Fourth-order Elements Research Articles

A spectral element-based phase field method for incompressible two-phase flows

In this paper, we propose a spectral element-based phase field method by solving the Navier–Stokes/Cahn–Hilliard equations for incompressible two-phase flows. The high-order differential nonlinear term of the Cahn–Hilliard equation poses a great challenge for obtaining numerical solutions, so the Newton–Raphson method is adopted to tackle this conundrum. Furthermore, we employ the time-stepping scheme to decouple the Navier–Stokes equations to favor the computations with large density and viscosity contrast, in terms of both stability and convergence efficiency. In addition, the continuum surface tension model is used to account for the tangential component of the interfacial force, and thus our numerical method has the ability to simulate thermocapillary flows. We present four examples to demonstrate the interface capture accuracy of the proposed method. The shape of the rotating Zalesak's disk is accurately preserved by the present method even for two periods, which implies less dissipation and higher accuracy at long time numerical simulation. It is also noted that the present method with fourth-order element can achieve similar accuracy with the consistent scheme by evaluating the convective fluxes with the fifth-order weighted essentially non-oscillatory scheme. Moreover, the proposed method appears to comply well with mass conservation. And the results of thermocapillary flow test show good agreement with theoretical prediction. Finally, the rising bubble and Rayleigh–Taylor instability are considered to verify the proposed method for complex changes in interfacial topology, as well as its performance under large density and viscosity contrasts and high Reynolds number conditions.

Dynamic behaviour of thick plates resting on Winkler foundation with fourth order element

This paper focuses on the study of dynamic analysis of thick plates resting on Winkler foundation. The governing equation is derived from Mindlin\'s theory. This study is a parametric analysis of the reflections of the thickness / span ratio, the aspect ratio and the boundary conditions on the earthquake excitations are studied. In the analysis, finite element method is used for spatial integration and the Newmark-B method is used for the time integration. While using finite element method, a new element is used. This element is 17-noded and it\'s formulation is derived from using higher order displacement shape functions. C++ program is used for the analyses. Graphs are presented to help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that the 17-noded finite element is used in the earthquake analysis of thick plates. It is shown that the changes in the aspect ratio are more effective than the changes in the aspect ratio. The center displacements of the reinforced concrete thick clamped plates for b/a=1, and t/a=0.2, and for b/a=2, and t/a=0.2, reached their absolute maximum values of 0.00244 mm at 3.48 s, and of 0.00444 mm at 3.48 s, respectively.

Choosing the most stable members of Kou’s family of iterative methods

In this manuscript, we analyze the dynamical anomalies of a parametric family of iterative schemes designed by Kou et al. It is known that its order of convergence is three for any arbitrary value of the parameter, but it has order four (and it is optimal in the sense of Kung–Traub’s conjecture) when a specific value is selected. Among all the elements of this family, one can choose this fourth-order element or any of the infinite members of third order of convergence, if only the speed of convergence is considered. However, the stability of the methods plays an important role in their reliability when they are applied on different problems. This is the reason why we analyze in this paper the dynamical behavior on quadratic polynomials of the mentioned family. The study of fixed points and their stability, joint with the critical points and their associated parameter planes, show the richness of the class and allow us to find members of it with excellent numerical properties, as well as other ones with very unstable behavior. Some test functions are analyzed for confirming the theoretical results.

Architecture of Glaciotectonic Complexes

Glaciotectonic studies are an integrated part of the Quaternary geological research carried out by the Danish geological survey. Almost all the hilly areas in Denmark were created or affected by glaciotectonic deformations, and the features are included in the mapping of surface near deposits. For the mapping and support of constructing 3D geological models a classification of architecture of glaciotectonic complexes is suggested. The important elements for classification of architecture are the surfaces. Four orders of surfaces are defined for glaciotectonic complexes: first-order surfaces are décollement surfaces and glaciotectonic unconformities; second-order surfaces are ramps and flats—the thrust faults; third-order surfaces are folded beds—anticlines and synclines; and fourth-order surfaces are small scale folds and faults—kink bands, conjugate faults, box folds, etc. The most important first-order surface is the décollement surface. This surface limits the glaciotectonic complex at its base and controls the extent of glaciotectonic disturbances. Below this surface, ordinary flat lying planar bedding occurs, whereas above the surface a number of structures are present characteristic of second- to fourth-order elements in the glaciotectonic architecture. The other first-order surface is the topographic top of the glaciotectonic complex, which eventually may be replaced by a truncating unconformity.

