Response And Stability Of A Stochastic Beam-Column Using Stochastic Fem

A beam-column resting on continuous Winkler foundation and discrete elastic supports is considered. The beam-column is of variable cross-section and the variation of sectional properties along the axis of the beam-column is deterministic. Young's modulus, mass per unit length and distributed axial loadings of the beam-column have a stochastic distribution. The foundation stiffness coefficient of the Winkler model, the stiffnesses of discrete elastic supports, stiffnesses of end springs and the end thrust, are all considered as random parameters. The material property fluctuations and distributed axial loadings are considered to constitute independent, one-dimension uni-variate homogeneous real stochastic fields in space. The foundation stiffness coefficient, stiffnesses of the discrete elastic supports, stiffnesses of end springs and the end thrust are considered to constitute independent random variables. Static response, free vibration and stability behaviour of the beam-column are studied. Hamilton's principle is used to formulate the problem using stochastic FEM. Sensitivity vectors of the response and stability parameters are evaluated. Using these statistics of free vibration frequencies, mode shapes, buckling parameters, etc., are evaluated. A numerical example is given.

A FEM-Based Forward Solver for Studying the Forward Problem of Electrical Impedance Tomography (EIT) with A Practical Biological Phantom

A Finite Element Method based forward solver is developed for solving the forward problem of a 2D-Electrical Impedance Tomography. The Method of Weighted Residual technique with a Galerkin approach is used for the FEM formulation of EIT forward problem. The algorithm is written in MatLAB7.0 and the forward problem is studied with a practical biological phantom developed. EIT governing equation is numerically solved to calculate the surface potentials at the phantom boundary for a uniform conductivity. An EIT-phantom is developed with an array of 16 electrodes placed on the inner surface of the phantom tank filled with KCl solution. A sinusoidal current is injected through the current electrodes and the differential potentials across the voltage electrodes are measured. Measured data is compared with the differential potential calculated for known current and solution conductivity. Comparing measured voltage with the calculated data it is attempted to find the sources of errors to improve data quality for better image reconstruction.

Free vibration of a stochastic beam-column using stochastic fem

The free vibrational characteristics of a beam-column, which is having randomly varying Young's modulus and mass density and subjected to randomly distributed axial loading is analysed. The material property fluctuations and axial loadings are considered to constitute independent one-dimensional, uni-variate, homogeneous real, spatially distributed stochastic fields. Hamilton's principle is used to formulate the problem using stochastic FEM. Vibration frequencies and mode shapes are analysed for their statistical descriptions. A numerical example is shown.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.

Automatic decomposition of unstructured meshes employing genetic algorithms for parallel FEM computations

Parallel execution of computational mechanics codes requires efficient mesh-partitioning techniques. These mesh-partitioning techniques divide the mesh into specified number of submeshes of approximately the same size and at the same time, minimise the interface nodes of the submeshes. This paper describes a new mesh partitioning technique, employing Genetic Algorithms. The proposed algorithm operates on the deduced graph (dual or nodal graph) of the given finite element mesh rather than directly on the mesh itself. The algorithm works by first constructing a coarse graph approximation using an automatic graph coarsening method. The coarse graph is partitioned and the results are interpolated onto the original graph to initialise an optimisation of the graph partition problem. In practice, hierarchy of (usually more than two) graphs are used to obtain the final graph partition. The proposed partitioning algorithm is applied to graphs derived from unstructured finite element meshes describing practical engineering problems and also several example graphs related to finite element meshes given in the literature. The test results indicate that the proposed GA based graph partitioning algorithm generates high quality partitions and are superior to spectral and multilevel graph partitioning algorithms.

Smooth DMS-FEM: A new approach to solving nearly incompressible nonlinear elasto-static problems

The DMS-FEM, which enables functional approximations with C(1) or still higher inter-element continuity within an FEM-based meshing of the domain, has recently been proposed by Sunilkumar and Roy [39,40]. Through numerical explorations on linear elasto-static problems, the method was found to have conspicuously superior convergence characteristics as well as higher numerical stability against locking. These observations motivate the present study, which aims at extending and exploring the DMS-FEM to (geometrically) nonlinear elasto-static problems of interest in solid mechanics and assessing its numerical performance vis-a-vis the FEM. In particular, the DMS-FEM is shown to vastly outperform the FEM (presently implemented through the commercial software ANSYS (R)) as the former requires fewer linearization and load steps to achieve convergence. In addition, in the context of nearly incompressible nonlinear systems prone to volumetric locking and with no special numerical artefacts (e.g. stabilized or mixed weak forms) employed to arrest locking, the DMS-FEM is shown to approach the incompressibility limit much more closely and with significantly fewer iterations than the FEM. The numerical findings are suggestive of the important role that higher order (uniform) continuity of the approximated field variables play in overcoming volumetric locking and the great promise that the method holds for a range of other numerically ill-conditioned problems of interest in computational structural mechanics. (C) 2011 Elsevier Ltd. All rights reserved.

