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This paper presents a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche's method which was initially designed for Dirichlet's condition. A main interesting characteristic is that this approximation produces well-posed space semi-discretizations contrary to standard finite element discretizations. This paper is then mainly devoted to present an analysis of the space semi-discretization in terms of consistency, well-posedness and energy conservation, and also to study the well-posedness of some time-marching schemes (θ-scheme, Newmark and a new hybrid scheme). The stability properties of the schemes and the corresponding numerical experiments can be found in a second paper [F. Chouly, P. Hild and Y. Renard, A Nitsche finite element method for dynamic contact. 2. Stability analysis and numerical experiments. ESAIM: M2AN 49 (2015) 503-528.]. [ABSTRACT FROM AUTHOR]

In this paper we establish a best approximation property of fully discrete Galerkin solutions of second-order parabolic problems on convex polygonal and polyhedral domains in the L∞(I;W1,∞(Ω)) norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors [D. Leykekhman and B. Vexler, Numer. Math., 135 (2017), pp. 923-952]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish an interior best approximation property that shows more local dependence of the error at a point. [ABSTRACT FROM AUTHOR]

Predicting the mechanical response of components requires simplifications and idealizations that affect the fidelity of the results and introduce errors. Some errors correspond to the limited knowledge of intrinsic physical attributes while others are introduced by the modeling framework and mathematical approximations. This paper studies the dependence of the force-displacement response of threaded fasteners on modeling attributes such as geometry, material, and friction resistance using finite element simulations. A systematic comparison of 1 D , 2.5 D or 3 D computational models demonstrates the influence of model properties and the limitations of the methodologies. Finally, the paper discusses the sources of model inputs and model form errors for threaded fasteners. [ABSTRACT FROM AUTHOR]

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element. [ABSTRACT FROM AUTHOR]

The paper is concerned with a low-order finite element method, namely the staggered cell-centered finite element method, which has been proposed and analyzed in Ong et al. (2015) for two-dimensional compressible and nearly incompressible linear elasticity problems. In this work, we extend the results to the three-dimensional case and focus on the creating of the meshes. In particular, from a general primal mesh M , we construct a polygonal dual mesh M ∗ and its submesh M ∗ ∗ in a way such that each dual control volume of M ∗ corresponds to a primal vertex and is a union (macro-element) of some fixed number of adjacent tetrahedral elements of M ∗ ∗ . The displacement is approximated by piecewise trilinear functions on the subdual mesh M ∗ ∗ and the pressure by piecewise constant functions on the dual mesh M ∗ . As for two-dimensional case, such construction of the meshes and approximation spaces satisfies the macroelement condition, which implies stability and convergence of the scheme. Numerical experiments are carried out to investigate the performance of the proposed method on various mesh types. [ABSTRACT FROM AUTHOR]

We consider the approximation of elliptic eigenvalue problem with an interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix–Raviart P 1 -nonconforming approximation. We show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem. We analyze the IFEM for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues. Numerical experiments demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper discusses classical and refined beam and plate theories based on the Carrera Unified Formulation (CUF). Attention is focussed on (but not limited to) a new refined beam element with enhanced kinematics based on Legendre polynomial expansions of the primary mechanical variables. By employing CUF, the governing equations and the related finite element arrays are written in a hierarchical, compact and general manner. Readily, these characteristics are used to arbitrarily tune the finite element model at the cross-sectional level, by locally enriching the theory kinematics up to the desired accuracy. The uncompromising accuracy of the present beam model is demonstrated by considering various numerical examples, including solid and thin-walled beams with open and close cross-sections as well as plate structures. The results are compared with those from classical and already established refined CUF models. Eventually, three-dimensional elasticity solutions by the commercial tool MSC Nastran are also given to underline the high accuracy of the present methodology. The numerical efficiency and the capabilities of the Legendre-based CUF beam models to deal with complex structures with no geometrical approximations result clear from the analyses conducted. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a finite element method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. We introduce an auxiliary variable w : = Δ u (with u representing the displacement of the plate) to write a variational formulation of the spectral problem. We propose a conforming discretization of the problem, where the unknowns are approximated by piecewise linear and continuous finite elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments supporting our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrange multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations. [ABSTRACT FROM AUTHOR]

