A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems

We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes.     

Influence of graphene reinforcement in conductive polymer: Synthesis and characterization

Lightweight conductive polymers are considered for lightning strike mitigation in composites by synthesizing intrinsically conductive polymers (ICPs) and by the inclusion of conductive fillers in insulating matrices. Conductive films based on polyaniline (PANI) and graphene have been developed to improve through‐thickness conductivity of polymer composites. The result shows that the conductivity of PANI enhanced by blending polyvinylpyrrolidone (PVP) and PANI in 3:1 ratio. Conductive composite thin films are prepared by dispersing graphene in PANI. The conductivity of composite films was found to increase by 40× at 20 wt% of graphene inclusion compared with PVP and PANI blend. Fourier‐transform‐infrared (FTIR) spectra confirmed in situ polymerization of the polymer blend. The inclusion of graphene also exhibits an increase in Tg by 21°C. Graphene additions also showed an increase in thermal stability by approximately 148°C in the composite films. The mechanical result obtained from DMA shows that inclusion of graphene increases the tensile strength by 48% at 20 wt% of graphene reinforcement. A thin, highly conductive surface that is compatible with a composite resin system can enhance the surface conductivity of composites, improving its lightning strike mitigation capabilities.     

Biorthogonal bases with local support and approximation properties

We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree . Working with Gauß-Lobatto nodes, we find an interesting connection between biorthogonality and quadrature formulas. One important application of these newly constructed biorthogonal basis functions are two-dimensional mortar finite elements. The weak continuity condition of the constrained mortar space is realized in terms of our new dual bases. As a result, local static condensation can be applied which is very attractive from the numerical point of view. Numerical results are presented for cubic mortar finite elements.

     

A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems

We present a symmetric version of the nonsymmetric mixed finite element method presented in (Lamichhane, ANZIAM J 50 (2008), C324–C338) for nearly incompressible elasticity. The displacement–pressure formulation of linear elasticity is discretized using a Petrov–Galerkin discretization for the pressure equation in (Lamichhane, ANZIAM J 50 (2008), C324–C338) leading to a non‐symmetric saddle point problem. A new three‐field formulation is introduced to obtain a symmetric saddle point problem which allows us to use a biorthogonal system. Working with a biorthogonal system, we can statically condense out all auxiliary variables from the saddle point problem arriving at a symmetric and positive‐definite system based only on the displacement. We also derive a residual based error estimator for the mixed formulation of the problem. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

A new three‐field formulation of the biharmonic problem and its finite element discretization

We consider a new three‐field formulation of the biharmonic problem. The solution, the gradient and the Lagrange multiplier are the three unknowns in the formulation. Adding a stabilization term in the discrete setting we can use the standard Lagrange finite element to discretize the solution, whereas we use the Raviart‐Thomas finite element to discretize the gradient. The Lagrange multipliers are constructed to achieve the optimal error estimate. Numerical results are presented to demonstrate the performance of our approach. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 199–217, 2017     

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. Copyright © 2013 John Wiley & Sons, Ltd.