Standard Lagrangian Elements Research Articles

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.

Numerical analysis of a time domain elastoacoustic problem

AbstractThis paper deals with the numerical analysis of a system of second order in time partial differential equations modeling the vibrations of a coupled system that consists of an elastic solid in contact with an inviscid compressible fluid. We analyze a weak formulation with the unknowns in both media being the respective displacement fields. For its numerical approximation, we propose first a semidiscrete in space discretization based on standard Lagrangian elements in the solid and Raviart–Thomas elements in the fluid. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an $\mathrm L^{\infty }(\mathrm L^2)$ optimal rate of convergence under minimal regularity assumptions of the solution, which are proved to hold for appropriate data of the problem. Then we consider a fully discrete approximation based on a family of implicit finite difference schemes in time, from which we obtain optimal error estimates for sufficiently smooth solutions. Finally, we report some numerical results, which allow us to assess the performance of the method. These results also show that the numerical solution is not polluted by spurious modes as is the case with other alternative approaches.

Blending isogeometric and Lagrangian elements in three-dimensional analysis

A method for blending three-dimensional Non-Uniform Rational B-Splines (NURBS) elements and Lagrangian elements is proposed. In the blended element, selected boundary facets of a NURBS patch are presented in Lagrangian form, enabling the patch to be directly connected to standard Lagrangian elements. The transformation from NURBS to Lagrangian preserves the original geometry and thus retains the geometric exactness. The transformation is utilized to develop blended models wherein a part of the domain is described in NURBS while the remaining domain is in traditional finite elements. A simply supported beam is used as the benchmark example to test the convergence of the 3D blended element model. The utility of the blended model is also demonstrated with an example of a gear with intricate geometry. The stress solutions are compared with those of the pure finite element model and the validity of the blended analysis is demonstrated.

Comparison of pressure and displacement formulations for finite elements in linear time-harmonic acoustics

In this article, we compare the finite element solution of the pressure and the displacement formulations for the acoustic problem in the frequency domain. The Helmholtz differential equation is solved for both, the spectral and the source problems, and each of these is formulated in terms of pressure and displacement. The pressure formulation allows to use standard Lagrangian elements whereas Raviart–Thomas elements are applied for the displacement formulation. Both formulations are tested in several examples. It is shown that Raviart–Thomas elements account for a reasonable alternative to Lagrangian elements in some cases.

Three-dimensional isogeometrically enriched finite elements for frictional contact and mixed-mode debonding

We present an isogeometric enrichment technique for three-dimensional finite element computations applied to frictional contact and mixed-mode debonding. This is an extension of previous work that focused on two-dimensional and frictionless problems. To offer a more complete view of the enriched element’s performance, a comparison of the results to tri-variate isogeometric discretizations and standard Lagrangian elements is also included here. The enrichment is applied by discretizing parts of the surface that require higher accuracy with isogeometric basis functions, while the rest of the body uses Lagrangian shape functions. By using an isogeometric surface representation, the higher continuity across element boundaries and higher order of interpolation can be exploited. At the same time, the generation of tri-variate isogeometric meshes is avoided.A convergence study without any surface effects, involving only volume integrals, shows that the enriched elements can also be beneficial for these problems. The major advantage of the isogeometric element enrichment over standard tri-linear elements is demonstrated in contact problems including normal and tangential tractions. For both, mixed-mode cohesive debonding and frictional contact, the enrichment increases robustness and leads to more accurate results than standard linear Lagrangian elements. All computations are also compared to results using tri-variate isogeometric discretizations to give a complete picture of the element’s performance. It is also shown that the proposed enrichment formulation has advantages in mesh generation.

The Equivalence of Standard and Mixed Finite Element Methods in Applications to Elasto-Acoustic Interaction

Two commonly used problem formulations for the description of acoustic wave propagation are investigated; one is based on fluid displacement, and the other one is based on the velocity potential as the primary variable. Their equivalence under general Neumann boundary conditions is shown on both the continuous and the discrete level. To obtain the equivalence in the discrete setting, a nonstandard mixed finite element formulation is introduced. Thus, the transfer of an already available analysis for coupled elasto-acoustic problems in the displacement formulation to the potential formulation can be achieved. For the applications, the potential formulation is of special interest because it allows the use of standard Lagrangian elements in both the stucture and the fluid subdomain. Moreover, the approach does not require any conformity of the subdomain meshes at the interface, which considerably simplifies the physics-adapted mesh generation. Several engineering examples demonstrate the applicability and efficiency of the resulting numerical scheme.

Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations

This paper deals with the numerical solution of a system of second-order in time partial differential equations modeling the vibrations of a coupling between an elastic solid and an inviscid compressible fluid. Both media are described in terms of their respective displacement fields. The problem is solved in the time domain by combining a Newmark's scheme for time discretization with a finite element method for space discretization. The latter combines standard Lagrangian elements in the solid with lowest-order Raviart–Thomas elements in the fluid. Stability is proved and numerical results showing the good behavior of the method are reported.

Numerically stable finite element methods for the Galerkin solution of eddy current problems

Two sources of error that contribute to the instability of Galerkin solutions of vector eddy-current problems are identified. The first results from a magnification of the error in modeling or integrating the solenoidal forcing function. The second arises from contamination of the deterministic solutions by spurious modes. To obtain stable finite-element solutions of vector eddy-current problems, it is therefore necessary to model the forcing function accurately and to use vector basis functions that do not produce spurious modes. Two such sets of stable basis functions are found to be standard Lagrangian elements on a common C/sup 1/ mesh and tangential vector elements on an arbitrary mesh.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>