In the present paper we consider a 1D Poisson model characterized by the presence of an interface, where a transmission condition arises due to jumps of the coefficients. We aim at studying finite element methods with meshes not fitting such an interface. It is well known that when the mesh does not fit the material discontinuities the resulting scheme provides in general lower order accurate solutions. We focus on so-called embedded approaches, frequently adopted to treat fluid–structure interaction problems, with the aim of recovering higher order of approximation also in presence of non fitting meshes; we implement several methods inspired by: the Immersed Boundary method, the Fictitious Domain method, and the Extended Finite Element method. In particular, we present four formulations in a comprehensive and unified format, proposing several numerical tests and discussing their performance. Moreover, we point out issues that may be encountered in the generalization to higher dimensions and we comment on possible solutions. [ABSTRACT FROM AUTHOR]

In this paper, we focus on efficient second-order in time approximations of the Allen–Cahn and Cahn–Hilliard equations. First of all, we present the equations, generic second-order schemes (based on a mid-point approximation of the diffusion term) and some schemes already introduced in the literature. Then, we propose new ways of deriving second-order in time approximations of the potential term (starting from the main schemes introduced in Guillén-González and Tierra (2013)), yielding to new second-order schemes. For these schemes and other second-order schemes previously introduced in the literature, we study the constraints on the physical and discrete parameters that can appear to assure the energy-stability, unique solvability and, in the case of nonlinear schemes, the convergence of Newton’s method to the nonlinear schemes. Moreover, in order to save computational cost we have developed a new adaptive time-stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step. Finally, we compare the behaviour of the schemes and the effectiveness of the adaptive time-stepping algorithm through several computational experiments. [ABSTRACT FROM AUTHOR]

In this work, we study, from the numerical point of view, a dynamic thermoviscoelastic problem involving micropolar materials. The model leads to a linear system composed of parabolic partial differential equations for the displacements, the microrotation and the temperature. Its weak form is written as a linear system made of first-order variational equations, in terms of the velocity field, the microrotation speed and the temperature. Fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence is derived under some additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution. [ABSTRACT FROM AUTHOR]

Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a 'kinetic under-cooling' regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions. [ABSTRACT FROM AUTHOR]

While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC finite element approach based on deterministic adaptive mesh refinement for the arising "pathwise" problems and outline a convergence theory in terms of desired accuracy and required computational cost. Our theoretical and heuristic reasoning together with the efficiency of our new approach are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we numerically study a dynamic viscoelastic problem. The variational formulation is written as a linear parabolic variational equation for the velocity field. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced using the implicit Euler scheme and the finite element method, for which some a priori error estimates are derived, leading to the linear convergence of the algorithm under suitable additional regularity conditions. Finally, some one- and three-dimensional numerical simulations are presented to show the accuracy of the algorithm and the behaviour of the solution, including a comparison with an experimental study. [ABSTRACT FROM AUTHOR]

We introduce a residual-based a posteriori error estimator for contact problems in two- and threedimensional linear elasticity, discretized with linear and quadratic finite elements and Nitsche's method. Efficiency and reliability of the estimator are proved under a saturation assumption. Numerical experiments illustrate the theoretical properties and the good performance of the estimator. [ABSTRACT FROM AUTHOR]

