Finite Elements Of Arbitrary Order Research Articles

Polytopal templates for semi-continuous vectorial finite elements of arbitrary order on triangulations and tetrahedralizations

The Hilbert spaces H(curl) and H(div) are employed in various variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Seeing as the well-posedness follows automatically from the continuous setting to the discrete setting in the presence of commuting interpolants as per Fortin’s criterion, the construction of conforming subspaces becomes a crucial step in the formulation of stable numerical schemes. This work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference element via template vectors associated with the geometric polytopes of the element and an underlying H1-conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying H1-conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. We prove a linearly independent construction of Nédélec elements of the first and second type, Brezzi–Douglas–Marini elements, and Raviart–Thomas elements on triangulations and tetrahedralizations. The application of the method is demonstrated with two examples in the relaxed micromorphic model.

A monolithic finite element formulation for the hydroelastic analysis of very large floating structures

AbstractIn this work we present a novel monolithic Finite Element method for the hydroelastic analysis of very large floating structures (VLFS) with arbitrary shapes that is stable, energy conserving, and overcomes the need of an iterative algorithm. The new formulation enables a fully monolithic solution of the linear free‐surface flow, described by linear potential flow, coupled with floating thin structures, described by the Euler–Bernoulli beam or Poisson–Kirchhoff plate equations. The formulation presented in this work is general in the sense that solutions can be found in the frequency and time domains, it overcomes the need of using elements with continuity by employing a continuous/discontinuous Galerkin approach, and it is suitable for finite elements of arbitrary order. We show that the proposed approach can accurately describe the hydroelastic phenomena of VLFS with a variety of tests, including structures with elastic joints, variable bathymetry, and arbitrary structural shapes.

Stable broken 𝐻(𝑐𝑢𝑟𝑙) polynomial extensions and 𝑝-robust a posteriori error estimates by broken patchwise equilibration for the curl–curl problem

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl–curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H ( curl ) \boldsymbol H(\boldsymbol {\operatorname {curl}}) -conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.

Isoparametric finite element analysis of a generalized Robin boundary value problem on curved domains

We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator on the boundary surface. The boundary is approximated with piecewise polynomial faces and we use isoparametric finite elements of arbitrary order for the discretization. We derive optimal-order error bounds for this non-conforming finite element method in both $L^2$- and $H^1$-norm. Numerical examples illustrate the theoretical results.

Finite Element Approximation of the Minimal Eigenvalue and the Corresponding Positive Eigenfunction of a Nonlinear Sturm—Liouville Problem

The problem of finding the minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm—Liouville problem for the ordinary differential equation with coefficients nonlinear depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. A sufficient condition for the existence of a minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm— Liouville problem is established. The original differential eigenvalue problem is approximated by the finite element method with Lagrangian finite elements of arbitrary order on a uniform grid. The error estimates of the approximate eigenvalue and the approximate positive eigenfunction to exact ones are proved. Investigations of this paper generalize well known results for the Sturm—Liouville problem with linear entrance on the spectral parameter.

Error of the Finite Element Approximation for a Differential Eigenvalue Problem with Nonlinear Dependence on the Spectral Parameter

The positive definite ordinary differential nonlinear eigenvalue problem of the second order with homogeneous Dirichlet boundary condition is considered. The problem is formulated as a symmetric variational eigenvalue problem with nonlinear dependence of the spectral parameter in a real infinite-dimensional Hilbert space. The variational eigenvalue problem consists in finding eigenvalues and corresponding eigenfunctions of the eigenvalue problem for a symmetric positive definite bounded bilinear form with respect to a symmetric positive definite completely continuous bilinear form in a real infinite-dimensional Hilbert space. The variational eigenvalue problem is approximated by the mesh scheme of the finite element method on the uniform grid. For constructing the mesh scheme, Lagrangian finite elements of arbitrary order are applied. Error estimates of approximate eigenvalues and error estimates of approximate eigenfunctions in the norm of initial real infinite-dimensional Hilbert space are established. These error estimates coincide in the order with error estimates of mesh scheme of the finite element method for linear eigenvalue problems. Moreover, superconvergence estimates for approximate eigenfunctions in the mesh norm with Gauss quadrature nodes are derived. Investigations of this paper generalize well known results for the eigenvalue problem with linear entrance on the spectral parameter.

