A stable nonconforming finite element on hexahedra

A new nonconforming brick element is introduced, which only has 13 DOFs locally and takes as its shape functions space. The vector‐valued version generates, together with a discontinuous approximation, an inf‐sup stable finite element pair of order 2 for the Stokes problem in the energy norm. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust and efficient monolithic fluid‐structure‐interaction solvers

We construct new robust and efficient preconditioned generalized minimal residual solvers for the monolithic linear systems of algebraic equations arising from the finite element discretization and Newton's linearization of the fully coupled fluid–structure interaction system of partial differential equations in the arbitrary Lagrangian–Eulerian formulation. We admit both linear elastic and nonlinear hyperelastic materials in the solid model and cover a large range of flows, for example, water, blood, and air, with highly varying density. The preconditioner is constructed in form of , where , , and are proper approximations to the matrices L, D, and U in the LDU block factorization of the fully coupled system matrix, respectively. The inverse of the corresponding Schur complement is approximated by applying a few cycles of a special class of algebraic multigrid methods to the perturbed fluid sub‐problem, which is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation to the exact perturbation coming from the sparse matrix–matrix multiplications. The numerical studies presented impressively demonstrate the robustness and the efficiency of the preconditioner proposed in the paper. Copyright © 2016 John Wiley & Sons, Ltd.     

A second‐order decoupled implicit/explicit method of the 3D primitive equations of ocean II: finite element spatial discretization

A fully discrete second‐order decoupled implicit/explicit method is proposed for solving 3D primitive equations of ocean in the case of Dirichlet boundary conditions on the side, where a second‐order decoupled implicit/explicit scheme is used for time discretization, and a finite element method based on the P₁(P₁) ? P₁?P₁(P₁) elements for velocity, pressure and density is used for spatial discretization of these primitive equations. Optimal H¹?L²?H¹ error estimates for numerical solution and an optimal L² error estimate for are established under the convergence condition of 0 < h≤β₁,0 < τ≤β₂, and τ≤β₃h for some positive constants β₁,β₂, and β₃. Furthermore, numerical computations show that the H¹?L²?H¹ convergence rate for numerical solution is of O(h + τ²) and an L² convergence rate for is O(h²+τ²) with the assumed convergence condition, where h is a mesh size and τ is a time step size. More practical calculations are performed as a further validation of the numerical method. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient adaptive integration of functions with sharp gradients and cusps in n‐dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a C⁰ function (derivative discontinuity at a point), a rate of convergence of n + 1 is obtained in . Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function—a strongly localized function that is used for modeling sharp gradients. Then the adaptive quadratures are employed in the enriched finite element solution of the all‐electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n‐dimensional parallelepiped elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method

The method of weighted residuals can efficiently enforce time‐periodic solutions of flexible structures experiencing unilateral contact. The harmonic balance method (HBM) based on Fourier expansion of the sought solution is a common formulation, although wavelet bases that can sparsely define nonsmooth solutions may be superior. This hypothesis is investigated using a full three‐dimensional blade with unilateral contact conditions on a set of N_c discrete contact points located at its tip. The unilateral contact conditions are first regularized, and a distributional formulation in time is introduced, allowing trial functions to properly approximate in the time domain the solution to the governing equations. The mixed wavelet Petrov–Galerkin solutions are found to yield consistent or better results than HBM, with higher convergence rates and seemingly more accurate contact force prediction. Copyright © 2014 John Wiley & Sons, Ltd.     

Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub‐triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the in case of the conventional polygonal FEM, while it scales as in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Copyright © 2015 John Wiley & Sons, Ltd.     

