Continuous Linear Elements Research Articles

Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations

<p style='text-indent:20px;'>In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the <inline-formula><tex-math id="M1">$ H^1 $</tex-math></inline-formula>-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.</p>

A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces

We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

A coupling FEM/BEM method with linear continuous elements for acoustic-structural interaction problems

The effects of dynamical load from acoustic waves to structural responses cannot be neglected for some cases such as a thin structure submerged in a heavy fluid. To deal with this kind of problem, a hybrid numerical method consisting of finite element method (FEM) and boundary element method (BEM) for the acoustic-structural coupling problems is proposed. The FEM is built on the first-order Reissner-Mindlin plate theory and the discrete shear gap (DSG) method is applied to construct a locking-free FEM. A collocation BEM with linear continuous element based on Burton-Miller formulation is adopted with an aim to produce a meshing conforming coupling method with the FEM not only on geometry but also on physical nodal quantities at the interface. In addition, analytical expressions of the singular integrals appeared in the Burton-Miller formulations are available to eliminate the numerical difficulties in the implementation of the BEM. Furthermore, the coupling matrix at the interface based on the conforming linear triangular mesh is obtained analytically to avoid the numerical quadrature. A solution scheme by absorbing the FEM equations to the BEM’s is proposed to take the advantage of less memory cost of the linear BEM. Optimized algorithms are designed to reduce the memory cost while keep the accuracy of the coupling method during the absorption of the structure effects to the acoustic domain. Numerical examples are setup to validate the accuracy and demonstrate the potential capability of the proposed method.

A collocation BEM for 3D acoustic problems based on a non-singular Burton–Miller formulation with linear continuous elements

It is well known that Burton–Miller formulation can overcome the problems of non-unique solutions of exterior problems at the fictitious frequencies of the corresponding interior problems. However, the increased singularity involved in the Burton–Miller formulation is one of the crucial troublesomeness in the implementation of BEM with linear continuous element for 3D acoustic problems. Among the prevailing regularization methods, analytical evaluation of the singular integrals is a more attractive option due to the higher accuracy and less computational cost. In this work, an explicitly non-singular Burton–Miller formulation is developed for the model discretized with planar triangular elements. To facilitate the derivation of non-singular expressions, an integration method based on the vertexes’ coordinates of the element is applied to the singular integrals. A collocation approach, which does not put the collocation points overlap with the nodes, is proposed to guarantee a C1,a Hölder continuity for canceling out the divergent terms appeared in obtaining the non-singular expression. By expanding the exponent function in the integrand into a Taylor series, analytical expressions are derived for the generic line integrals left in the non-singular Burton–Miller formulation with respect to the polar angle. At last, a collocation BEM with linear continuous element is built for 3D acoustic problems, which is more accurate and efficient in solution, as well as less in memory consumption than that of the BEM with constant element. Numerical examples are elaborately designed to validate the formulations and procedure of the linear BEM.

Stabilized Lowest-Order Finite Element Approximation for Linear Three-Field Poroelasticity

A stabilized conforming mixed finite element method for the three-field (displacement, fluid flux, and pressure) poroelasticity problem is developed and analyzed. We use the lowest possible approximation order, namely piecewise constant approximation for the pressure and piecewise linear continuous elements for the displacements and fluid flux. By applying a local pressure jump stabilization term to the mass conservation equation, we ensure stability and avoid pressure oscillations. Importantly, the discretization leads to a symmetric linear system. For the fully discretized problem we prove existence and uniqueness, an energy estimate, and an optimal a priori error estimate, including an error estimate for the divergence of the fluid flux. Numerical experiments in two and three dimensions illustrate the convergence of the method, show its effectiveness in overcoming spurious pressure oscillations, and evaluate the added mass effect of the stabilization term.

A nonconforming finite element method for the Stokes equations using the Crouzeix-Raviart element for the velocity and the standard linear element for the pressure

We present a finite element method for Stokes equations using the Crouzeix-Raviart element for the velocity and the continuous linear element for the pressure. We show that the inf-sup condition is satisfied for this pair. Two numerical experiments are presented to support the theoretical results.

Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints

In this paper we consider a model elliptic optimal control problem with finitely many state constraints in two and three dimensions. Such problems are challenging due to low regularity of the adjoint variable. For the discretization of the problem we consider continuous linear elements on quasi-uniform and graded meshes separately. Our main result establishes optimal a priori error estimates for the state, adjoint, and the Lagrange multiplier on the two types of meshes. In particular, in three dimensions the optimal second order convergence rate for all three variables is possible only on properly refined meshes. Numerical examples at the end of the paper support our theoretical results.

Shape optimal design of structures with dual boundary elements

This paper is concerned with the numerical implementation of the dual boundary element method for shape optimal design of two-dimensional linear elastic structures. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The dual boundary element method, used for the discretization of the state problem, applies the stress equation for collocation on the design boundary and the displacement equation for collocation on other boundaries. The use of the stress boundary integral equation, discretized with discontinuous quadratic elements, allows an efficient and accurate computation of stresses on the design boundary. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node. The optimization problem is solved by the modified method of feasible directions available in the program ADS. Examples of a plate with a hole are analyzed with the present method, for different loading conditions. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of shape optimal design of structures.

A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails

This paper deals with the mathematical and numerical analysis of a simplified two-dimensional model for the interaction between the wind and a sail. The wind is modeled as a steady irrotational plane flow past the sail, satisfying the Kutta-Joukowski condition. This condition guarantees that the flow is not singular at the trailing edge of the sail. Although for the present analysis the position of the sail is taken as data, the final aim of this research is to develop tools to compute the sail shape under the aerodynamic pressure exerted by the wind. This is the reason why we propose a fictitious domain formulation of the problem, involving the wind velocity stream function and a Lagrange multiplier; the latter allows computing the force density exerted by the wind on the sail. The Kutta-Joukowski condition is imposed in integral form as an additional constraint. The resulting problem is proved to be well posed under mild assumptions. For the numerical solution, we propose a finite element method based on piecewise linear continuous elements to approximate the stream function and piecewise constant ones for the Lagrange multiplier. Error estimates are proved for both quantities and a couple of numerical tests confirming the theoretical results are reported. Finally the method is used to determine the sail shape under the action of the wind. © EDP Sciences, SMAI, 2011.

Hybrid finite element analysis of fluid-structure systems with coupling on curved interfaces

This paper deals with a finite element approximation of the vibration modes of fluid–structure systems coupled on curved interfaces. It is based on a displacement formulation for both the fluid and the solid. Lowest order Raviart–Thomas elements are used for the fluid and standard continuous linear elements for the solid. Compatibility conditions are weakly imposed at a polygonal approximation of the fluid–solid interface by means of a Lagrange multiplier. Convergence, nonexistence of spurious or circulation nonzero frequency modes and optimal order error estimates for eigenfunctions/eigenvalues are proved. To do this, we use recent results about spectral approximations for noncompact operators to nonconforming hybrid finite element methods. The validity of a discrete compactness property for the discrete spaces considered here is also discussed.

Numerical analysis of a finite element method to compute the vibration modes of a Reissner-Mindlin laminated plate

This paper deals with the finite element approximation of the vibration modes of a laminated plate modeled by the Reissner-Mindlin equations; DL3 elements are used for the bending terms and standard piecewise linear continuous elements for the in-plane displacements. An a priori estimate of the regularity of the solution, independent of the plate thickness, is proved for the corresponding load problem. This allows using the abstract approximation theory for spectral problems to study the convergence of the proposed finite element method. Thus, optimal order error estimates including a double order for the vibration frequencies are obtained under appropriate assumptions. These estimates are independent of the plate thickness, which leads to the conclusion that the method is locking-free. Numerical tests are reported to assess the performance of the method.

Modeling ocean circulation on unstructured meshes: comparison of two horizontal discretizations

Finite-element models on unstructured meshes are frequently formulated in terms of continuous linear elements, which suffer from pressure modes and require stabilization. Alternatively, horizontal velocities may be represented with linear nonconforming elements. While the latter formulation uses three times more degrees of freedom for the velocity, it does not support pressure modes. The effects of stabilization are estimated by comparing the performance of continuous linear and nonconforming versions of the finite-element ocean circulation model (FEOM) in two simple configurations: a Munk gyre and baroclinic turbulence in a zonally reentrant channel. It is shown that, outside the free slip boundary layers, the presence of stabilization does not lead to noticeable effects if its strength is kept within certain limits. In order to evaluate the performance of FEOM, the baroclinic turbulence test is repeated with the MIT general circulation model (MITgcm), which serves as a benchmark, and reasonable agreement between different model codes is found. The two versions of FEOM have a similar computational cost, but both are significantly slower (per node) than the regular-mesh MITgcm. The paper also provides a brief description of the implementation of the nonconforming version of FEOM.

Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements

This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.

Analysis of Finite Element Domain Embedding Methods for Curved Domains using Uniform Grids

We analyze the error of a finite element domain embedding method for elliptic equations on a domain $\omega$ with curved boundary. The domain is embedded in a rectangle R on which uniform mesh and linear continuous elements are employed. The numerical scheme is based on an extension of the differential equation from $\omega$ to R by regularization with a small parameter $\epsilon$ (for Neumann and Robin problems), or penalty with a large parameter $\epsilon^{-1}$ (for the Dirichlet problem), or a mixture of these (for a mixed boundary value problem). For Neumann and Robin problems, we prove that when $\epsilon\le h$ (the mesh size), the error in the $H^1(\omega)$ norm is of the optimal order $\mathcal{O}(h)$. For the Dirichlet problem, when $\epsilon\le h^{1/2}$, the error is $\mathcal{O}(h^{1/2})$ that is not optimal. If the mesh is adjusted around $\partial\omega$ to fit it, then the optimal convergence rate $\mathcal{O}(h)$ holds for the Dirichlet problem if $\omega$ is convex and $\epsilon\le h$. If $\omega$ is not convex, then the convergence rate can only be improved to $\mathcal{O}(h^{2/3})$ by such mesh adjustment, with the parameter being $\epsilon=h^{2/3}$. In this latter case, a parameter smaller than $h^{2/3}$ thwarts the convergence rate, which is verified by a numerical result.

Weak formulation of axi-symmetric frictionless contact problems with boundary elements: Application to interface cracks

A boundary element formulation of the axi-symmetric elastic problem, including contact conditions for multi-domain problems, is presented in this paper and applied to the analysis of interface cracks. A weak formulation of the equilibrium and compatibility equations has been developed to allow non-conforming discretizations to be used in the contacting boundaries. In order to obtain a high accuracy in the solution (employing continuous linear elements), the singular terms appearing in the axi-symmetric boundary integral equations have been integrated analytically. The numerical analysis of a penny-shaped crack at the interface of two dissimilar materials and of a fibre–matrix debonding crack in the single fibre fragmentation test is presented.

Analysis of a coupled BEM/FEM eigensolver for the hydroelastic vibrations problem

A coupled finite/boundary element method to approximate the free vibration modes of an elastic structure containing an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of one of the most usual procedures in engineering practice: an added mass formulation, which is posed in terms of boundary integral equations. Piecewise linear continuous elements are used to discretize the solid displacements and the fluid-solid interface variables. Spectral convergence is proved and error estimates are settled for the approximate eigenfunctions and their corresponding vibration frequencies. Implementation issues are also discussed and numerical experiments are reported.

Finite element approximation of spectral problems with Neumann boundary conditions on curved domains

This paper deals with the finite element approximation of the spectral problem for the Laplace equation with Neumann boundary conditions on a curved nonconvex domain Ω. Convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain Ωh ⊄ Ω in tile framework of the abstract spectral approximation theory.

A finite element solution of an added mass formulation for coupled fluid-solid vibrations

A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed.

The application of a FEM–BEM model to predict acoustic silencer performance and the BEM model to predict vented earmold dynamics

A directly coupled boundary element and finite element model has been applied to the dynamics of acoustic silencers exhibiting flexible surfaces. In this coupled BEM–FEM formulation, the BEM model used a three-node isoparametric triangular linear continuous element to discretize the acoustic cavity. The BEM model was coupled to a flat shell FEM model. The formulated model was then used to predict the performance of various silencer configurations. In a parallel study, the BEM model was used to study the various parameters affecting the performance of vented earmolds coupled to a headset driver and the ear canal. The preliminary findings from this effort will be presented. [See NOISE-CON Proceedings for full paper.]