Sundararajan Natarajan

In this work, we propose a new n+1 integration scheme over arbitrary polygonal elements based on centroid approximation and Richardson extrapolation scheme. For the purpose of numerical integration, the polygonal element is divided into quadrilateral subcells by connecting the centroid of the polygon with the mid-point of the edges. The bilinear form is then computed in a two-stage approximation: as a first approximation, the bilinear form is computed at the centroid of the given polygonal element and in the second approximation, it is computed at the center of the quadrilateral cells. Both steps can be computed independently and thus parallelization is possible. When compared to commonly used approach, numerical integration based on sub-triangulation, the proposed scheme requires less computational time and fewer integration points. The accuracy, convergence properties and the efficiency are demonstrated with a few standard benchmark problems in two dimensional linear elasto-statics. From the systematic numerical study, it can be inferred that the proposed numerical scheme converges with an optimal rate in both L2 norm and H1 semi-norm at a fraction of computational time when compared to existing approaches, without compromising the accuracy.

A novel cell-based smoothed finite element method is proposed for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields. The domain is discretized with arbitrary polygons and on each side of the polygonal element, discrete shear constraints are considered to relate the kinematical and the independent shear strains. The plate is made of functionally graded material with effective properties computed using the rule of mixtures. The influence of various parameters, viz., the plate aspect ratio and the material gradient index on the static bending response and the first fundamental frequency is numerically studied. It is seen that the proposed element: (a) has proper rank; (b) does not require derivatives of shape functions and hence no isoparametric mapping required; (c) independent of shape and size of elements and (d) is free from shear locking.

In this paper, an adaptive phase-field scaled boundary finite element method for fracture in functionally graded material (FGM) is presented. The model accounts for spatial variation in the material and fracture properties. The quadtree decomposition is adopted for refinement, and the refinement is based on an error indicator evaluated directly from the solutions of the scaled boundary finite element method. This combination makes it a suitable choice to study fracture using the phase field method, as it reduces the mesh burden. A few standard benchmark numerical examples are solved to demonstrate the improvement in computational efficiency in terms of the number of degrees of freedom.

Piezo-electric energy harvesters (PEHs) are widely used to recycle waste vibrations for powering small electronic devices. In this work, the performance of PEHs of non-conventional shapes is studied with the aim to optimize their design. A PEH is modeled as a bimorph cantilever Kirchhoff-Love plate subjected to base acceleration. The shape is described by Non Uniform Rational B-Splines (NURBS), and the problem is solved by the isogeometric analysis (IGA). Accuracy of the IGA solutions is demonstrated on two benchmark problems. Then, we conduct a parametric study to investigate the effect of shape perturbations on the fundamental frequency, the peak amplitude of the frequency response function (FRF) and the peak amplitude of the FRF per unit of area. Next, we build the Kriging meta-model to obtain the boundary shapes that maximize the peak amplitude of the FRF and the peak amplitude of the FRF per unit of area for a target frequency. The optimization is performed for three cases: with a fixed tip mass, without a tip mass and with a tip mass being one of the design parameters.

In this study, three variants of strain smoothing technique, viz. the cell-based, edge-based, and node-based smoothed finite element method, are employed for structural topology optimization. The salient features of the strain smoothing technique are: (i) does not require an explicit form of shape functions and (ii) less sensitive to mesh distortion. Within the proposed framework, the structural materials are modelled as the relative material density powered by the power-law approach. An optimum structural topology is estimated from the condition that minimizes the total strain energy of the structures of interest. The efficacy and the robustness of the strain smoothing technique, when applied to topology optimization, are demonstrated with a few standard benchmark problems. A systematic parametric study is done to find suitable and optimal control parameters for the topology optimization, viz. filter size, tuning parameter, and move limit. The relative performance of different strain smoothing techniques for structural topology optimization is also presented.

This contribution discusses surrogate models that emulate the solution field(s) in the entire simulation domain. The surrogate uses the most characteristic modes of the solution field(s), in combination with neural networks to emulate the coefficients of each mode. This type of surrogate is well known to rapidly emulate flow simulations, but rather new for simulations of elastoplastic solids. The surrogate avoids the iterative process of constructing and solving the linearized governing equations of rate-independent elastoplasticity, as necessary for direct numerical simulations or (hyper-)reduced-order-models. Instead, the new plastic variables are computed only once per increment, resulting in substantial time savings. The surrogate uses a recurrent neural network to treat the path dependency of rate-independent elastoplasticity within the neural network itself. Because only a few of these surrogates have been developed for elastoplastic simulations, their potential and limitations are not yet well studied. The aim of this contribution is to shed more light on their numerical capabilities in the context of elastoplasticity. Although more widely applicable, the investigation focuses on a representative volume element, because these surrogates have the ability to both emulate the macroscale stress-deformation relation (which drives the multiscale simulation), as well as to recover all microstructural quantities within each representative volume element.

At the Inter-Noise 2017 conference, the authors had proposed employing the semi-analytical framework based scaled boundary finite element method (SBFEM), to study interior acoustics problems in the mid-frequency range. The method was shown to be quite accurate in computing the natural frequencies and modes, well into the mid-frequency range, with significantly less computational resources than the conventional finite element method (FEM). However, only the Dirichlet and Neumann type boundary conditions were used in the study. In most acoustic cavity numerical simulations impedance boundary conditions need to be imposed, to represent acoustic treatments on the walls. In this paper, the SBFEM formulation is extended to handle such boundary conditions as well as wall excitations. The boundary is represented by a Lagrangian description; other choices such as iso-geometric representation are also possible. The proposed method is validated with an analytical solution for a constant impedance boundary condition. For practical complex valued impedance cases, the results from the present approach are compared with conventional finite element method. The computational efficiency of SBFEM is demonstrated over conventional FEM for the mid-frequency range, using a two-dimensional rectangular cavity.

The scaled boundary finite element method is extended to model fracture in functionally graded materials (FGM) under coupled thermo-mechanical loads. The governing equations of coupled thermo-mechanical equilibrium are discretized using scaled boundary shape functions enriched with the thermal load terms. The material gradient is modeled as a series of power functions, and the stiffness matrix is calculated semi-analytically. Stress intensity factors and T−stress are directly calculated from their definition without any need for additional post-processing techniques. Arbitrary-sided polygon elements are employed for flexible mesh generation. Several numerical examples for isotropic and orthotropic FGMs are presented to validate the proposed technique.

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 L2 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.

Isogeometric Analysis (IGA) provides an alternative to Lagrange based finite element methods by representing the geometry and field with the same Non-Uniform Rational B-Splines (NURBS) shape functions within a weak Galerkin formulation. IGA has proven to be highly efficient in solving the Helmholtz equation, due to the ease with which the order and continuity of the approximation space can be increased, as well as the geometrical exactness enabled by the use of NURBS. In Atroshchenko et al. (2018), we generalize IGA, by allowing an independent representation of the geometry and fields (Geometry Independent Field approximaTion, GIFT or Generalized IGA Marussig et al., 2015). GIFT with NURBS and PHT-splines (for the geometry and the field, respectively) allows to keep original coarse parameterization of CAD geometry and enables adaptive local refinement of the solution. In the present work, we investigate the possibility to further improve the approach by enriching the PHT-splines field approximation with a set of plane-waves propagating in different directions. Plane wave enrichment is commonly used to capture oscillatory behaviour of the solution and achieve smaller error on coarser meshes. The performance of PUPHT-splines for varying frequencies, degree of PHT-splines, number of plane waves, different refinement strategies is demonstrated on three benchmark problems.