Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

We consider a natural class of composite finite elements that provide the mth-order smoothness of the resulting piecewise polynomial function on a triangulated domain and do not require any information on neighboring elements. It is known that, to provide a required convergence rate in the finite element method, the “smallest angle condition” must be often imposed on the triangulation of the initial domain; i.e., the smallest possible values of the smallest angles of the triangles must be lower bounded. On the other hand, the negative role of the smallest angle can be weakened (but not eliminated completely) by choosing appropriate interpolation conditions. As shown earlier, for a large number of methods of choosing interpolation conditions in the construction of simple (noncomposite) finite elements, including traditional conditions, the influence of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for a number of derivatives of order 2 and higher for m ≥ 1. In the present paper, a similar result is proved for some class of composite finite elements.

Finite element methods for the Stokes problem on complicated domains

In this work, we combine an hp–adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction.Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp–adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Composite element method for the seepage analysis of rock masses containing fractures and drainage holes

A parallel computing procedure for computing the bounds on the {\it J}-integral in functionally graded materials is presented based on a Neumann element a-posteriori error bound. The finite element solution of {\it J}-integral is first obtained on a coarser finite element mesh, then a-posteriori bounding procedure based on the finite element error estimate is used to compute the lower and upper bounds on the {\it J}-integral. The computation of the error estimate is performed by solving independent elemental Neumann sub problems decomposed from the finite element model, thus the computing procedure is parallel and potential to solve large scale structural problems. An example is given in the end of paper to compute the lower and upper bounds on the {\it J}-integral of functionally graded materials.

Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)

Objectives: The present paper demonstrates a study on properties of epoxy-polyurethane composite pipe reinforced with specially treated Nanoclay as reinforcement using finite element modeling under static conditions. Methods/Statistical Analysis: The analysis of composite pipe and the resulting properties of pipeline with uniform distribution of particles have been studied by the systematic procedure using the finite element model. The modeling is performed as per the pipeline construction standards and the properties were compared with test values as per ASTM standards. The pipeline for various particle orientations is studied using improved mechanical property criteria. Findings: Epoxy being good in mechanical properties, blended in the ratio of 60:40 percent with polyurethane for better compatibility and hence improved mechanical properties by providing with interpenetrating resin matrix. The amine treated Nanoclay increases the compression strength of the existing polyurethane pipelineby about 38% for the 3% reinforcement and shear strength for 1% Nanoclay. This also supported by Finite element analysis and also it has been shown that for 0 0 orientation of Nanoclay the composites exhibit better properties. Application/Improvements: The polymer composite pipes are used in varieties of applications and they have been widely used in industries such as oil,gas,transport and offshore applications. Polyurethane coated pipelines are most popular for industrial applications.

Instability of laminated composite thin-walled open-section beams

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H1(Ω)‐norm, L2(Ω)‐norm and L2(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.

We design and analyze $C^0$ interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints.MSC codeselliptic distributed optimal control problemspointwise state constraintsnonconvex domainsvariational inequalitiesdiscontinuous Galerkin methodsMSC codes49J2065K1565N30

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

A posteriori error estimation for the method of lumped masses applied to second-order hyperbolic problems

A three node two-dimensional laminated composite curved beam finite element formulation for linear static analysis is presented where the displacement approximation for the laminate is piecewise hierarchical and is derived based on p-version. The displacement approximation for the element is developed by establishing a hierarchical displacement approximation for each lamina of the laminate and then by imposing interlamina continuity conditions of displacement at the interfaces between the laminas. The approximation functions and the nodal variables for each lamina are derived directly from the Lagrange family of interpolation functions of order pξand pη. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three and one node equivalent configurations that correspond to pξ+1 and pη+1 equally spaced nodes in the ξ and η directions and then taking their products. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the interlamina continuity conditions of displacements. The element formulation ensures C0 continuity of displacement across the interelement as well as interlamina boundaries.

Nonstationary random response of laminated composite structures by a hybrid strain-based laminated flat triangular shell finite element

The prediction and analysis of response of laminated composite shell structures under nonstationary random excitation is of considerable interest to design engineers in aerospace and automobile engineering fields. However, it seems that there is no known comprehensive published work on such an analysis that employs the versatile finite element method. Thus, the main focus of the investigation reported in this paper is the application of the hybrid strain-based laminated composite flat triangular shell finite element, that has been developed by the authors, for the analysis of laminated composite shell structures under a relatively wide class of nonstationary random excitations. Representative results of a simply-supported laminated composite cylindrical panel subjected to a point nonstationary random excitation are included.