Extended finite element method for plastic limit load computation of cracked structures

The extended finite element method is extended to allow computation of the limit load of cracked structures. In the paper, it is demonstrated that the linear elastic tip enrichment basis with and without radial term may be used in the framework of limit analysis, but the six‐function enrichment basis based on the well‐known Hutchinson–Rice–Rosengren asymptotic fields appears to be the best. The discrete kinematic formulation is cast in the form of a second‐order cone problem, which can be solved using highly efficient interior‐point solvers. Finally, the proposed numerical procedure is applied to various benchmark problems, showing that the present results are in good agreement with those in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Geodesic finite elements on simplicial grids

We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v : Ω → M, where Ω is an open subset of and M is a Riemannian manifold. These geodesic finite elements naturally generalize standard first‐order finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for d = 1 in a previous publication of the author. Our formulation is equivariant under isometries of M and, hence, preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds Mⁿ. These can be solved efficiently using a Riemannian trust‐region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example, we numerically compute harmonic maps from a domain in to S². Copyright © 2012 John Wiley & Sons, Ltd.     

An explicit residual‐type error estimator for ‐quadrilateral extended finite element method in two‐dimensional linear elastic fracture mechanics

This paper deals with explicit residual a posteriori error estimation analysis for ‐quadrilateral extended finite element method (XFEM) discretizations applied to the two‐dimensional problem of linear elastic fracture mechanics. The result is twofold. First, to enable estimation procedures with application to XFEM, a specific quasi‐interpolation operator of averaging type is constructed. The main challenge here arises from the different types of enrichments implemented, and hence, to impose the constant‐preserving property of the interpolation operator on an element, we use the idea of an extension operator. An upper bound on the discretization error measured in the energy norm and associated local error indicators are then constructed and analyzed. The second result follows from the error analysis and concerns an alternative choice of branch functions used in XFEM applications. In particular, the branch functions have to be chosen to fulfill the divergence‐free conditions within the crack tip element and traction‐free boundary conditions on the crack faces. Then, the corresponding XFEM solution gains a better accuracy with less degrees of freedom. Finally, numerical examples are provided with comparative results. Copyright © 2012 John Wiley & Sons, Ltd.     

An interpolation‐based fast multipole method for higher‐order boundary elements on parametric surfaces

In this article, a black‐box higher‐order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher‐order methods is not limited by approximation errors of the surface. An element‐wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either ‐ or ‐matrices. In fact, several simplifications in the construction of ‐matrices are pointed out, which are a by‐product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational homogenization of soft matter friction: Isogeometric framework and elastic boundary layers

A computational contact homogenization framework is established for the modeling and simulation of soft matter friction. The main challenges toward the realization of the framework are (1) the establishment of a frictional contact algorithm that displays an optimal combination of accuracy, efficiency, and robustness and plays a central role in (2) the construction of a micromechanical contact test within which samples of arbitrary size may be embedded and which is not restricted to a single deformable body. The former challenge is addressed through the extension of mixed variational formulations of contact mechanics to a mortar‐based isogeometric setting where the augmented Lagrangian approach serves as the constraint enforcement method. The latter challenge is addressed through the concept of periodic embedding, with which a periodically replicated ‐continuous interface topography is realized across which not only pending but also ensuing contact among simulation cells will be automatically captured. Two‐dimensional and three‐dimensional investigations with unilateral/bilateral periodic/random roughness on two elastic micromechanical samples demonstrate the overall framework and the nature of the macroscopic frictional response. Copyright © 2014 John Wiley & Sons, Ltd.     

Dispersion and isogeometric analyses of second‐order and fourth‐order implicit gradient‐enhanced plasticity models

Implicit gradient plasticity models incorporate higher‐order spatial gradients via an additional Helmholtz type equation for the plastic multiplier. So far, the enrichment has been limited to second‐order spatial gradients, resulting in a formulation that can be discretised using ‐continuous finite elements. Herein, an implicit gradient plasticity model is formulated that includes a fourth‐order gradient term as well. A comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second‐order gradient plasticity model is made using a dispersion analysis. The higher‐order continuity requirement for the fourth‐order implicit gradient plasticity model has been met by exploiting the higher‐order continuity property of isogeometric analysis, which uses nonuniform rational B‐splines as shape functions instead of Lagrange polynomials. The discretised variables, displacements, and plastic multiplier may require different orders of interpolation, an issue that is also addressed. Numerical results show that both formulations can be used as a localisation limiter, but that quantitative differences occur, and a different evolution of the localisation band is obtained for 2‐dimensional problems.     