On a general implementation of h- and p-adaptive curl-conforming finite elements

Edge (or Nédélec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, especially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR . We cover into detail how to implement arbitrary order (i.e., p-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., h-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation.

Error Investigation of a Finite Element Approximation for a Nonlinear Sturm–Liouville Problem

A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter is studied. The problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms depending nonlinearly on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues that correspond to a normalized system of eigenfunctions. The variational problem is approximated by a finite element mesh scheme on a uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are proved depending on the mesh size and the eigenvalue size. The results obtained are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.

A partition of unity finite element method for computational diffusion MRI

The Bloch–Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries.Many approaches have been taken to solve the Bloch–Torrey equation but an efficient high performance computing framework is still missing. In this paper, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations are based on the partition of unity concept which allows for a discontinuous solution across interfaces conforming with the mesh with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as an operator splitting for the exterior boundary conditions. The method is straightforward to implement and it is available in FEniCS for moderate-scale simulations and in FEniCS-HPC for large-scale simulations. The order of accuracy of the resulting method is validated in numerical tests and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available and we performed simulations for a realistic model of a neuron to show that the method can be used for complex geometries.

Bernstein–Bézier bases for tetrahedral finite elements

We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2-conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1-conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H1-conforming Bernstein–Bézier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis.Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piecewise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems.

Quadrature finite element method for elliptic eigenvalue problems

A positive semi-definite eigenvalue problem for second-order self-adjoint elliptic differential operator definedon a bounded domain in the planewith smooth boundary and Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange curved triangular finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are established.

On variational formulations with rigid-body constraints for finite elasticity: Applications to 2D and 3D finite element simulations

We investigate a recently proposed variational principle with rigid-body constraints and present an extension of its implementation in three dimensional finite elasticity problems. Through numerical examples, we illustrate that the proposed variational principle with rigid-body constraints is applicable to both single field and mixed finite elements of arbitrary order and geometry, e.g. triangular/tetrahedral and quadrilateral/hexagonal elements, in two and three dimensions. Moreover, we demonstrate that, as compared to the commonly adopted approach of discretizing the rigid domains with standard finite elements, the proposed formulation requires neither discretization nor numerical integration in the interior of each rigid domain. As a comparative result, the variational formulation may reduce the total number of degrees of freedom of the resulting finite element system and provide improved accuracy.

Polygonal Finite Elements of Arbitrary Order

Finite elements with the shape of arbitrary polygons have been previously described. Originally they were first-order, i.e., able to represent exactly all polynomials of degree 1 in the space coordinates, but recently polygonal finite elements (PFEs) up to order 3 have been reported. Here, we propose a general theory for generating PFEs of arbitrary order. They are hierarchical, allowing mixing of orders. Results for a wave problem and two magnetic field problems show the effectiveness of the elements up to order 5.

A variational formulation with rigid-body constraints for finite elasticity: theory, finite element implementation, and applications

This paper presents a new variational principle in finite elastostatics applicable to arbitrary elastic solids that may contain constitutively rigid spatial domains (e.g., rigid inclusions). The basic idea consists in describing the constitutive rigid behavior of a given spatial domain as a set of kinematic constraints over the boundary of the domain. From a computational perspective, the proposed formulation is shown to reduce to a set of algebraic constraints that can be implemented efficiently in terms of both single-field and mixed finite elements of arbitrary order. For demonstration purposes, applications of the proposed rigid-body-constraint formulation are illustrated within the context of elastomers, reinforced with periodic and random distributions of rigid filler particles, undergoing finite deformations.