Wrinkled and slack membranes: nonlinear 3D elasticity solutions via smooth DMS-FEM and experiment

The smooth DMS-FEM, recently proposed by the authors, is extended and applied to the geometrically nonlinear and ill-posed problem of a deformed and wrinkled/slack membrane. A key feature of this work is that three-dimensional nonlinear elasticity equations corresponding to linear momentum balance, without any dimensional reduction and the associated approximations, directly serve as the membrane governing equations. Domain discretization is performed with triangular prism elements and the higher order (C1 or more) interelement continuity of the shape functions ensures that the errors arising from possible jumps in the first derivatives of the conventional C0 shape functions do not propagate because the ill-conditioned tangent stiffness matrices are iteratively inverted. The present scheme employs no regularization and exhibits little sensitivity to h-refinement. Although the numerically computed deformed membrane profiles do show some sensitivity to initial imperfections (nonplanarity) in the membrane profile needed to initiate transverse deformations, the overall patterns of the wrinkles and the deformed shapes appear to be less so. Finally, the deformed profiles, computed through the DMS FEM-based weak formulation, are compared with those obtained through an experiment on an ultrathin Kapton membrane, wherein wrinkles form because of the applied boundary displacement conditions. Comparisons with a reported experiment on a rectangular membrane are also provided. These exercises lend credence to the feasibility of the DMS FEM-based numerical route to computing post-wrinkled membrane shapes. Copyright (c) 2012 John Wiley & Sons, Ltd.

An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order

This paper presents a singular edge-based smoothed finite element method (sES-FEM) for mechanics problems with singular stress fields of arbitrary order. The sES-FEM uses a basic mesh of three-noded linear triangular (T3) elements and a special layer of five-noded singular triangular elements (sT5) connected to the singular-point of the stress field. The sT5 element has an additional node on each of the two edges connected to the singular-point. It allows us to represent simple and efficient enrichment with desired terms for the displacement field near the singular-point with the satisfaction of partition-of-unity property. The stiffness matrix of the discretized system is then obtained using the assumed displacement values (not the derivatives) over smoothing domains associated with the edges of elements. An adaptive procedure for the sES-FEM is proposed to enhance the quality of the solution with minimized number of nodes. Several numerical examples are provided to validate the reliability of the present sES-FEM method. (C) 2012 Elsevier B.V. All rights reserved.

Semi-analytical finite element analysis of a strip footing on an elastic reinforced soil

A plane strain elastic interaction analysis of a strip footing resting on a reinforced soil bed has been made by using a combined analytical and finite element method (FEM). In this approach the stiffness matrix for the footing has been obtained using the FEM, For the reinforced soil bed (halfplane) the stiffness matrix has been obtained using an analytical solution. For the latter, the reinforced zone has been idealised as (i) an equivalent orthotropic infinite strip (composite approach) and (ii) a multilayered system (discrete approach). In the analysis, the interface between the strip footing and reinforced halfplane has been assumed as (i) frictionless and (ii) fully bonded. The contact pressure distribution and the settlement reduction have been given for different depths of footing and scheme of reinforcement in soil. The load-deformation behaviour of the reinforced soil obtained using the above modelling has been compared with some available analytical and model test results. The equivalent orthotropic approach proposed in this paper is easy to program and is shown to predict the reinforcing effects reasonably well.

Influence of friction during forming processes-a study using a numerical simulation technique

Friction has an important influence in metal forming operations, as it contributes to the success or otherwise of the process. In the present investigation, the effect of friction on metal forming was studied by simulating compression tests on cylindrical Al-Mg alloy using the finite element method (FEM) technique. Three kinds of compression tests were considered wherein a constant coefficient of friction was employed at the upper die-work-piece interface. However, the coefficient of friction between the lower die-work-piece interfaces was varied in the tests. The simulation results showed that a difference in metal flow occurs near the interfaces owing to the differences in the coefficient of friction. It was concluded that the variations in the coefficient of friction between the dies and the work-piece directly affect the stress distribution and shape of the work-piece, having implications on the microstructure of the material being processed.