We introduce a new finite element method for the approximation of the three-dimensional Brinkman problem formulated in terms of the velocity, vorticity and pressure fields. The proposed strategy exhibits the advantage that, at the continuous level, a complete decoupling of vorticity and pressure can be established under the assumption of sufficient regularity. The velocity is then obtained as a simple postprocess from vorticity and pressure, using the momentum equation. Well-posedness follows straightforwardly by the Lax-Milgram theorem. The Galerkin scheme is based on Nédélec and piecewise continuous finite elements of degree k≥1 for vorticity and pressure, respectively. The discrete setting uses the very same ideas as in the continuous case, and the error analysis for the vorticity scheme is carried out first. As a byproduct of these error bounds and the problem decoupling, the convergence rates for the pressure and velocity are readily obtained in the natural norms with constants independent of the viscosity. We also present details about how the analysis of the method is modified for axisymmetric, meridian Brinkman flows; and modify the decoupling strategy to incorporate the case of Dirichlet boundary conditions for the velocity. A set of numerical examples in two and three spatial dimensions illustrate the robustness and accuracy of the finite element method, as well as its competitive computational cost compared with recent fully mixed and augmented formulations of incompressible flows. [ABSTRACT FROM AUTHOR]

The automatic generation of meshes for the finite element (FE) method can be an expensive computational burden, especially in structural problems with localized stress peaks. The use of meshless methods can address such an issue, as these techniques do not require the existence of an underlying connection among the particles selected in a general domain. This study advances a numerical strategy that blends the FE method with the meshless local Petrov-Galerkin technique in structural mechanics, with the aim at exploiting the most attractive features of each procedure. The idea relies on the use of FEs to compute a background solution that is locally improved by enriching the approximation space with the basis functions associated to a few meshless points, thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid. Adding the meshless particles only where needed avoids the cost of mesh refining, or even of remeshing, without the prohibitive computational cost of a thoroughly meshfree approach. In the present implementation, an efficient integration strategy for the computation of the coefficients taking into account the mutual FE-meshless local Petrov-Galerkin interactions is introduced. Moreover, essential boundary conditions are enforced separately on both FEs and meshless particles, thus allowing for an overall accuracy improvement also when the enriched region is close to the domain boundary. Numerical examples in structural problems show that the proposed approach can significantly improve the solution accuracy at a local level, with no remeshing effort, and at a low computational cost. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of Darcy equations in a fractured domain, with highly discontinuous permeability tensors. In order to avoid the meshing of the fracture, we work with the asymptotic model obtained as the fracture’s thickness goes toand the fracture’s permeability goes to infinity in the weak formulation. This leads to treat a non-standard transmission condition between the two domains delimited by the fracture, instead of solving the equations in the fracture. Then, we propose an approximation of the asymptotic model based on Nitsche’s extended finite element method, which allows to consider fractures not aligned with the mesh. We prove consistency and stability of the discrete formulation and we present some numerical tests showing the relevance of the method. [ABSTRACT FROM AUTHOR]

This is a study of the well-posedness and asymptotic stability of a "degenerately damped" PDE that models a vibrating elastic string. The coefficient of the damping may vanish at small amplitudes thus weakening the effect of the dissipation. It is shown that the resulting dynamical system has strictly monotonically decreasing energy and uniformly decaying lower-order norms of the solutions, however, it is not uniformly stable on the associated finite-energy space. These theoretical findings were motivated by numerical simulations of this model using an FEM scheme and successive approximations. A description of the numerical approach and sample plots of energy decay are supplied. In addition, for certain initial data the solution can be determined in closed form up to a dissipative nonlinear ordinary differential equation. These special solutions can be used to assess the accuracy of the numerical examples. [ABSTRACT FROM AUTHOR]

We consider a fully practical finite element approximation of the nonlinear parabolic Cahn-Hilliard system [ABSTRACT FROM AUTHOR]

We present a parametric finite element approximation of two-phase flow with insoluble surfactant. This free boundary problem is given by the Navier-Stokes equations for the two-phase flow in the bulk, which are coupled to the transport equation for the insoluble surfactant on the interface that separates the two phases.We combine the evolving surface finite element method with an approach previously introduced by the authors for two-phase Navier-Stokes flow, which maintains good mesh properties. The derived finite element approximation of two-phase flow with insoluble surfactant can be shown to be stable. Several numerical simulations demonstrate the practicality of our numerical method. [ABSTRACT FROM AUTHOR]