Gradient correction for polygonal and polyhedral finite elements

SummaryPrevious studies have shown that the commonly used quadrature schemes for polygonal and polyhedral finite elements lead to consistency errors that persist under mesh refinement and subsequently render the approximations non‐convergent. In this work, we consider minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover optimal convergence rates when the weak form integrals are evaluated using quadrature. For finite elements of arbitrary order, we state the accuracy requirements on the underlying volumetric and boundary quadrature rules and discuss the properties of the resulting corrected gradient operator. We compare the proposed approach with the pseudo‐derivative method developed by Belytschko and co‐workers and, for linear elliptic problems, with our previous remedy that involves splitting of polynomial and non‐polynomial of elemental energy bilinear form. We present several numerical results for linear and nonlinear elliptic problems in two and three dimensions that not only confirm the recovery of optimal convergence rates but also suggest that the global error levels are close to those of approximations obtained from exact evaluation of the weak form integrals. Copyright © 2015 John Wiley & Sons, Ltd.

Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Cea lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an $$H^1$$H1-type Finsler norm and with the $$H^1$$H1-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem.

Higher-order linked interpolation in triangular thick plate finite elements

Purpose – In this work, the authors aim to employ the so-called linked-interpolation concept already tested on beam and quadrilateral plate finite elements in the design of displacement-based higher-order triangular plate finite elements and test their performance. Design/methodology/approach – Starting from the analogy between the Timoshenko beam theory and the Mindlin plate theory, a family of triangular linked-interpolation plate finite elements of arbitrary order are designed. The elements are tested on the standard set of examples. Findings – The derived elements pass the standard patch tests and also the higher-order patch tests of an order directly related to the order of the element. The lowest-order member of the family of developed elements still suffers from shear locking for very coarse meshes, but the higher-order elements turn out to be successful when compared to the elements from literature for the problems with the same total number of the degrees of freedom. Research limitations/implications – The elements designed perform well for a number of standard benchmark tests, but the well-known Morley's skewed plate example turns out to be rather demanding, i.e. the proposed design principle cannot compete with the mixed-type approach for this test. Work is under way to improve the proposed displacement-based elements by adding a number of internal bubble functions in the displacement and rotation fields, specifically chosen to satisfy the basic patch test and enable a softer response in the bench-mark test examples. Originality/value – A new family of displacement-based higher-order triangular Mindlin plate finite elements has been derived. The higher-order elements perform very well, whereas the lowest-order element requires improvement.

Anisotropic “Goal-Oriented” Mesh Adaptivity for Elliptic Problems

We propose in this paper an anisotropic, adaptive, finite element algorithm for steady, linear advection-diffusion-reaction problems with strong anisotropic features. The error analysis is based on the dual weighted residual methodology, allowing us to perform “goal-oriented” adaptation of a certain functional $J(u)$ of the solution and derive an “optimal” metric tensor for local mesh adaptation with linear and quadratic finite elements. As a novelty, and to evaluate the weights of the error estimator on unstructured meshes composed of anisotropic triangles, we make use of a patchwise, higher-order interpolation recovery readily extendable to finite elements of arbitrary order. We carry out a number of numerical experiments in two dimensions so as to prove the capabilities of the goal-oriented adaptive method. We compute the convergence rate and the effectivity index for a series of output functionals of the solution. The results show the good performance of the algorithm with linear as well as quadratic ...

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.

Optimally adapted meshes for finite elements of arbitrary order and W 1, p norms

Given a function f defined on a bounded polygonal domain $${\Omega \subset \mathbb{R}^2}$$ and a number N > 0, we study the properties of the triangulation $${\mathcal{T}_N}$$ that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the W 1, p semi-norm for 1 ≤ p < ∞, and we consider Lagrange finite elements of arbitrary polynomial order m − 1. We establish sharp asymptotic error estimates as N → +∞ when the optimal anisotropic triangulation is used. A similar problem has been studied in Babenko et al. (East J Approx. 12(1):71–101, 2006), Cao (J Numer Anal. 45(6):2368–2391, 2007), Chen et al. (Math Comput. 76:179–204, 2007), Cohen (Multiscale, Nonlinear and Adaptive Approximation. Springer, Berlin, 2009), Mirebeau (Constr Approx. 32(2):339–383, 2010), but with the error measured in the L p norm. The extension of this analysis to the W 1, p norm is required in order to match more closely the needs of numerical PDE analysis, and it is not straightforward. In particular, the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle, a second feature that does not appear for the L p error norm. Our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant.