A high order isoparametric finite element method for the computation of waveguide modes

We have studied the approximation of optical waveguide eigenvalues by a high order isoparametric vector finite element method. Isoparametric mappings are used for the approximation of domains with curved boundaries or curved material interfaces. Eigenvalue convergence for curved elements is investigated. Numerical results verify the predicted order of convergence and show the remarkable accuracy of the method.     

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.     

A penalty model for the analysis of curved composite beams

A simple one-dimensional mechanical model for curved laminated beams is presented. The laminae composing the beam are modelled as Timoshenko beams, perfectly bonded at the interfaces. Because the laminae can rotate differently from one to the other, the cross-sections of the composite beam can warp. The elasto-static problem of the beam is formulated through the principle of stationary potential energy, imposing constraint conditions between the displacements of adjacent laminae by a penalty technique. This approach produces an approximation of radial and tangential interactions between adjacent laminae. By using four-node isoparametric finite elements, numerical values of interlaminar stresses in straight and curved laminated beams are given. They are compared with the results obtained by other authors under different conditions.     

A new fractional Chebyshev FDM: an application for solving the fractional differential equations generated by optimisation problem

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge–Kutta method is given.     

Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces

We address a two-phase Stokes problem, namely the coupling of two fluids with different kinematic viscosities. The domain is crossed by an interface corresponding to the surface separating the two fluids. We observe that the interface conditions allow the pressure and the velocity gradients to be discontinuous across the interface. The eXtended Finite Element Method (XFEM) is applied to accommodate the weak discontinuity of the velocity field across the interface and the jump in pressure on computational meshes that do not fit the interface. Numerical evidence shows that the discrete pressure approximation may be unstable in the neighborhood of the interface, even though the spatial approximation is based on inf-sup stable finite elements. It means that XFEM enrichment locally violates the satisfaction of the stability condition for mixed problems. For this reason, resorting to pressure stabilization techniques in the region of elements cut by the unfitted interface is mandatory. In alternative, we consider the application of stabilized equal order pressure/velocity XFEM discretizations and we analyze their approximation properties. On one side, this strategy increases the flexibility on the choice of velocity and pressure approximation spaces. On the other side, symmetric pressure stabilization operators, such as local pressure projection methods or the Brezzi–Pitkaranta scheme, seem to be effective to cure the additional source of instability arising from the XFEM approximation. We will show that these operators can be applied either locally, namely only in proximity of the interface, or globally, that is on the whole domain when combined with equal order approximations. After analyzing the stability, approximation properties and the conditioning of the scheme, numerical results on benchmark cases will be discussed, in order to thoroughly compare the performance of different variants of the method.     

A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes–Darcy coupled problem

In this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We consider dual-mixed formulations in both media, which yields the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the traces of the porous media pressure and the fluid velocity on the interface, as the resulting unknowns. The set of feasible finite element subspaces includes Raviart–Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the traces. We derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. The proof of reliability makes use of the global inf–sup condition, Helmholtz decompositions in both media, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported.     

Studied X-FEM enrichment to handle material interfaces with higher order finite element

An approach to improve the geometrical representation of surfaces with the eXtended Finite Element Method is proposed. Surfaces are implicitly represented using the level set method. The finite element approximation is enriched by additional functions through the notion of partition of unity, to track material interfaces. Optimal rate of convergence is achieved with curved geometries, using linear elements and linear level set in elements. In order to accelerate the convergence, the order of approximation shape functions is increased, while keeping the same computational mesh. The level set is represented on a finer sub-mesh than the finite element mesh. A special attention to integration procedure is necessary. A new enrichment function is introduced to represent the behavior of curved material interfaces. Numerical examples including free surfaces and material interfaces in 2-D linear elasticity are presented to study convergence rates.     

Convergence of a lowest-order finite element method for the transmission eigenvalue problem

The transmission eigenvalue problem arises in scattering theory. The main difficulty in its analysis is the fact that, depending on the chosen formulation, it leads either to a quadratic eigenvalue problem or to a non-classical mixed problem. In this paper we prove the convergence of a mixed finite element approximation. This approach, which is close to the Ciarlet–Raviart discretization of biharmonic problems, is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. The convergence analysis is based on classical abstract spectral approximation result and the theory of mixed finite element methods for solving the stream function–vorticity formulation of the Stokes problem. Numerical experiments are reported in order to assess the efficiency of the method.     

On Wachspress quadrilateral patches

Wachspress initiated the study of rational basis functions for finite element construction over quadrilaterals and more general polygonal and curved elements. Later Apprato et al. (1979) and Gout (1979, 1985) studied the interpolatory and convergence properties of lower degree rational finite elements and their applications in solving second order boundary value problems. In the present paper we introduce higher degree Wachspress functions by an iterative technique and study their properties from the point of view of applications to surface fitting problems. It is indeed remarkable to note that these functions possess properties similar to tensor product Bernstein polynomials and hence could be effectively used to generate quadrilateral patches.     

Construction of multi-wavelets from compactly supported refinable super functions

In this work, a new method for constructing multi-wavelets with arbitrary approximation order is presented. The approach uses refinable super functions which enable the formulation of approximation order condition in terms of a generalized eigenvalue equation. The generalized left eigenvectors of the resulting finite down-sampled convolution matrix are recognized as the coefficients that enter the finite linear combination of multi-scaling functions that produce the desired super function. The method is demonstrated by constructing a specific class of multi-wavelets with approximation order 2, which includes Geronimo–Hardin–Massopust (GHM) multi-wavelet as one of its parameterized solutions. A new multi-wavelet, possessing approximation order three and multiplicity two with support [0,2] and [0,3] is also constructed.     

