algorithm is easy to implement and computationally inexpensive. • A two-level finite element method is introduced and theoretically analyzed. • The solution space is as a group space with continuous and discontinuous parts. • The method is consistent, stable and provides control over the streamline derivative. • A cost-inexpensive iterative solution scheme is proposed to compute the solutions. • The proposed two-level method prevents too much numerical smoothing. [ABSTRACT FROM AUTHOR]

In Gil and Ortigosa (2016), Gil and Ortigosa introduced a new convex multi-variable framework for the numerical simulation of Electro Active Polymers (EAPs) in the presence of extreme deformations and electric fields. This extends the concept of polyconvexity to strain energies which depend on non-strain based variables. The consideration of the new concept of multi-variable convexity guarantees the well posedness of generalised Gibbs’ energy density functionals and, hence, opens up the possibility of a new family of mixed variational principles. The aim of this paper is to present, as an example, the Finite Element implementation of two of these mixed variational principles. These types of enhanced methodologies are known to be necessary in scenarios in which the simpler displacement-potential based formulation yields non-physical results, such as volumetric locking, bending and shear locking, pressure oscillations and electro-mechanical locking, to name but a few. Crucially, the use of interpolation spaces in which some of the unknown fields are described as piecewise discontinuous across elements can be used in order to efficiently condense these fields out. This results in mixed formulations with a computational cost comparable to that of the displacement-potential based approach, yet far more accurate. Finally, a series of very challenging numerical examples are presented in order to demonstrate the accuracy, robustness and efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

This paper presents a novel technique for coupling non-matching finite element meshes, based on the use of special finite elements termed coupling finite elements (CFEs), which share nodes with non-matching meshes. The main features of the proposed technique are: (i) no additional degree of freedom is introduced to the problem; (ii) non-rigid coupling can be considered to describe the nonlinear behavior of interfaces similar to cohesive models; (iii) non-matching meshes of any dimension and any type of finite elements can be coupled, and (iv) overlapping and non-overlapping meshes can be considered. The applicability of the proposed technique is illustrated by a variety of 2D and 3D examples with different non-matching mesh configurations. The results demonstrate that the technique is able to enforce the continuity of displacements in the case of rigid coupling, and to properly transfer the interaction forces across the non-matching interfaces, according to any chosen interface model, in the case of non-rigid coupling. [ABSTRACT FROM AUTHOR]

We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bES-FEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak ( W 2 ) procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf–sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM. [ABSTRACT FROM AUTHOR]

In this paper we study the numerical stability of equilibrium configurations of an elastic rod with unilateral frictionless self-contact. We use a finite-element method with penalty and augmented Lagrangian approaches to account for the self-penetration constraint. The numerical solution is carried out using the arclength continuation method. In the case where the energy expression satisfies some hypotheses, we have proved that any isolated minimum of the continuous self-contact problem is the strong limit of a sequence of local minima of the discrete problem. Our stability analysis is based on detecting the zeros of the determinant of a symmetrized version of the stiffness matrix and whether there is a sign change of the determinant or not. [ABSTRACT FROM AUTHOR]

This paper proposes a novel way to solve transient linear, and non-linear solid dynamics for compressible, nearly incompressible, and incompressible material in the updated Lagrangian framework for tetrahedral unstructured finite elements. It consists of a mixed formulation in both displacement and pressure, where the momentum equation of the continuum is complemented with a pressure equation that handles incompressibility inherently. It is obtained through the deviatoric and volumetric split of the stress, that enables us to solve the problem in the incompressible limit. A linearization of the deviatoric part of the stress is implemented as well. The Variational Multi-Scale method (VMS) is developed based on the orthogonal decomposition of the variables, which damps out spurious pressure fields for piece wise linear tetrahedral elements. Various numerical examples are presented to assess the robustness, accuracy and capabilities of our scheme in bending dominated problems, and for complex geometries. • Simulation of transient solid dynamics in the updated Lagrangian framework. • Ability to handle arbitrary geometries through tetrahedral unstructured finite elements. • Deviatoric and volumetric split of the stress to be able to handle materials in the incompressible regime. • Variational Multi-Scale stabilization for piece-wise linear finite elements to avoid pressure locking. • The framework is tested successfully on several numerical simple and complex 2D and 3D examples. [ABSTRACT FROM AUTHOR]

The paper presents an extension of the Certain Generalized Stresses Method (CGSM) for the static finite element analysis of homogeneous and laminated shells with variability. The basic assumption is that the generalized stresses do not depend on input parameters perturbation. The CGSM is a non-intrusive method that requires only one finite element analysis with some load cases to calculate the variability of mechanical quantities of interest. The uncertain input parameters are material and physical properties. Uniform random parameters as well as random fields are considered. The displacements statistical results: mean value, standard deviation and distribution are obtained by Monte Carlo simulations, using a semi-analytical formula. Two examples are treated: the Scordelis–Lo shell roof and an automotive windscreen. The method provides results of good quality and is very economical from a computational time point of view. • An extension of the CGSM for the stochastic analysis of homogeneous and laminated shells is presented. • The CGSM requires only one finite element analysis with several load cases. • The first example is the homogeneous and isotropic Scordelis–Lo shell roof. • The second example is a car windscreen, which is a sandwich structure. • The CGSM is efficient in terms of quality of results and computational time. [ABSTRACT FROM AUTHOR]