Computation of three-dimensional standing water waves

We develop a method for computing three-dimensional gravity-driven water waves, which we use to search for time-periodic standing wave solutions. We simulate an inviscid, irrotational, incompressible fluid bounded below by a flat wall, and above by an evolving free surface. The computations make use of spectral derivatives on the surface, but also require computing a velocity potential in the bulk, which we carry out using a finite element method with fourth-order elements that are curved to match the free surface. This computationally expensive step is solved using a parallel multigrid algorithm, which is discussed in detail. Time-periodic solutions are searched for using a previously developed overdetermined shooting method. Several families of large-amplitude three-dimensional standing waves are found in both shallow and deep regimes, and their physical characteristics are examined and compared to previously known two-dimensional solutions.

Comparison of hierarchical basis functions for efficient multilevel solvers

V-cycle and non-standard W-cycle versions of a multiplicative Schwarz preconditioner based on hierarchical basis functions are presented. It is used together with a Krylov subspace method to efficiently solve the linear system that results from higher-order finite element (FE) discretisations of time-harmonic Maxwell equations. A recently developed hierarchical basis for H(curl)-conforming (vector-valued and tangentially continuous) FE spaces on tetrahedral meshes is also briefly presented. On this basis, a certain amount of orthogonality between basis functions of different orders is obtained through the requirement that the Nédélec interpolation of higher-order basis functions vanishes in lower-order FE spaces. Numerical experiments are used to show the good performance of the presented schemes. In these experiments, the performance obtained with the presented basis is compared with the performance obtained by several other hierarchical bases found in the literature. For third-order elements, it is observed that most bases give very similar performance and that the V-cycle preconditioner typically requires about 30% more computing time than the W-cycle one. For fourth-order elements, the new basis combined with the non-standard W-cycle preconditioner leads to the best performance. The computing times for the other combinations are about 40% longer, at best.

Treatment of two-dimensional E-plane bandstop filters using the finite element method

The method developed recently for analysing two-dimensional waveguide bandpass filters in the H-Plane with the finite element method has been extended to bandstop filters in the E-plane. The treatment is even simpler in the bandstop filter case. Only 20 fourth-order elements are needed to obtain an accurate analysis of a five-section bandstop filter.

Eigensusceptances and eigenreactances of symmetrical waveguide inductive obstacles using the finite element method

The eigenadmittance or eigenreactance approach has often been used to simplify the treatment of symmetrical microwave networks. The author shows that it may also be used to simplify the evaluation of symmetrical waveguide obstacles by the finite element method. An accurate value for the susceptance B of a rectangular obstacle is obtained with only 40 fourth-order elements.

A new technique for measuring the constitutive parameters of planar materials

A technique for measuring the constitutive parameters of materials over a broad range of frequencies is developed. The technique is specifically designed for planar samples. Planar samples can be placed in the measurement fixture without any special preparation; this makes the technique more convenient than other techniques which require that the sample has to be machined to fit into a measurement fixture. Using fourth-order elements, the finite-element method provides a general, very accurate solution. The technique is experimentally verified by measuring the constitutive parameters of two materials with known properties. The measured results are compared to those of other investigators and are shown to be in good agreement.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Integration of triangular finite elements containing corrective rational functions

AbstractRational corrective functions are often supplemented to complete polynomials in order to create conforming triangular finite elements for plate bending. Although such elements containing complete polynomials of degree 3 or 4 have long been known, conforming triangular elements for plate bending, which contain complete polynomials of arbitrary degree p ⩽ 5 and new corrective rational functions, have been given only recently.Integration of rational functions usually requires numerical quadrature. It is shown here that these new corrective rational functions fall into two classes. For those in one class, explicit closed form integration formulas are available. For those in the other class, integration can be performed very efficiently by summing series whose general term is O(kd) for large k, where d ⩽ 5. These formulas also apply to the older third and fourth order elements.