Nonconforming rotated Q1 tetrahedral element with explicit time stepping for elastodynamics

In this paper, we apply a rotated bilinear tetrahedral element to elastodynamics in . This element performs superior to the constant strain element in bending and, unlike the conforming linear strain tetrahedron, allows for row‐sum lumping of the mass matrix. We study the effect of different choices of approximation (pointwise continuity versus edge average continuity) as well as lumping versus consistent mass in the setting of eigenvibrations. We also use the element in combination with the leapfrog method for time domain computations and make numerical comparisons with the constant strain and linear strain tetrahedra. Copyright © 2012 John Wiley & Sons, Ltd.     

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

We consider an efficient preconditioner for a boundary integral equation (BIE) formulation of the two‐dimensional Stokes equations in porous media. While BIEs are well‐suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of ‐matrices and uses low‐rank compressions to approximate certain matrix blocks. It has a tunable accuracy ε and a computational cost that scales as . We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block‐diagonal preconditioner, especially for pipe flow problems involving many pores.     

Establishing stable time‐steps for DEM simulations of non‐collinear planar collisions with linear contact laws

The discrete element method, developed by Cundall and Strack, typically uses some variations of the central difference numerical integration scheme. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time‐step. The current methods for estimating appropriate discrete element method time‐steps are based on many assumptions; therefore, large factors of safety are usually applied to the time‐step to ensure stability, which substantially increases the computational cost of a simulation. This work introduces a general framework for estimating critical time‐steps for any planar rigid body subject to linear damping and forcing. A numerical investigation of how system damping, coupled with non‐collinear impact, affects the critical time‐step is also presented. It is shown that the critical time‐step is proportional to if a linear contact model is adopted, where m and k represent mass and stiffness, respectively. The term which multiplies this factor is a function of known physical parameters of the system. The stability of a system is independent of the initial conditions. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

Universal meshes for smooth surfaces with no boundary in three dimensions

We introduce a method to mesh the boundary Γ of a smooth, open domain in immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh 1 for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.     

An algorithm for triangulating smooth three-dimensional domains immersed in universal meshes

We describe an algorithm to recover a boundary-fitting triangulation for a bounded C²-regular domain immersed in a nonconforming background mesh of tetrahedra. The algorithm consists in identifying a polyhedral domain ω_h bounded by facets in the background mesh and morphing ω_h into a boundary-fitting polyhedral approximation Ω_h of Ω. We discuss assumptions on the regularity of the domain, on element sizes and on specific angles in the background mesh that appear to render the algorithm robust. With the distinctive feature of involving just small perturbations of a few elements of the background mesh that are in the vicinity of the immersed boundary, the algorithm is designed to benefit numerical schemes for simulating free and moving boundary problems. In such problems, it is now possible to immerse an evolving geometry in the same background mesh, called a universal mesh, and recover conforming discretizations for it. In particular, the algorithm entirely avoids remeshing-type operations and its complexity scales approximately linearly with the number of elements in the vicinity of the immersed boundary. We include detailed examples examining its performance.     

A new multi‐scale dispersive gradient elasticity model with micro‐inertia: Formulation and ‐finite element implementation

Motivated by nano‐scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi‐scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro‐inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain‐gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro‐inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth‐order equations are rewritten in two sets of symmetric second‐order equations so that ‐continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi‐scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

High‐order space‐time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space‐time coupling matrices are diagonalizable over for r ?100, and this means that the time‐coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG‐in‐time methodology, for the first time, to second‐order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high‐order (up to degree 7) temporal and spatio‐temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.