Curved shell elements for heat conduction with p-approximation in the shell thickness direction

A finite element formulation is presented for the curved shell elements for heat conduction where the element temperature approximation in the shell thickness direction can be of an arbitrary polynomial order p. This is accomplished by introducing additional nodal variables in the element approximation corresponding to the complete Lagrange interpolating polynomials in the shell thickness direction. This family of elements has the important hierarchical property, i.e. the element properties corresponding to an approximation order p are a subset of the element properties corresponding to an approximation order p + 1. The formulation also enforces continuity or smoothness of temperature across the inter-element boundaries, i.e. C⁰ continuity is guaranteed.

The curved shell geometry is constructed using the co-ordinates of the nodes lying on the middle surface of the shell and the nodal point normals to the middle surface. The element temperature field is defined in terms of hierarchical element approximation functions, nodal temperatures and the derivatives of the nodal temperatures in the element thickness direction corresponding to the complete Lagrange interpolating polynomials. The weak formulation (or the quadratic functional) of the three-dimensional Fourier heat conduction equation is constructed in the Cartesian co-ordinate space. The element properties of the curved shell elements are then derived using the weak formulation (or the quadratic functional) and the hierarchical element approximation. The element matrices and the equivalent heat vectors (resulting from distributed heat flux, convective boundaries and internal heat generation) are all of hierarchical nature. The element formulation permits any desired order of temperature distribution through the shell thickness.

A number of numerical examples are presented to demonstrate the superiority, efficiency and accuracy of the present formulation and the results are also compared with the analytical solutions. For the first three examples, the h-approximation results are also presented for comparison purposes.     

Static and dynamic calculation of a Francis turbine runner with some remarks on accuracy

The work presented in this paper describes both the static deformation under centrifugal forces and water pressure, and the natural vibrations of a hydraulic runner of the Francis type. First, the method of calculation is described; it uses finite elements and considers only one sector of the structure with appropriate boundary conditions. Second, preliminary test calculations with two simple bodies are shown; these tests were performed in order that we might familiarise ourselves with the method, in particular with the plane and curved shell elements used, and they had to provide information about the accuracy that could be expected from the results. Finally, several calculations of the Francis runner are presented and their main features are discussed.     

Accuracy of some finite element models for arch problems

This paper presents a theoretical accuracy study of some finite element models for thin arches. As is well known, the selection of finite elements for curved members is quite a delicate problem. We obtain order estimates of errors of finite element solutions by means of the perturbation theory of mixed models and the technique of asymptotic expansion. In particular, we theoretically show that certain finite element models may suffer from the so-called locking phenomenon. Numerical results are also given to be compared with the theoretical error estimates.     

Construction of near optimal meshes for 3D curved domains with thin sections and singularities for p-version method

The adaptive variable p- and hp-version finite element method can achieve exponential convergence rate when a near optimal finite element mesh is provided. For general 3D domains, near optimal p-version meshes require large curved elements over the smooth portions of the domain, geometrically graded curved elements to the singular edges and vertices, and a controlled layer of curved prismatic elements in the thin sections. This paper presents a procedure that accepts a CAD solid model as input and creates a curved mesh with the desired characteristics. One key component of the procedure is the automatic identification of thin sections of the model through a set of discrete medial surface points computed from an Octree-based tracing algorithm and the generation of prismatic elements in the thin directions in those sections. The second key component is the identification of geometric singular edges and the generation of geometrically graded meshes in the appropriate directions from the edges. Curved local mesh modification operations are applied to ensure the mesh can be curved to the geometry to the required level of geometric approximation.     

Finite Pointset Method for biharmonic equations

Finite pointset method is one of the grid free methods that is used to solve differential equations arising from physical problems. It is a local iterative procedure based on weighted least square approximation technique. In this paper, biharmonic equation with simply supported, clamped and Cahn–Hilliard type boundary conditions, is solved using the finite pointset method. Numerical examples illustrate the efficiency of the method.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

Some results on the curved plate bending problem solved with non-conforming finite elements

Convergence properties are studied for two non-conforming finite elements of degree two and three for the plate bending problem, which were introduced by Morley and Fraeijs de Veubeke. Unlike some other non-conforming elements, they turn out to be very suitable for the case of curved boundaries since success in the patchtest is guaranteed without any restrictions on the shape of the plate. Two kind of support conditions are examined: for the clamped plate we prove that for both elements no kind of curved elements are needed to attain the same rates of convergence derived byLascaux & Lesaint for the polygonal case. In the case of the simply supported plate, the conclusion is the same for Morley's element. For Fraeijs de Veubeke's triangle however this is only possible with the use of second degree curved elements. In addition to the above results, an indication is given for the optimal choice of Poisson's coefficient to be used in the variational formulation for clamped plates and numerical examples are shown. Work supported by Catholic University of Rio de Janeiro.     

Improved ADI Scheme for Linear Hyperbolic Equations: Extension to Nonlinear Cases and Compact ADI Schemes

This paper studies the Galerkin finite element approximation of time-fractional Navier–Stokes equations. The discretization in space is done by the mixed finite element method. The time Caputo-fractional derivative is discretized by a finite difference method. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Under some certain conditions that the solution and initial value satisfy, we give the error estimates for both semidiscrete and fully discrete schemes. Finally, a numerical example is presented to demonstrate the effectiveness of our numerical methods.     

Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D

A finite difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimate of nearly second order. In a standard patched adaptive refinement method certain slave nodes appear where the approximation is done by interpolating the values of the approximate solution at adjacent nodes. This deteriorates the accuracy of truncation error. In order to avoid the slave points we change the stencil at the interface points from a cross to a skew one. The efficiency of this technique is illustrated by numerical experiments in 2D.