We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

In this paper, the bending and free flexural vibration behaviour of sandwich plates with carbon nanotube (CNT) reinforced facesheets are investigated using QUAD-8 shear flexible element developed based on higher-order structural theory. This theory accounts for the realistic variation of the displacements through the thickness, and the possible discontinuity in slope at the interface, and the thickness stretch affecting the transverse deflection. The in-plane and rotary inertia terms are considered in the formulation. The governing equations obtained using Lagrange's equation of motions are solved for static and dynamic analyses considering a sandwich plate with homogeneous core and CNT reinforced face sheets. The accuracy of the present formulation is tested considering the problems for which solutions are available. A detailed numerical study is carried out based on various higher-order models deduced from the present theory to examine the influence of the volume fraction of the CNT, core-to-face sheet thickness and the plate thickness ratio on the global/local response of different sandwich plates.

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