Finite Element Solution Of Surface-Tension Driven Flows In Laser Surface-Melting

Lasers are very efficient in heating localized regions and hence they find a wide application in surface treatment processes. The surface of a material can be selectively modified to give superior wear and corrosion resistance. In laser surface-melting and welding problems, the high temperature gradient prevailing in the free surface induces a surface-tension gradient which is the dominant driving force for convection (known as thermo-capillary or Marangoni convection). It has been reported that the surface-tension driven convection plays a dominant role in determining the melt pool shape. In most of the earlier works on laser-melting and related problems, the finite difference method (FDM) has been used to solve the Navier Stokes equations [1]. Since the Reynolds number is quite high in these cases, upwinding has been used. Though upwinding gives physically realistic solutions even on a coarse grid, the results are inaccurate. McLay and Carey have solved the thermo-capillary flow in welding problems by an implicit finite element method [2]. They used the conventional Galerkin finite element method (FEM) which requires that the pressure be interpolated by one order lower than velocity (mixed interpolation). This restricts the choice of elements to certain higher order elements which need numerical integration for evaluation of element matrices. The implicit algorithm yields a system of nonlinear, unsymmetric equations which are not positive definite. Computations would be possible only with large mainframe computers.Sluzalec [3] has modeled the pulsed laser-melting problem by an explicit method (FEM). He has used the six-node triangular element with mixed interpolation. Since he has considered the buoyancy induced flow only, the velocity values are small. In the present work, an equal order explicit FEM is used to compute the thermo-capillary flow in the laser surface-melting problem. As this method permits equal order interpolation, there is no restriction in the choice of elements. Even linear elements such as the three-node triangular elements can be used. As the governing equations are solved in a sequential manner, the computer memory requirement is less. The finite element formulation is discussed in this paper along with typical numerical results.

Transmission loss analysis of rectangular expansion chamber with arbitrary location of inlet/outlet by means of Green's functions

Transmission loss of a rectangular expansion chamber, the inlet and outlet of which are situated at arbitrary locations of the chamber, i.e., the side wall or the face of the chamber, are analyzed here based on the Green's function of a rectangular cavity with homogeneous boundary conditions. The rectangular chamber Green's function is expressed in terms of a finite number of rigid rectangular cavity mode shapes. The inlet and outlet ports are modeled as uniform velocity pistons. If the size of the piston is small compared to wavelength, then the plane wave excitation is a valid assumption. The velocity potential inside the chamber is expressed by superimposing the velocity potentials of two different configurations. The first configuration is a piston source at the inlet port and a rigid termination at the outlet, and the second one is a piston at the outlet with a rigid termination at the inlet. Pressure inside the chamber is derived from velocity potentials using linear momentum equation. The average pressure acting on the pistons at the inlet and outlet locations is estimated by integrating the acoustic pressure over the piston area in the two constituent configurations. The transfer matrix is derived from the average pressure values and thence the transmission loss is calculated. The results are verified against those in the literature where use has been made of modal expansions and also numerical models (FEM fluid). The transfer matrix formulation for yielding wall rectangular chambers has been derived incorporating the structural–acoustic coupling. Parametric studies are conducted for different inlet and outlet configurations, and the various phenomena occurring in the TL curves that cannot be explained by the classical plane wave theory, are discussed.

Estimation of gear tooth deflection by the finite element method

The finite element method (FEM) is used to determine for pitch-point, mid-point and tip loading, the deflection curve of a Image 1 diamentral pitch (DP) standard spur gear tooth corresponding to number of teeth of 14, 21, 26 and 34. In all these cases the deflection of the gear tooth at the point of loading obtained by FEM is in good agreement with the experimental value. The contraflexure in the deflection curve at the point of loading observed experimentally in the cases of pitch-point and mid-point loading, is predicted correctly by the FEM analysis.

Thermal characterization of microheaters from the dynamic response

Thermal characterization of surface-micromachined microheaters is carried out from their dynamic response to electrothermal excitations. An electrical equivalent circuit model is developed for the thermo-mechanical system. The mechanical parameters are extracted from the frequency response obtained using a laser Doppler vibrometer. The resonant frequencies of the microheaters are measured and compared with FEM simulations. The thermal time constants are obtained from the electrical equivalent model by fitting the model response to the measured frequency response. Microheaters with an active area of 140 µm × 140 µm have been realized on two different layers (poly-1 and poly-2) with two different air gaps (2 µm and 2.75 µm). The effective time constants, combining thermal and mechanical responses, are in the range of 0.13–0.22 ms for heaters on the poly-1 layer and 1.9 µs–0.15 ms for microheaters on the poly-2 layer. The thermal time constants of the microheaters are in the range of a few microseconds, thus making them suitable for sensor applications that need a faster thermal response.